[ { "meta_data": { "dataset": "creative_math", "id": "1992_AJHSME_Problems_25", "eval_func": null }, "input": { "text": "One half of the water is poured out of a full container. Then one third of the remainder is poured out. Continue the process: one fourth of the remainder for the third pouring, one fifth of the remainder for the fourth pouring, etc. After how many pourings does exactly one tenth of the original water remain?\n$\\text{(A)}\\ 6 \\qquad \\text{(B)}\\ 7 \\qquad \\text{(C)}\\ 8 \\qquad \\text{(D)}\\ 9 \\qquad \\text{(E)}\\ 10$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "1)Model the amount left in the container as follows:\nAfter the first pour $\\frac12$ remains, after the second $\\frac12 \\times \\frac23$ remains, etc.\nThis becomes the product $\\frac12 \\times \\frac23 \\times \\frac34 \\times \\cdots \\times \\frac{9}{10}$.\nNote that the terms cancel out leaving $\\frac{1}{10}$.\nNow all that remains is to count the number of terms or pourings, as the numerators form an arithmetic sequence with a common difference of 1 and endpoints (1,9), the number of pourings is $\\boxed{\\text{(D)}\\ 9}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2016_AMC_8_Problems_15", "eval_func": null }, "input": { "text": "What is the largest power of $2$ that is a divisor of $13^4 - 11^4$?\n$\\textbf{(A)}\\mbox{ }8\\qquad \\textbf{(B)}\\mbox{ }16\\qquad \\textbf{(C)}\\mbox{ }32\\qquad \\textbf{(D)}\\mbox{ }64\\qquad \\textbf{(E)}\\mbox{ }128$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "First, we use difference of squares on $13^4 - 11^4 = (13^2)^2 - (11^2)^2$ to get $13^4 - 11^4 = (13^2 + 11^2)(13^2 - 11^2)$. Using difference of squares again and simplifying, we get $(169 + 121)(13+11)(13-11) = 290 \\cdot 24 \\cdot 2 = (2\\cdot 8 \\cdot 2) \\cdot (3 \\cdot 145)$. Realizing that we don't need the right-hand side because it doesn't contain any factor of 2, we see that the greatest power of $2$ that is a divisor $13^4 - 11^4$ is $\\boxed{\\textbf{(C)}\\ 32}$.", "Just like in the above solution, we use the difference-of-squares factorization, but only once to get $13^4-11^4=(13^2-11^2)(13^2+11^2).$ We can then compute that this is equal to $48\\cdot290.$ Note that $290=2\\cdot145$ (we don't need to factorize any further as $145$ is already odd) thus the largest power of $2$ that divides $290$ is only $2^1=2,$ while $48=2^4\\cdot3,$ so the largest power of $2$ that divides $48$ is $2^4=16.$ Hence, the largest power of $2$ that is a divisor of $13^4-11^4$ is $2\\cdot16=\\boxed{\\textbf{(C)}~32}.$", "Let $n=13^4-11^4.$ We wish to find the largest power of $2$ that divides $n$.\nDenote $v_p(k)$ as the largest exponent of $p$ in the prime factorization of $n$. In this problem, we have $p=2$. \nBy the Lifting the Exponent Lemma on $n$, \n\\[v_2(13^4-11^4)=v_2(13-11)+v_2(4)+v_2(13+11)-1\\]\n\\[=v_2(2)+v_2(4)+v_2(24)-1\\]\n\\[=1+2+3-1=5.\\]\nTherefore, exponent of the largest power of $2$ that divids $13^4-11^4$ is $5,$ so the largest power of $2$ that divides this number is $2^5=\\boxed{\\textbf{(C)} 32}$.", "We can simply take 13 to the 4th power, which is 28561. We subtract that by 11 to the 4th power, which is 14641 (You can use Pascal's Triangle to find this). Finally, subtract the numbers to get 13920. \nTo test the options, since we need the largest one, we can go from top down. Testing, we see that both D and E are decimals,\nand 32 works. So, our answer is $\\boxed{\\textbf{(C)}~32}.$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1991_AJHSME_Problems_25", "eval_func": null }, "input": { "text": "An equilateral triangle is originally painted black. Each time the triangle is changed, the middle fourth of each black triangle turns white. After five changes, what fractional part of the original area of the black triangle remains black?\n[asy] unitsize(36); fill((0,0)--(2,0)--(1,sqrt(3))--cycle,gray); draw((0,0)--(2,0)--(1,sqrt(3))--cycle,linewidth(1)); fill((4,0)--(6,0)--(5,sqrt(3))--cycle,gray); fill((5,0)--(9/2,sqrt(3)/2)--(11/2,sqrt(3)/2)--cycle,white); draw((5,sqrt(3))--(4,0)--(5,0)--(9/2,sqrt(3)/2)--(11/2,sqrt(3)/2)--(5,0)--(6,0)--cycle,linewidth(1)); fill((8,0)--(10,0)--(9,sqrt(3))--cycle,gray); fill((9,0)--(17/2,sqrt(3)/2)--(19/2,sqrt(3)/2)--cycle,white); fill((17/2,0)--(33/4,sqrt(3)/4)--(35/4,sqrt(3)/4)--cycle,white); fill((9,sqrt(3)/2)--(35/4,3*sqrt(3)/4)--(37/4,3*sqrt(3)/4)--cycle,white); fill((19/2,0)--(37/4,sqrt(3)/4)--(39/4,sqrt(3)/4)--cycle,white); draw((9,sqrt(3))--(35/4,3*sqrt(3)/4)--(37/4,3*sqrt(3)/4)--(9,sqrt(3)/2)--(35/4,3*sqrt(3)/4)--(33/4,sqrt(3)/4)--(35/4,sqrt(3)/4)--(17/2,0)--(33/4,sqrt(3)/4)--(8,0)--(9,0)--(17/2,sqrt(3)/2)--(19/2,sqrt(3)/2)--(9,0)--(19/2,0)--(37/4,sqrt(3)/4)--(39/4,sqrt(3)/4)--(19/2,0)--(10,0)--cycle,linewidth(1)); label(\"Change 1\",(3,3*sqrt(3)/4),N); label(\"$\\Longrightarrow $\",(3,5*sqrt(3)/8),S); label(\"Change 2\",(7,3*sqrt(3)/4),N); label(\"$\\Longrightarrow $\",(7,5*sqrt(3)/8),S); [/asy]\n$\\text{(A)}\\ \\frac{1}{1024} \\qquad \\text{(B)}\\ \\frac{15}{64} \\qquad \\text{(C)}\\ \\frac{243}{1024} \\qquad \\text{(D)}\\ \\frac{1}{4} \\qquad \\text{(E)}\\ \\frac{81}{256}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "With each change, $\\frac{3}{4}$ of the black space from the previous stage remains. Since there are $5$ changes, the fractional part of the triangle that remains black is $\\left( \\frac{3}{4} \\right)^5=\\frac{243}{1024}\\rightarrow \\boxed{\\text{C}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2003_AMC_8_Problems_18", "eval_func": null }, "input": { "text": "Each of the twenty dots on the graph below represents one of Sarah's classmates. Classmates who are friends are connected with a line segment. For her birthday party, Sarah is inviting only the following: all of her friends and all of those classmates who are friends with at least one of her friends. How many classmates will not be invited to Sarah's party?\n[asy] pair a=(102,256), b=(68,131), c=(162,101), d=(134,150); pair e=(269,105), f=(359,104), g=(303,12), h=(579,211); pair i=(534, 342), j=(442,432), k=(374,484), l=(278,501); pair m=(282,411), n=(147,451), o=(103,437), p=(31,373); pair q=(419,175), r=(462,209), s=(477,288), t=(443,358); pair oval=(282,303); draw(l--m--n--cycle); draw(p--oval); draw(o--oval); draw(b--d--oval); draw(c--d--e--oval); draw(e--f--g--h--i--j--oval); draw(k--oval); draw(q--oval); draw(s--oval); draw(r--s--t--oval); dot(a); dot(b); dot(c); dot(d); dot(e); dot(f); dot(g); dot(h); dot(i); dot(j); dot(k); dot(l); dot(m); dot(n); dot(o); dot(p); dot(q); dot(r); dot(s); dot(t); filldraw(yscale(.5)*Circle((282,606),80),white,black); label(scale(0.75)*\"Sarah\", oval);[/asy]\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 5\\qquad\\textbf{(D)}\\ 6\\qquad\\textbf{(E)}\\ 7$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "There are $3$ people who are friends with only each other who won't be invited, plus $1$ person who has no friends, and $2$ people who are friends of friends of friends who won\u2019t be invited. So the answer is $\\boxed{\\textbf{(D)}\\ 6}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1987_AJHSME_Problems_17", "eval_func": null }, "input": { "text": "Abby, Bret, Carl, and Dana are seated in a row of four seats numbered #1 to #4. Joe looks at them and says:\n Bret is next to Carl.\n Abby is between Bret and Carl. \nHowever each one of Joe's statements is false. Bret is actually sitting in seat #3. Who is sitting in seat #2?\n$\\text{(A)}\\ \\text{Abby} \\qquad \\text{(B)}\\ \\text{Bret} \\qquad \\text{(C)}\\ \\text{Carl} \\qquad \\text{(D)}\\ \\text{Dana} \\qquad \\text{(E)}\\ \\text{There is not enough information to be sure.}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "\\[\\text{-----}\\qquad \\text{-----} \\qquad \\text{Bret} \\qquad \\text{-----}\\]\nWe know that Carl does not sit next to Bret, so he must sit in seat #1. Since Abby is not between Bret and Carl, she must sit in seat #4. Finally, Dana has to take the last seat available, which is #2.\n$\\boxed{\\text{D}}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2019_AMC_8_Problems_9", "eval_func": null }, "input": { "text": "Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are $6$ cm in diameter and $12$ cm high. Felicia buys cat food in cylindrical cans that are $12$ cm in diameter and $6$ cm high. What is the ratio of the volume of one of Alex's cans to the volume of one of Felicia's cans?\n$\\textbf{(A) }1:4\\qquad\\textbf{(B) }1:2\\qquad\\textbf{(C) }1:1\\qquad\\textbf{(D) }2:1\\qquad\\textbf{(E) }4:1$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Using the formula for the volume of a cylinder, we get Alex, $108\\pi$, and Felicia, $216\\pi$. We can quickly notice that $\\pi$ cancels out on both sides and that Alex's volume is $1/2$ of Felicia's leaving $1/2 = \\boxed{1:2}$ as the answer. ", "Using the formula for the volume of a cylinder, we get that the volume of Alex's can is $3^2\\cdot12\\cdot\\pi$, and that the volume of Felicia's can is $6^2\\cdot6\\cdot\\pi$. Now, we divide the volume of Alex's can by the volume of Felicia's can, so we get $\\frac{1}{2}$, which is $\\boxed{\\textbf{(B)}\\ 1:2}$.", "The ratio of the numbers is $1/2$. Looking closely at the formula $r^2 * h * \\pi$, we see that the $r * h * \\pi$ will cancel, meaning that the ratio of them will be $\\frac{1(2)}{2(2)}$ = $\\boxed{\\textbf{(B)}\\ 1:2}$.", "The second can is $\\cdot 2$ size in each of 2 dimensions, and $\\cdot 1/2$ size in 1 dimension. $2^2/2 = \\boxed{\\textbf{(B)}\\ 1:2}$.", "Without calculating much, you can do\n($\\pi ra^2) \\cdot ha$ <-- which is Alex's volume, with ra being Alex's radius$(1/2 \\cdot$ diameter), and $ha$ being her cylinders height$(\\pi rf^2) \\cdot hf <--$which is Felicia's volume, with $rf$ being Felicia's radius, and $hf$ being her cylinders height.\nSince we need the ratio between Alexa's and Felicias, we can do $(\\pi ra^2)\\cdot ha/(\\pi rf^2)\\cdot hf$ The $\\pi$ cancel out, then substitute back in the numbers, which gives you:\n$(3^2 \\cdot 12)/(6^2 \\cdot 6) = (9 \\cdot 12)/(36 \\cdot 6) = 18/36 = 1/2 = 1:2$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2010_AMC_8_Problems_9", "eval_func": null }, "input": { "text": "Ryan got $80\\%$ of the problems correct on a $25$-problem test, $90\\%$ on a $40$-problem test, and $70\\%$ on a $10$-problem test. What percent of all the problems did Ryan answer correctly? \n$\\textbf{(A)}\\ 64 \\qquad\\textbf{(B)}\\ 75\\qquad\\textbf{(C)}\\ 80\\qquad\\textbf{(D)}\\ 84\\qquad\\textbf{(E)}\\ 86$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Ryan answered $(0.8)(25)=20$ problems correct on the first test, $(0.9)(40)=36$ on the second, and $(0.7)(10)=7$ on the third. This amounts to a total of $20+36+7=63$ problems correct. The total number of problems is $25+40+10=75.$ Therefore, the percentage is $\\dfrac{63}{75} = 84\\% \\rightarrow \\boxed{\\textbf{(D)}\\ 84}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2008_AMC_8_Problems_24", "eval_func": null }, "input": { "text": "Ten tiles numbered $1$ through $10$ are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?\n$\\textbf{(A)}\\ \\frac{1}{10}\\qquad\\textbf{(B)}\\ \\frac{1}{6}\\qquad\\textbf{(C)}\\ \\frac{11}{60}\\qquad\\textbf{(D)}\\ \\frac{1}{5}\\qquad\\textbf{(E)}\\ \\frac{7}{30}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The numbers can at most multiply to be $60$. The squares less than $60$ are $1,4,9,16,25,36,$ and $49$. The possible pairs are $(1,1),(1,4),(2,2),(4,1),(3,3),(9,1),(4,4),(8,2),(5,5),(6,6),$ and $(9,4)$. There are $11$ choices and $60$ possibilities giving a probability of $\\boxed{\\textbf{(C)}\\ \\frac{11}{60}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2015_AMC_8_Problems_21", "eval_func": null }, "input": { "text": "In the given figure, hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\\triangle KBC$?\n[asy] draw((-4,6*sqrt(2))--(4,6*sqrt(2))); draw((-4,-6*sqrt(2))--(4,-6*sqrt(2))); draw((-8,0)--(-4,6*sqrt(2))); draw((-8,0)--(-4,-6*sqrt(2))); draw((4,6*sqrt(2))--(8,0)); draw((8,0)--(4,-6*sqrt(2))); draw((-4,6*sqrt(2))--(4,6*sqrt(2))--(4,8+6*sqrt(2))--(-4,8+6*sqrt(2))--cycle); draw((-8,0)--(-4,-6*sqrt(2))--(-4-6*sqrt(2),-4-6*sqrt(2))--(-8-6*sqrt(2),-4)--cycle); label(\"$I$\",(-4,8+6*sqrt(2)),dir(100)); label(\"$J$\",(4,8+6*sqrt(2)),dir(80)); label(\"$A$\",(-4,6*sqrt(2)),dir(280)); label(\"$B$\",(4,6*sqrt(2)),dir(250)); label(\"$C$\",(8,0),W); label(\"$D$\",(4,-6*sqrt(2)),NW); label(\"$E$\",(-4,-6*sqrt(2)),NE); label(\"$F$\",(-8,0),E); draw((4,8+6*sqrt(2))--(4,6*sqrt(2))--(4+4*sqrt(3),4+6*sqrt(2))--cycle); label(\"$K$\",(4+4*sqrt(3),4+6*sqrt(2)),E); draw((4+4*sqrt(3),4+6*sqrt(2))--(8,0),dashed); label(\"$H$\",(-4-6*sqrt(2),-4-6*sqrt(2)),S); label(\"$G$\",(-8-6*sqrt(2),-4),W); label(\"$32$\",(-10,-8),N); label(\"$18$\",(0,6*sqrt(2)+2),N); [/asy]\n$\\textbf{(A) }6\\sqrt{2}\\quad\\textbf{(B) }9\\quad\\textbf{(C) }12\\quad\\textbf{(D) }9\\sqrt{2}\\quad\\textbf{(E) }32$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Clearly, since $\\overline{FE}$ is a side of a square with area $32$, $\\overline{FE} = \\sqrt{32} = 4 \\sqrt{2}$. Now, since $\\overline{FE} = \\overline{BC}$, we have $\\overline{BC} = 4 \\sqrt{2}$.\nNow, $\\overline{JB}$ is a side of a square with area $18$, so $\\overline{JB} = \\sqrt{18} = 3 \\sqrt{2}$. Since $\\Delta JBK$ is equilateral, $\\overline{BK} = 3 \\sqrt{2}$.\nLastly, $\\Delta KBC$ is a right triangle. We see that $\\angle JBA + \\angle ABC + \\angle CBK + \\angle KBJ = 360^{\\circ} \\rightarrow 90^{\\circ} + 120^{\\circ} + \\angle CBK + 60^{\\circ} = 360^{\\circ} \\rightarrow \\angle CBK = 90^{\\circ}$, so $\\Delta KBC$ is a right triangle with legs $3 \\sqrt{2}$ and $4 \\sqrt{2}$. Now, its area is $\\dfrac{3 \\sqrt{2} \\cdot 4 \\sqrt{2}}{2} = \\dfrac{24}{2} = \\boxed{\\textbf{(C)}~12}$.", "Since $\\overline{FE} = \\sqrt{32}$, and $\\overline{FE} = \\overline{BC}$, $\\overline{BC} = 4\\sqrt{2}$. Meanwhile, $\\overline{JB} = 3\\sqrt{2}$, and since $\\triangle JBK$ is equilateral, $\\overline{BK} = 3\\sqrt{2}$. If $ABCDEF$ is equiangular, $\\angle ABC = \\frac{180 \\cdot (n-2)}{n} = 120^{\\circ}$, where $n$ is the number of sides of the shape. Adding all the angles around $B$ gives $270^{\\circ}$, so $\\angle KBC = 360 - 90 = 270^{\\circ}$. Because $\\triangle KBC$ is right, the area of $\\triangle KBC = \\frac{4\\sqrt{2} \\cdot 3\\sqrt{2}} {2} = 12$. Therefore, the answer is $\\boxed{\\textbf{(C)}~12}$. " ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1991_AJHSME_Problems_9", "eval_func": null }, "input": { "text": "How many whole numbers from $1$ through $46$ are divisible by either $3$ or $5$ or both?\n$\\text{(A)}\\ 18 \\qquad \\text{(B)}\\ 21 \\qquad \\text{(C)}\\ 24 \\qquad \\text{(D)}\\ 25 \\qquad \\text{(E)}\\ 27$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "There are $\\left\\lfloor \\frac{46}{3}\\right\\rfloor =15$ numbers divisible by $3$, $\\left\\lfloor\\frac{46}{5}\\right\\rfloor =9$ numbers divisible by $5$, so at first we have $15+9=24$ numbers that are divisible by $3$ or $5$, except we counted the multiples of $\\text{LCM}(3,5)=15$ twice, once for $3$ and once for $5$.\nThere are $\\left\\lfloor \\frac{46}{15}\\right\\rfloor =3$ numbers divisible by $15$, so there are $24-3=21$ numbers divisible by $3$ or $5$. $\\rightarrow \\boxed{\\text{B}}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2023_AMC_8_Problems_2", "eval_func": null }, "input": { "text": "A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures?\n[asy] size(0,3cm); path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; path rh = (-0.125,-0.125)--(0.5,-0.5)--(0.5,0.5)--(-0.125,0.875)--cycle; path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle; path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle; path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle; path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; filldraw(sq,mediumgrey,black); draw((0.75,0)--(1.25,0),currentpen+1,Arrow(size=6)); //folding path sqside = (-0.5,-0.5)--(0.5,-0.5); path rhside = (-0.125,-0.125)--(0.5,-0.5); transform fld = shift((1.75,0))*scale(0.5); draw(fld*sq,black); int i; for(i=0; i<10; i=i+1) { draw(shift(0,0.05*i)*fld*sqside,deepblue); } path rhedge = (-0.125,-0.125)--(-0.125,0.8)--(-0.2,0.85)--cycle; filldraw(fld*rhedge,grey); path sqedge = (-0.5,-0.5)--(-0.5,0.4475)--(-0.575,0.45)--cycle; filldraw(fld*sqedge,grey); filldraw(fld*rh,white,black); int i; for(i=0; i<10; i=i+1) { draw(shift(0,0.05*i)*fld*rhside,deepblue); } draw((2.25,0)--(2.75,0),currentpen+1,Arrow(size=6)); //cutting transform cut = shift((3.25,0))*scale(0.5); draw(shift((-0.01,+0.01))*cut*sq); draw(cut*sq); filldraw(shift((0.01,-0.01))*cut*sq,white,black); int j; for(j=0; j<10; j=j+1) { draw(shift(0,0.05*j)*cut*sqside,deepblue); } draw(shift((0.01,-0.01))*cut*(0,-0.5)--shift((0.01,-0.01))*cut*(0.5,0),dashed); //Answers Below, but already Separated //filldraw(sqA,grey,black); //filldraw(sqB,grey,black); //filldraw(sq,grey,black); //filldraw(sqC,white,black); //filldraw(sq,grey,black); //filldraw(trD,white,black); //filldraw(sq,grey,black); //filldraw(sqE,white,black); [/asy]\n[asy] size(0,7.5cm); path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; path rh = (-0.125,-0.125)--(0.5,-0.5)--(0.5,0.5)--(-0.125,0.875)--cycle; path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle; path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle; path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle; path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; //ANSWERS real sh = 1.5; label(\"$\\textbf{(A)}$\",(-0.5,0.5),SW); label(\"$\\textbf{(B)}$\",shift((sh,0))*(-0.5,0.5),SW); label(\"$\\textbf{(C)}$\",shift((2sh,0))*(-0.5,0.5),SW); label(\"$\\textbf{(D)}$\",shift((0,-sh))*(-0.5,0.5),SW); label(\"$\\textbf{(E)}$\",shift((sh,-sh))*(-0.5,0.5),SW); filldraw(sqA,mediumgrey,black); filldraw(shift((sh,0))*sqB,mediumgrey,black); filldraw(shift((2*sh,0))*sq,mediumgrey,black); filldraw(shift((2*sh,0))*sqC,white,black); filldraw(shift((0,-sh))*sq,mediumgrey,black); filldraw(shift((0,-sh))*trD,white,black); filldraw(shift((sh,-sh))*sq,mediumgrey,black); filldraw(shift((sh,-sh))*sqE,white,black); [/asy]", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Notice that when we unfold the paper along the vertical fold line, we get the following shape:\n [asy] size(90); path sq = (-0.5,0)--(0.5,0)--(0.5,0.5)--(-0.5,0.5)--cycle; path trE = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; real sh = 1.5; filldraw(shift((sh,-sh))*sq,mediumgrey,black); filldraw(shift((sh,-sh))*trE,white,black); size(90); path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; real sh = 1.5; filldraw(shift((sh,-sh))*sq,mediumgrey,black); filldraw(shift((sh,-sh))*sqE,white,black); [/asy]\nIt is clear that the answer is $\\boxed{\\textbf{(E)}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1987_AJHSME_Problems_18", "eval_func": null }, "input": { "text": "Half the people in a room left. One third of those remaining started to dance. There were then $12$ people who were not dancing. The original number of people in the room was\n$\\text{(A)}\\ 24 \\qquad \\text{(B)}\\ 30 \\qquad \\text{(C)}\\ 36 \\qquad \\text{(D)}\\ 42 \\qquad \\text{(E)}\\ 72$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let the original number of people in the room be $x$. Half of them left, so $\\frac{x}{2}$ of them are left in the room.\nAfter that, one third of this group is dancing, so $\\frac{x}{2}-\\frac{1}{3}\\left( \\frac{x}{2}\\right) =\\frac{x}{3}$ people are not dancing.\nThis is given to be $12$, so \\[\\frac{x}{3}=12\\Rightarrow x=36\\]\n$\\boxed{\\text{C}}$", "First note that of the $\\frac{1}{2}$ people remaining in the room, $\\frac{2}{3}$ are not dancing. Therefore $\\frac{1}{2}\\cdot\\frac{2}{3}= \\frac{1}{3}$ of the original amount of people in the room is $12$. The answer is $\\boxed{C}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2006_AMC_8_Problems_4", "eval_func": null }, "input": { "text": "Initially, a spinner points west. Chenille moves it clockwise $2 \\dfrac{1}{4}$ revolutions and then counterclockwise $3 \\dfrac{3}{4}$ revolutions. In what direction does the spinner point after the two moves?\n[asy]size(96); draw(circle((0,0),1),linewidth(1)); draw((0,0.75)--(0,1.25),linewidth(1)); draw((0,-0.75)--(0,-1.25),linewidth(1)); draw((0.75,0)--(1.25,0),linewidth(1)); draw((-0.75,0)--(-1.25,0),linewidth(1)); label(\"$N$\",(0,1.25), N); label(\"$W$\",(-1.25,0), W); label(\"$E$\",(1.25,0), E); label(\"$S$\",(0,-1.25), S); draw((0,0)--(-0.5,0),EndArrow);[/asy]\n$\\textbf{(A)}\\ \\text{north} \\qquad \\textbf{(B)}\\ \\text{east} \\qquad \\textbf{(C)}\\ \\text{south} \\qquad \\textbf{(D)}\\ \\text{west} \\qquad \\textbf{(E)}\\ \\text{northwest}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "If the spinner goes clockwise $2 \\dfrac{1}{4}$ revolutions and then counterclockwise $3 \\dfrac{3}{4}$ revolutions, it ultimately goes counterclockwise $1 \\dfrac{1}{2}$ which brings the spinner pointing $\\boxed{\\textbf{(B)}\\ \\text{east}}$.", "Note that full revolutions do not matter, so this is equivalent to going clockwise $\\dfrac{1}{4}$ revolutions and then counterclockwise $\\dfrac{3}{4}$ revolutions, making it ultimately go counterclockwise $\\dfrac{1}{2}$, having the spinner point $\\boxed{\\textbf{(B)}\\ \\text{east}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1994_AJHSME_Problems_7", "eval_func": null }, "input": { "text": "If $\\angle A = 60^\\circ$, $\\angle E = 40^\\circ$ and $\\angle C = 30^\\circ$, then $\\angle BDC =$\n[asy] pair A,B,C,D,EE; A = origin; B = (2,0); C = (5,0); EE = (1.5,3); D = (1.75,1.5); draw(A--C--D); draw(B--EE--A); dot(A); dot(B); dot(C); dot(D); dot(EE); label(\"$A$\",A,SW); label(\"$B$\",B,S); label(\"$C$\",C,SE); label(\"$D$\",D,NE); label(\"$E$\",EE,N); [/asy]\n$\\text{(A)}\\ 40^\\circ \\qquad \\text{(B)}\\ 50^\\circ \\qquad \\text{(C)}\\ 60^\\circ \\qquad \\text{(D)}\\ 70^\\circ \\qquad \\text{(E)}\\ 80^\\circ$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The sum of the angles in a triangle is $180^\\circ$. We can find $\\angle ABE = 80^\\circ$, so $\\angle CBD = 180-80=100^\\circ$.\n\\[\\angle BDC = 180-100-30=\\boxed{\\text{(B)}\\ 50^\\circ}\\]" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2004_AMC_8_Problems_11", "eval_func": null }, "input": { "text": "The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules:1. The largest isn't first, but it is in one of the first three places. \n 2. The smallest isn't last, but it is in one of the last three places. \n 3. The median isn't first or last. \nWhat is the average of the first and last numbers?\n$\\textbf{(A)}\\ 3.5 \\qquad \\textbf{(B)}\\ 5 \\qquad \\textbf{(C)}\\ 6.5 \\qquad \\textbf{(D)}\\ 7.5 \\qquad \\textbf{(E)}\\ 8$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "From rule 1, the largest number, $12$, can be second or third. From rule 2, because there are five places, the smallest number $-2$ can either be third or fourth. The median, $6$ can be second, third, or fourth. Because we know the middle three numbers, the first and last numbers are $4$ and $9$, disregarding their order. Their average is $(4+9)/2 = \\boxed{\\textbf{(C)}\\ 6.5}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2023_AMC_8_Problems_13", "eval_func": null }, "input": { "text": "Along the route of a bicycle race, $7$ water stations are evenly spaced between the start and finish lines,\nas shown in the figure below. There are also $2$ repair stations evenly spaced between the start and\nfinish lines. The $3$rd water station is located $2$ miles after the $1$st repair station. How long is the race\nin miles?\n[asy] usepackage(\"mathptmx\"); size(10cm); filldraw((11,4.5)--(171,4.5)--(171,17.5)--(11,17.5)--cycle,mediumgray*0.4 + lightgray*0.6); draw((11,11)--(171,11),linetype(\"2 2\")+white+linewidth(1.2)); draw((0,0)--(11,0)--(11,22)--(0,22)--cycle); draw((171,0)--(182,0)--(182,22)--(171,22)--cycle); draw((31,4.5)--(31,0)); draw((51,4.5)--(51,0)); draw((151,4.5)--(151,0)); label(scale(.85)*rotate(45)*\"Water 1\", (23,-13.5)); label(scale(.85)*rotate(45)*\"Water 2\", (43,-13.5)); label(scale(.85)*rotate(45)*\"Water 7\", (143,-13.5)); filldraw(circle((103,-13.5),.2)); filldraw(circle((98,-13.5),.2)); filldraw(circle((93,-13.5),.2)); filldraw(circle((88,-13.5),.2)); filldraw(circle((83,-13.5),.2)); label(scale(.85)*rotate(90)*\"Start\", (5.5,11)); label(scale(.85)*rotate(270)*\"Finish\", (176.5,11)); [/asy]\n$\\textbf{(A)}\\ 8 \\qquad \\textbf{(B)}\\ 16 \\qquad \\textbf{(C)}\\ 24 \\qquad \\textbf{(D)}\\ 48 \\qquad \\textbf{(E)}\\ 96$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Suppose that the race is $d$ miles long. The water stations are located at \\[\\frac{d}{8}, \\frac{2d}{8}, \\ldots, \\frac{7d}{8}\\] miles from the start, and the repair stations are located at \\[\\frac{d}{3}, \\frac{2d}{3}\\] miles from the start.\nWe are given that $\\frac{3d}{8}=\\frac{d}{3}+2,$ from which \n\\begin{align*} \\frac{9d}{24}&=\\frac{8d}{24}+2 \\\\ \\frac{d}{24}&=2 \\\\ d&=\\boxed{\\textbf{(D)}\\ 48}. \\end{align*}" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1988_AJHSME_Problems_2", "eval_func": null }, "input": { "text": "The product $8\\times .25\\times 2\\times .125 =$\n$\\text{(A)}\\ \\frac18 \\qquad \\text{(B)}\\ \\frac14 \\qquad \\text{(C)}\\ \\frac12 \\qquad \\text{(D)}\\ 1 \\qquad \\text{(E)}\\ 2$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Converting the decimals to fractions, this is \n\\begin{align*} 8\\times \\frac{1}{4} \\times 2\\times \\frac{1}{8} &= \\frac{8\\times 2}{4\\times 8} \\\\ &= \\frac{16}{32} \\\\ &= \\frac{1}{2} \\rightarrow \\boxed{\\text{C}} \\end{align*}" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2008_AMC_8_Problems_1", "eval_func": null }, "input": { "text": "Susan had 50 dollars to spend at the carnival. She spent 12 dollars on food and twice as much on rides. How many dollars did she have left to spend?\n$\\textbf{(A)}\\ 12 \\qquad \\textbf{(B)}\\ 14 \\qquad \\textbf{(C)}\\ 26 \\qquad \\textbf{(D)}\\ 38 \\qquad \\textbf{(E)}\\ 50$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "If Susan spent 12 dollars, then twice that much on rides, then she spent $12+12 \\times 2=36$ dollars in total. We subtract $36$ from $50$ to get $\\boxed{\\textbf{(B)}\\ 14}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2023_AMC_8_Problems_14", "eval_func": null }, "input": { "text": "Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of $5$-cent, $10$-cent, and $25$-cent stamps, with exactly $20$ of each type. What is the greatest number of stamps Nicolas can use to make exactly $$7.10$ in postage?\n(Note: The amount $$7.10$ corresponds to $7$ dollars and $10$ cents. One dollar is worth $100$ cents.)\n$\\textbf{(A)}\\ 45 \\qquad \\textbf{(B)}\\ 46 \\qquad \\textbf{(C)}\\ 51 \\qquad \\textbf{(D)}\\ 54\\qquad \\textbf{(E)}\\ 55$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let's use the most stamps to make $7.10.$ We have $20$ of each stamp, $5$-cent (nickels), $10$-cent (dimes), and $25$-cent (quarters).\nIf we want the highest number of stamps, we must have the highest number of the smaller value stamps (like the coins above). We can use $20$ nickels and $20$ dimes to bring our total cost to $7.10 - 3.00 = 4.10$. However, when we try to use quarters, the $25$ cents don\u2019t fit evenly, so we have to give back $15$ cents to make the quarter amount $4.25$. The most efficient way to do this is to give back a $10$-cent (dime) stamp and a $5$-cent (nickel) stamp to have $38$ stamps used so far. Now, we just use $\\frac{425}{25} = 17$ quarters to get a grand total of $38 + 17 = \\boxed{\\textbf{(E)}\\ 55}$.", "The value of his entire stamp collection is $8$ dollars. To make $$7.10$ with stamps, he should remove $90$ cents worth of stamps with as few stamps as possible. To do this, he should start by removing as many $25$ cent stamps as possible as they have the greatest denomination. He can remove at most $3$ of these stamps. He still has to remove $90-25\\cdot3=15$ cents worth of stamps. This can be done with one $5$ and $10$ cent stamp. In total, he has $20\\cdot3=60$ stamps in his entire collection. As a result, the maximum number of stamps he can use is $20\\cdot3-5=\\boxed{\\textbf{(E)}\\ 55}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1996_AJHSME_Problems_20", "eval_func": null }, "input": { "text": "Suppose there is a special key on a calculator that replaces the number $x$ currently displayed with the number given by the formula $1/(1-x)$. For example, if the calculator is displaying 2 and the special key is pressed, then the calculator will display -1 since $1/(1-2)=-1$. Now suppose that the calculator is displaying 5. After the special key is pressed 100 times in a row, the calculator will display \n$\\text{(A)}\\ -0.25 \\qquad \\text{(B)}\\ 0 \\qquad \\text{(C)}\\ 0.8 \\qquad \\text{(D)}\\ 1.25 \\qquad \\text{(E)}\\ 5$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We look for a pattern, hoping this sequence either settles down to one number, or that it forms a cycle that repeats.\nAfter $1$ press, the calculator displays $\\frac{1}{1 - 5} = -\\frac{1}{4}$\nAfter $2$ presses, the calculator displays $\\frac{1}{1 - (-\\frac{1}{4})} = \\frac{1}{\\frac{5}{4}} = \\frac{4}{5}$\nAfter $3$ presses, the calculator displays $\\frac{1}{1 - \\frac{4}{5}} = \\frac{1}{\\frac{1}{5}} = 5$\nThus, every three presses, the display will be $5$. On press $3\\cdot 33 = 99$, the display will be $5$. One more press will give $-\\frac{1}{4}$, which is answer $\\boxed{A}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1988_AJHSME_Problems_14", "eval_func": null }, "input": { "text": "$\\diamondsuit$ and $\\Delta$ are whole numbers and $\\diamondsuit \\times \\Delta =36$. The largest possible value of $\\diamondsuit + \\Delta$ is \n$\\text{(A)}\\ 12 \\qquad \\text{(B)}\\ 13 \\qquad \\text{(C)}\\ 15 \\qquad \\text{(D)}\\ 20\\ \\qquad \\text{(E)}\\ 37$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Since it doesn't take too long, we can just make a table with all the possible values of the sum: \n\\[\\begin{array}{|c|c|c|} \\multicolumn{3}{}{} \\\\ \\hline \\diamondsuit & \\Delta & \\diamondsuit + \\Delta \\\\ \\hline 36 & 1 & 37 \\\\ \\hline 18 & 2 & 20 \\\\ \\hline 12 & 3 & 15 \\\\ \\hline 9 & 4 & 13 \\\\ \\hline 6 & 6 & 12 \\\\ \\hline \\end{array}\\]\nClearly $37\\rightarrow \\boxed{\\text{E}}$ is the largest." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1988_AJHSME_Problems_22", "eval_func": null }, "input": { "text": "Tom's Hat Shoppe increased all original prices by $25\\%$. Now the shoppe is having a sale where all prices are $20\\%$ off these increased prices. Which statement best describes the sale price of an item?\n$\\text{(A)}\\ \\text{The sale price is }5\\% \\text{ higher than the original price.}$\n$\\text{(B)}\\ \\text{The sale price is higher than the original price, but by less than }5\\% .$\n$\\text{(C)}\\ \\text{The sale price is higher than the original price, but by more than }5\\% .$\n$\\text{(D)}\\ \\text{The sale price is lower than the original price.}$\n$\\text{(E)}\\ \\text{The sale price is the same as the original price.}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let the original price of an item be $x$. The shoppe originally increased this to $1.25x$. The sale brings it down to $.8(1.25x)=x$, which is the same as the original $\\rightarrow \\boxed{\\text{E}}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2010_AMC_8_Problems_20", "eval_func": null }, "input": { "text": "In a room, $2/5$ of the people are wearing gloves, and $3/4$ of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove? \n$\\textbf{(A)}\\ 3 \\qquad\\textbf{(B)}\\ 5\\qquad\\textbf{(C)}\\ 8\\qquad\\textbf{(D)}\\ 15\\qquad\\textbf{(E)}\\ 20$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $x$ be the number of people wearing both a hat and a glove. Since the number of people wearing a hat or a glove must be whole numbers, the number of people in the room must be a multiple of (4,5), but we do lcm of 4,5 = 20. Since we are trying to find the minimum $x$, we must use the smallest possible value for the number of people in the room. Similarly, we can assume that there are no people present who are wearing neither of the two items since this would unnecessarily increase the number of people in the room. Thus, we can say that there are $20$ people in the room, all of which are wearing at least a hat or a glove.\nIt follows that there are $\\frac{2}{5}\\cdot 20 = 8$ people wearing gloves and $\\frac{3}{4}\\cdot 20 = 15$ people wearing hats. Then by applying the Principle of Inclusion Exclusion (PIE), the total number of people in the room wearing either a hat or a glove or both is $8+15-x = 23-x$. Since we know that this equals $20$, it follows that $23-x = 20$, which implies that $x=3$. Thus, $\\boxed{\\textbf{(A)}\\ 3}$ is the correct answer." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1997_AJHSME_Problems_7", "eval_func": null }, "input": { "text": "The area of the smallest square that will contain a circle of radius 4 is\n$\\text{(A)}\\ 8 \\qquad \\text{(B)}\\ 16 \\qquad \\text{(C)}\\ 32 \\qquad \\text{(D)}\\ 64 \\qquad \\text{(E)}\\ 128$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Draw a square circumscribed around the circle. (Alternately, the circle is inscribed in the square.) If the circle has radius $4$, it has diameter $8$. Two of the diameters of the circle will run parallel to the sides of the square. Thus, the smallest square that contains it has side length $8$, and area $8\\times8=64$.\n$\\boxed{\\textbf{(D)}}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2002_AMC_8_Problems_4", "eval_func": null }, "input": { "text": "The year 2002 is a palindrome (a number that reads the same from left to right as it does from right to left). What is the product of the digits of the next year after 2002 that is a palindrome?\n$\\text{(A)}\\ 0 \\qquad \\text{(B)}\\ 4 \\qquad \\text{(C)}\\ 9 \\qquad \\text{(D)}\\ 16 \\qquad \\text{(E)}\\ 25$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The palindrome right after 2002 is 2112. The product of the digits of 2112 is $\\boxed{\\text{(B)}\\ 4}$.", "The palindrome formula is to add 110 to the number in order to get the next palindrome, a palindrome needs to be in the form as ABBA . We can use this in this case to get 2112. 2*1*1*2=4. Therefore, the answer is $\\boxed{\\text{(B)}\\ 4}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2009_AMC_8_Problems_19", "eval_func": null }, "input": { "text": "Two angles of an isosceles triangle measure $70^\\circ$ and $x^\\circ$. What is the sum of the three possible values of $x$?\n$\\textbf{(A)}\\ 95 \\qquad \\textbf{(B)}\\ 125 \\qquad \\textbf{(C)}\\ 140 \\qquad \\textbf{(D)}\\ 165 \\qquad \\textbf{(E)}\\ 180$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "There are 3 cases: where $x^\\circ$ is a base angle with the $70^\\circ$ as the other angle, where $x^\\circ$ is a base angle with $70^\\circ$ as the vertex angle, and where $x^\\circ$ is the vertex angle with $70^\\circ$ as a base angle.\nCase 1: $x^\\circ$ is a base angle with the $70^\\circ$ as the other angle:\nHere, $x=70$, since base angles are congruent.\nCase 2: $x^\\circ$ is a base angle with $70^\\circ$ as the vertex angle:\nHere, the 2 base angles are both $x^\\circ$, so we can use the equation $2x+70=180$, which simplifies to $x=55$.\nCase 3: $x^\\circ$ is the vertex angle with $70^\\circ$ as a base angle:\nHere, both base angles are $70^\\circ$, since base angles are congruent. Thus, we can use the equation $x+140=180$, which simplifies to $x=40$.\nAdding up all the cases, we get $70+55+40=165$, so the answer is $\\boxed{\\textbf{(D)}\\ 165}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1999_AMC_8_Problems_8", "eval_func": null }, "input": { "text": "Six squares are colored, front and back, (R = red, B = blue, O = orange, Y = yellow, G = green, and W = white). They are hinged together as shown, then folded to form a cube. The face opposite the white face is\n[asy] draw((0,2)--(1,2)--(1,1)--(2,1)--(2,0)--(3,0)--(3,1)--(4,1)--(4,2)--(2,2)--(2,3)--(0,3)--cycle); draw((1,3)--(1,2)--(2,2)--(2,1)--(3,1)--(3,2)); label(\"R\",(.5,2.3),N); label(\"B\",(1.5,2.3),N); label(\"G\",(1.5,1.3),N); label(\"Y\",(2.5,1.3),N); label(\"W\",(2.5,.3),N); label(\"O\",(3.5,1.3),N); [/asy]\n$\\text{(A)}\\ \\text{B} \\qquad \\text{(B)}\\ \\text{G} \\qquad \\text{(C)}\\ \\text{O} \\qquad \\text{(D)}\\ \\text{R} \\qquad \\text{(E)}\\ \\text{Y}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "When G is arranged to be the base, B is the back face and W is\nthe front face. Thus, $\\boxed{\\text{(A)}\\ B}$ is opposite W.", "Let Y be the top and fold G, O, and W down.\nThen $\\boxed{\\text{(A)}\\ B}$ will fold to become the back face and be\nopposite W." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2001_AMC_8_Problems_25", "eval_func": null }, "input": { "text": "There are 24 four-digit whole numbers that use each of the four digits 2, 4, 5 and 7 exactly once. Only one of these four-digit numbers is a multiple of another one. Which of the following is it?\n$\\textbf{(A)}\\ 5724 \\qquad \\textbf{(B)}\\ 7245 \\qquad \\textbf{(C)}\\ 7254 \\qquad \\textbf{(D)}\\ 7425 \\qquad \\textbf{(E)}\\ 7542$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We begin by narrowing down the possibilities. If the larger number were twice the smaller number, then the smallest possibility for the larger number is $2457\\times2=4914$, since $2457$ is the smallest number in the set. The largest possibility would have to be twice the largest number in the set such that when it is multiplied by $2$, it is less than or equal to $7542$, the largest number in the set. This happens to be $2754\\times2=5508$. Therefore, the number would have to be between $4914$ and $5508$, and also even. The only even numbers in the set and in this range are $5472$ and $5274$. A quick check reveals that neither of these numbers is twice a number in the set. The number can't be quadruple or more another number in the set since $2457\\times4=9828$, well past the range of the set. Therefore, the number must be triple another number in the set. The least possibility is $2457\\times3=7371$ and the greatest is $2475\\times3=7425$, since any higher number in the set multiplied by $3$ would be out of the range of the set. Reviewing, we find that the upper bound does in fact work, so the multiple is $2475\\times3=7425, \\boxed{\\textbf{(D)}}$\n\nOr, since the greatest number possible divided by the smallest number possible is slightly greater than 3, divide all of the choices by 2, then 3 and see if the resulting answer contains 2,4,5 and 7. Doing so, you find that $7425/3 = 2475, \\boxed{\\textbf{(D)}}$", "There are only 5 options for the problem so we can just try them. It is easy since that we only need try to use $2$, $3$ to divide them. Even $5$ will leads to a solution start with $1$ which we don't need.\n$5724/2=2862$, $5724/3=1908$. $7245/3=2415$. $7254/2=3612$, $7254/3=2418$. $7425/3=2475$. The answer is $\\boxed{\\textbf{(D)}}$. You can obtain the answer in only 6 calculations." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1996_AJHSME_Problems_12", "eval_func": null }, "input": { "text": "What number should be removed from the list\n\\[1,2,3,4,5,6,7,8,9,10,11\\]\nso that the average of the remaining numbers is $6.1$?\n$\\text{(A)}\\ 4 \\qquad \\text{(B)}\\ 5 \\qquad \\text{(C)}\\ 6 \\qquad \\text{(D)}\\ 7 \\qquad \\text{(E)}\\ 8$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Adding all of the numbers gives us $\\frac{11\\cdot12}{2}=66$ as the current total. Since there are $11$ numbers, the current average is $\\frac{66}{11}=6$. We need to take away a number from the total and then divide the result by $10$ because there will only be $10$ numbers left to give an average of $6.1$. Setting up the equation:\n$\\frac{66-x}{10}=6.1$\n$66 - x = 61$\n$x = 5$\nThus, the answer is $\\boxed{B}$", "Similar to the first solution, the current total is $66$. Since there are $11$ numbers on the list, taking $1$ number away will leave $10$ numbers. If those $10$ numbers have an average of $6.1$, then those $10$ numbers must have a sum of $10 \\times 6.1 = 61$. Thus, the number that was removed must be $66 - 61 = 5$, and the answer is $\\boxed{B}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1985_AJHSME_Problems_24", "eval_func": null }, "input": { "text": "In a magic triangle, each of the six $10-15$ is placed in one of the so that the sum, $S$, of the three numbers on each side of the is the same. The largest possible value for $S$ is\n[asy] draw(circle((0,0),1)); draw(dir(60)--6*dir(60)); draw(circle(7*dir(60),1)); draw(8*dir(60)--13*dir(60)); draw(circle(14*dir(60),1)); draw((1,0)--(6,0)); draw(circle((7,0),1)); draw((8,0)--(13,0)); draw(circle((14,0),1)); draw(circle((10.5,6.0621778264910705273460621952706),1)); draw((13.5,0.86602540378443864676372317075294)--(11,5.1961524227066318805823390245176)); draw((10,6.9282032302755091741097853660235)--(7.5,11.258330249197702407928401219788)); [/asy]\n$\\text{(A)}\\ 36 \\qquad \\text{(B)}\\ 37 \\qquad \\text{(C)}\\ 38 \\qquad \\text{(D)}\\ 39 \\qquad \\text{(E)}\\ 40$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let the number in the top circle be $a$ and then $b$, $c$, $d$, $e$, and $f$, going in clockwise order. Then, we have \\[S=a+b+c\\] \\[S=c+d+e\\] \\[S=e+f+a\\]\nAdding these equations together, we get \n\\begin{align*} 3S &= (a+b+c+d+e+f)+(a+c+e) \\\\ &= 75+(a+c+e) \\\\ \\end{align*}\nwhere the last step comes from the fact that since $a$, $b$, $c$, $d$, $e$, and $f$ are the numbers $10-15$ in some order, their is $10+11+12+13+14+15=75$\nThe left hand side is divisible by $3$ and $75$ is divisible by $3$, so $a+c+e$ must be divisible by $3$. The largest possible value of $a+c+e$ is then $15+14+13=42$, and the corresponding value of $S$ is $\\frac{75+42}{3}=39$, which is choice $\\boxed{\\text{D}}$. \nIt turns out this sum is attainable if you let \\[a=15\\] \\[b=10\\] \\[c=14\\] \\[d=12\\] \\[e=13\\] \\[f=11\\]", "To make the sum the greatest, put the three largest numbers $(13,14$ and $15)$ in the corners. Then, balance the sides by putting the least integer $(10)$ between the greatest sum $(14$ and $15)$. Then put the next least integer $(11)$ between the next greatest sum ($13 +15$). Fill in the last integer $(12)$ and you can see that the sum of any three numbers on a side is (for example) $14 +10 + 15 = 39$\n$\\boxed{\\text{D}}$. " ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1986_AJHSME_Problems_6", "eval_func": null }, "input": { "text": "$\\frac{2}{1-\\frac{2}{3}}=$\n$\\text{(A)}\\ -3 \\qquad \\text{(B)}\\ -\\frac{4}{3} \\qquad \\text{(C)}\\ \\frac{2}{3} \\qquad \\text{(D)}\\ 2 \\qquad \\text{(E)}\\ 6$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Just simplify the bottom as $\\frac{3}{3}-\\frac{2}{3}=\\frac{1}{3}$, getting us $\\frac{2}{\\frac{1}{3}}$, with which we multiply top and bottom by 3, we get $\\frac{6}{1}$, or $6$\n$\\boxed{\\text{E}}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2003_AMC_8_Problems_19", "eval_func": null }, "input": { "text": "How many integers between 1000 and 2000 have all three of the numbers 15, 20, and 25 as factors?\n$\\textbf{(A)}\\ 1 \\qquad \\textbf{(B)}\\ 2 \\qquad \\textbf{(C)}\\ 3 \\qquad \\textbf{(D)}\\ 4 \\qquad \\textbf{(E)}\\ 5$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Find the least common multiple of $15, 20, 25$ by turning the numbers into their prime factorization. \\[15 = 3 * 5, 20 = 2^2 * 5, 25 = 5^2\\] Gather all necessary multiples \n$3, 2^2, 5^2$ when multiplied gets $300$. The multiples of $300 - 300, 600, 900, 1200, 1500, 1800, 2100$. The number of multiples between 1000 and 2000 is $\\boxed{\\textbf{(C)}\\ 3}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1994_AJHSME_Problems_20", "eval_func": null }, "input": { "text": "Let $W,X,Y$ and $Z$ be four different digits selected from the set\n$\\{ 1,2,3,4,5,6,7,8,9\\}.$\nIf the sum $\\dfrac{W}{X} + \\dfrac{Y}{Z}$ is to be as small as possible, then $\\dfrac{W}{X} + \\dfrac{Y}{Z}$ must equal\n$\\text{(A)}\\ \\dfrac{2}{17} \\qquad \\text{(B)}\\ \\dfrac{3}{17} \\qquad \\text{(C)}\\ \\dfrac{17}{72} \\qquad \\text{(D)}\\ \\dfrac{25}{72} \\qquad \\text{(E)}\\ \\dfrac{13}{36}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "\\[\\frac{W}{X} + \\frac{Y}{Z} = \\frac{WZ+XY}{XZ}\\]\nSmall fractions have small numerators and large denominators. To maximize the denominator, let $X=8$ and $Z=9$.\n\\[\\frac{9W+8Y}{72}\\]\nTo minimize the numerator, let $W=1$ and $Y=2$.\n\\[\\frac{9+16}{72} = \\boxed{\\text{(D)}\\rightarrow \\frac{25}{72}}\\]", "To make the smallest fraction, you need the lowest numerator and the highest denominator. So, take the first $2$ and last $2$ digits of the set, which are $1,2$ and $8,9$. Balance the equations to be \"even\". Since $1$ is smaller than $2$, put it over $8$. You get $\\frac{1}{8}+\\frac{2}{9}=\\frac{25}{72}$, or $\\boxed{D}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2005_AMC_8_Problems_14", "eval_func": null }, "input": { "text": "The Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many conference games are scheduled?\n$\\textbf{(A)}\\ 80\\qquad\\textbf{(B)}\\ 96\\qquad\\textbf{(C)}\\ 100\\qquad\\textbf{(D)}\\ 108\\qquad\\textbf{(E)}\\ 192$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Within each division, there are $\\binom {6}{2} = 15$ pairings, and each of these games happens twice. The same goes for the other division so that there are $4(15)=60$ games within their own divisions. The number of games between the two divisions is $(6)(6)=36$. Together there are $60+36=\\boxed{\\textbf{(B)}\\ 96}$ conference games.", "Each team plays 10 games in its own division and 6 games against teams in the other division. So each of the 12 teams plays 16 conference games. Because each game involves two teams, there are $\\frac{12\\times 16}{2}=\\boxed{96}$ games scheduled. " ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2019_AMC_8_Problems_20", "eval_func": null }, "input": { "text": "How many different real numbers $x$ satisfy the equation \\[(x^{2}-5)^{2}=16?\\]\n$\\textbf{(A) }0\\qquad\\textbf{(B) }1\\qquad\\textbf{(C) }2\\qquad\\textbf{(D) }4\\qquad\\textbf{(E) }8$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We have that $(x^2-5)^2 = 16$ if and only if $x^2-5 = \\pm 4$. If $x^2-5 = 4$, then $x^2 = 9 \\implies x = \\pm 3$, giving 2 solutions. If $x^2-5 = -4$, then $x^2 = 1 \\implies x = \\pm 1$, giving 2 more solutions. All four of these solutions work, so the answer is $\\boxed{\\textbf{(D) }4}$. Further, the equation is a in $x$, so by the , there can be at most four real solutions.", "We can expand $(x^2-5)^2$ to get $x^4-10x^2+25$, so now our equation is $x^4-10x^2+25=16$. Subtracting $16$ from both sides gives us $x^4-10x^2+9=0$. Now, we can factor the left hand side to get $(x^2-9)(x^2-1)=0$. If $x^2-9$ and/or $x^2-1$ equals $0$, then the whole left side will equal $0$. Since the solutions can be both positive and negative, we have $4$ solutions: $-3,3,-1,1$ (we can find these solutions by setting $x^2-9$ and $x^2-1$ equal to $0$ and solving for $x$). So, the answer is $\\boxed{\\textbf{(D) }4}$. ", "Subtract 16 from both sides and factor using difference of squares: \n\n\\[(x^2 - 5)^2 = 16\\]\n\\[(x^2 - 5)^2 - 16 =0\\]\n\\[(x^2 - 5)^2 - 4^2 = 0\\]\n\\[[(x^2 - 5)-4][(x^2 - 5) + 4] = 0\\]\n\\[(x^2 - 9)(x^2 - 1) =0\\]\n\\[(x+3)(x-3)(x+1)(x-1) = 0\\]\n\nQuite obviously, this equation has $\\boxed{\\textbf{(D) }4}$ solutions. \n" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2007_AMC_8_Problems_8", "eval_func": null }, "input": { "text": "In trapezoid $ABCD$, $\\overline{AD}$ is perpendicular to $\\overline{DC}$,\n$AD = AB = 3$, and $DC = 6$. In addition, $E$ is on $\\overline{DC}$, and $\\overline{BE}$ is parallel to $\\overline{AD}$. Find the area of $\\triangle BEC$.\n[asy] defaultpen(linewidth(0.7)); pair A=(0,3), B=(3,3), C=(6,0), D=origin, E=(3,0); draw(E--B--C--D--A--B); draw(rightanglemark(A, D, C)); label(\"$A$\", A, NW); label(\"$B$\", B, NW); label(\"$C$\", C, SE); label(\"$D$\", D, SW); label(\"$E$\", E, NW); label(\"$3$\", A--D, W); label(\"$3$\", A--B, N); label(\"$6$\", E, S); [/asy]\n$\\textbf{(A)}\\ 3 \\qquad \\textbf{(B)}\\ 4.5 \\qquad \\textbf{(C)}\\ 6 \\qquad \\textbf{(D)}\\ 9 \\qquad \\textbf{(E)}\\ 18$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Clearly, $ABED$ is a square with side-length $3.$ By segment subtraction, we have $EC = DC - DE = 6 - 3 = 3.$\nThe area of $\\triangle BEC$ is \\[\\frac12\\cdot EC\\cdot BE = \\frac12\\cdot3\\cdot3 = \\boxed{\\textbf{(B)}\\ 4.5}.\\]", "Clearly, $ABED$ is a square with side-length $3.$\nLet the brackets denote areas. We apply area subtraction to find the area of $\\triangle BEC:$\n\\begin{align*} [BEC]&=[ABCD]-[ABED] \\\\ &=\\frac{AB+CD}{2}\\cdot AD - AB^2 \\\\ &=\\frac{3+6}{2}\\cdot 3 - 3^2 \\\\ &=\\boxed{\\textbf{(B)}\\ 4.5}. \\end{align*}" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2002_AMC_8_Problems_12", "eval_func": null }, "input": { "text": "A board game spinner is divided into three regions labeled $A$, $B$ and $C$. The probability of the arrow stopping on region $A$ is $\\frac{1}{3}$ and on region $B$ is $\\frac{1}{2}$. The probability of the arrow stopping on region $C$ is:\n\n$\\text{(A)}\\ \\frac{1}{12}\\qquad\\text{(B)}\\ \\frac{1}{6}\\qquad\\text{(C)}\\ \\frac{1}{5}\\qquad\\text{(D)}\\ \\frac{1}{3}\\qquad\\text{(E)}\\ \\frac{2}{5}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Since the arrow must land in one of the three regions, the sum of the probabilities must be 1. Thus the answer is $1-\\frac{1}{2}-\\frac{1}{3}=\\boxed{\\text{(B)}\\ \\frac16}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1996_AJHSME_Problems_25", "eval_func": null }, "input": { "text": "A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region?\n$\\text{(A)}\\frac{1}{4} \\qquad \\text{(B)}\\frac{1}{3} \\qquad \\text{(C)}\\frac{1}{2} \\qquad \\text{(D)}\\frac{2}{3} \\qquad \\text{(E)}\\frac{3}{4}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Draw a circle with a radius of $2$. Draw a concentric circle with radius $1$. The circumference of this inner circle is the set of all points that are $1$ from the center, and $1$ from the outer circle meaning the set of points are equidistant from the center of the circles to the outside of the big circle. \nThe inside of smaller circle is the set of all points that are closer to the center of the region than to the boundary of the outer circle. The \"washer\" region that is outside the circle of radius $1$, but inside the circle of radius $2$, is the set of all points that are closer to the boundary than to the center of the circle.\nIf you select a random point in a region of area $B$, the probability that the point is in a smaller subregion $A$ is the ratio $\\frac{A}{B}$. In this case, $B = \\pi\\cdot 2^2 = 4\\pi$, and $A = \\pi\\cdot 1^2 = \\pi$, and the ratio of areas is $\\frac{\\pi}{4\\pi} = \\frac{1}{4}$, and the answer is $\\boxed{A}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1996_AJHSME_Problems_24", "eval_func": null }, "input": { "text": "The measure of angle $ABC$ is $50^\\circ$, $\\overline{AD}$ bisects angle $BAC$, and $\\overline{DC}$ bisects angle $BCA$. The measure of angle $ADC$ is\n[asy] pair A,B,C,D; A = (0,0); B = (9,10); C = (10,0); D = (6.66,3); dot(A); dot(B); dot(C); dot(D); draw(A--B--C--cycle); draw(A--D--C); label(\"$A$\",A,SW); label(\"$B$\",B,N); label(\"$C$\",C,SE); label(\"$D$\",D,N); label(\"$50^\\circ $\",(9.4,8.8),SW); [/asy]\n$\\text{(A)}\\ 90^\\circ \\qquad \\text{(B)}\\ 100^\\circ \\qquad \\text{(C)}\\ 115^\\circ \\qquad \\text{(D)}\\ 122.5^\\circ \\qquad \\text{(E)}\\ 125^\\circ$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $\\angle CAD = \\angle BAD = x$, and let $\\angle ACD = \\angle BCD = y$\nFrom $\\triangle ABC$, we know that $50 + 2x + 2y = 180$, leading to $x + y = 65$.\nFrom $\\triangle ADC$, we know that $x + y + \\angle D = 180$. Plugging in $x + y = 65$, we get $\\angle D = 180 - 65 = 115$, which is answer $\\boxed{C}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1996_AJHSME_Problems_1", "eval_func": null }, "input": { "text": "How many positive factors of 36 are also multiples of 4?\n$\\text{(A)}\\ 2 \\qquad \\text{(B)}\\ 3 \\qquad \\text{(C)}\\ 4 \\qquad \\text{(D)}\\ 5 \\qquad \\text{(E)}\\ 6$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The factors of $36$ are $1, 2, 3, 4, 6, 9, 12, 18,$ and $36$.\nThe multiples of $4$ up to $36$ are $4, 8, 12, 16, 20, 24, 28, 32$ and $36$.\nOnly $4, 12$ and $36$ appear on both lists, so the answer is $3$, which is option $\\boxed{B}$.", "$36 = 4^1 \\cdot 3^2$. All possible factors of $36$ will be here, except for ones divisible by $2$ and not by $4$. $(1+1)\\cdot (2+1) = 6$. Subtract factors not divisible by $4$, which are $1$, $3^1$, and $3^2$. $6-3=3$, which is $\\boxed{B}$.", "Divide $36$ by $4$, and the remaining factors, when multiplied by $4$, will be factors of $36$.\n$36 \\div 4 = 9$, which has $3$ factors, giving us option $\\boxed{B}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1993_AJHSME_Problems_14", "eval_func": null }, "input": { "text": "The nine squares in the table shown are to be filled so that every row and every column contains each of the numbers $1,2,3$. Then $A+B=$\n\\[\\begin{tabular}{|c|c|c|}\\hline 1 & &\\\\ \\hline & 2 & A\\\\ \\hline & & B\\\\ \\hline\\end{tabular}\\]\n$\\text{(A)}\\ 2 \\qquad \\text{(B)}\\ 3 \\qquad \\text{(C)}\\ 4 \\qquad \\text{(D)}\\ 5 \\qquad \\text{(E)}\\ 6$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The square connected both to 1 and 2 cannot be the same as either of them, so must be 3.\n\\[\\begin{tabular}{|c|c|c|}\\hline 1 & 3 &\\\\ \\hline & 2 & A\\\\ \\hline & & B\\\\ \\hline\\end{tabular}\\]\nThe last square in the top row cannot be either 1 or 3, so it must be 2.\n\\[\\begin{tabular}{|c|c|c|}\\hline 1 & 3 & 2\\\\ \\hline & 2 & A\\\\ \\hline & & B\\\\ \\hline\\end{tabular}\\]\nThe other two squares in the rightmost column with A and B cannot be two, so they must be 1 and 3 and therefore have a sum of $1+3=\\boxed{\\text{(C)}\\ 4}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2010_AMC_8_Problems_11", "eval_func": null }, "input": { "text": "The top of one tree is $16$ feet higher than the top of another tree. The heights of the two trees are in the ratio $3:4$. In feet, how tall is the taller tree? \n$\\textbf{(A)}\\ 48 \\qquad\\textbf{(B)}\\ 64 \\qquad\\textbf{(C)}\\ 80 \\qquad\\textbf{(D)}\\ 96\\qquad\\textbf{(E)}\\ 112$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let the height of the taller tree be $h$ and let the height of the smaller tree be $h-16$. Since the ratio of the smaller tree to the larger tree is $\\frac{3}{4}$, we have $\\frac{h-16}{h}=\\frac{3}{4}$. Solving for $h$ gives us $h=64 \\Rightarrow \\boxed{\\textbf{(B)}\\ 64}$", "To answer this problem, you have to make it so that we have the same proportion as 3:4, but the difference between them is 16. Since the two numbers are consecutive, if we multiply both of them by 16, we would get a difference of 16 between them. So, it would be 48:64 and since we need to find the height of the taller tree, we get $h=64 \\Rightarrow \\boxed{\\textbf{(B)}\\ 64}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2016_AMC_8_Problems_6", "eval_func": null }, "input": { "text": "The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names?\n[asy] unitsize(0.9cm); draw((-0.5,0)--(10,0), linewidth(1.5)); draw((-0.5,1)--(10,1)); draw((-0.5,2)--(10,2)); draw((-0.5,3)--(10,3)); draw((-0.5,4)--(10,4)); draw((-0.5,5)--(10,5)); draw((-0.5,6)--(10,6)); draw((-0.5,7)--(10,7)); label(\"frequency\",(-0.5,8)); label(\"0\", (-1, 0)); label(\"1\", (-1, 1)); label(\"2\", (-1, 2)); label(\"3\", (-1, 3)); label(\"4\", (-1, 4)); label(\"5\", (-1, 5)); label(\"6\", (-1, 6)); label(\"7\", (-1, 7)); filldraw((0,0)--(0,7)--(1,7)--(1,0)--cycle, black); filldraw((2,0)--(2,3)--(3,3)--(3,0)--cycle, black); filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, black); filldraw((6,0)--(6,4)--(7,4)--(7,0)--cycle, black); filldraw((8,0)--(8,4)--(9,4)--(9,0)--cycle, black); label(\"3\", (0.5, -0.5)); label(\"4\", (2.5, -0.5)); label(\"5\", (4.5, -0.5)); label(\"6\", (6.5, -0.5)); label(\"7\", (8.5, -0.5)); label(\"name length\", (4.5, -1)); [/asy]\n$\\textbf{(A) }3\\qquad\\textbf{(B) }4\\qquad\\textbf{(C) }5\\qquad\\textbf{(D) }6\\qquad \\textbf{(E) }7$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We first notice that the median name will be the $(19+1)/2=10^{\\mbox{th}}$ name. The $10^{\\mbox{th}}$ name is $\\boxed{\\textbf{(B)}\\ 4}$.", "To find the median length of a name from a bar graph, we must add up the number of names. Doing so gives us $7 + 3 + 1 + 4 + 4 = 19$. Thus the index of the median length would be the 10th name. Since there are $7$ names with length $3$, and $3$ names with length $4$, the $10$th name would have $4$ letters. Thus our answer is $\\boxed{\\textbf{(B)}\\ 4}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2002_AMC_8_Problems_8", "eval_func": null }, "input": { "text": "Problems 8,9 and 10 use the data found in the accompanying paragraph and table:\n[asy] size(3inch, 1.5inch); for ( int y = 0; y <= 5; ++y ) { draw((0,y)--(18,y)); } draw((0,0)--(0,5)); draw((6,0)--(6,5)); draw((9,0)--(9,5)); draw((12,0)--(12,5)); draw((15,0)--(15,5)); draw((18,0)--(18,5)); draw(scale(0.8)*\"50s\", (7.5,4.5)); draw(scale(0.8)*\"4\", (7.5,3.5)); draw(scale(0.8)*\"8\", (7.5,2.5)); draw(scale(0.8)*\"6\", (7.5,1.5)); draw(scale(0.8)*\"3\", (7.5,0.5)); draw(scale(0.8)*\"60s\", (10.5,4.5)); draw(scale(0.8)*\"7\", (10.5,3.5)); draw(scale(0.8)*\"4\", (10.5,2.5)); draw(scale(0.8)*\"4\", (10.5,1.5)); draw(scale(0.8)*\"9\", (10.5,0.5)); draw(scale(0.8)*\"70s\", (13.5,4.5)); draw(scale(0.8)*\"12\", (13.5,3.5)); draw(scale(0.8)*\"12\", (13.5,2.5)); draw(scale(0.8)*\"6\", (13.5,1.5)); draw(scale(0.8)*\"13\", (13.5,0.5)); draw(scale(0.8)*\"80s\", (16.5,4.5)); draw(scale(0.8)*\"8\", (16.5,3.5)); draw(scale(0.8)*\"15\", (16.5,2.5)); draw(scale(0.8)*\"10\", (16.5,1.5)); draw(scale(0.8)*\"9\", (16.5,0.5)); label(scale(0.8)*\"Country\", (3,4.5)); label(scale(0.8)*\"Brazil\", (3,3.5)); label(scale(0.8)*\"France\", (3,2.5)); label(scale(0.8)*\"Peru\", (3,1.5)); label(scale(0.8)*\"Spain\", (3,0.5)); label(scale(0.9)*\"Juan's Stamp Collection\", (9,0), S); label(scale(0.9)*\"Number of Stamps by Decade\", (9,5), N);[/asy]\nHow many of his European stamps were issued in the '80s? \n$\\text{(A)}\\ 9 \\qquad \\text{(B)}\\ 15 \\qquad \\text{(C)}\\ 18 \\qquad \\text{(D)}\\ 24 \\qquad \\text{(E)}\\ 42$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "France and Spain are European countries. The number of '80s stamps from France is $15$ and the number of '80s stamps from Spain is $9$. The total number of stamps is $15+9=\\boxed{\\text{(D)}\\ 24}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1989_AJHSME_Problems_21", "eval_func": null }, "input": { "text": "Jack had a bag of $128$ apples. He sold $25\\%$ of them to Jill. Next he sold $25\\%$ of those remaining to June. Of those apples still in his bag, he gave the shiniest one to his teacher. How many apples did Jack have then?\n$\\text{(A)}\\ 7 \\qquad \\text{(B)}\\ 63 \\qquad \\text{(C)}\\ 65 \\qquad \\text{(D)}\\ 71 \\qquad \\text{(E)}\\ 111$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "First he gives $128\\times .25 = 32$ apples to Jill, so he has $128-32=96$ apples left. Then he gives $96\\times .25 = 24$ apples to June, so he has $96-24=72$ left.\nFinally, he gives one to the teacher, leaving $71\\rightarrow \\boxed{\\text{D}}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2008_AMC_8_Problems_13", "eval_func": null }, "input": { "text": "Mr. Harman needs to know the combined weight in pounds of three boxes he wants to mail. However, the only available scale is not accurate for weights less than $100$ pounds or more than $150$ pounds. So the boxes are weighed in pairs in every possible way. The results are $122$, $125$ and $127$ pounds. What is the combined weight in pounds of the three boxes?\n$\\textbf{(A)}\\ 160\\qquad \\textbf{(B)}\\ 170\\qquad \\textbf{(C)}\\ 187\\qquad \\textbf{(D)}\\ 195\\qquad \\textbf{(E)}\\ 354$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Each box is weighed twice during this, so the combined weight of the three boxes is half the weight of these separate measures:\n\\[\\frac{122+125+127}{2} = \\frac{374}{2} = \\boxed{\\textbf{(C)}\\ 187}.\\]" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2012_AMC_8_Problems_23", "eval_func": null }, "input": { "text": "An equilateral triangle and a regular hexagon have equal perimeters. If the triangle's area is 4, what is the area of the hexagon?\n$\\textbf{(A)}\\hspace{.05in}4\\qquad\\textbf{(B)}\\hspace{.05in}5\\qquad\\textbf{(C)}\\hspace{.05in}6\\qquad\\textbf{(D)}\\hspace{.05in}4\\sqrt3\\qquad\\textbf{(E)}\\hspace{.05in}6\\sqrt3$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let the perimeter of the equilateral triangle be $3s$. The side length of the equilateral triangle would then be $s$ and the sidelength of the hexagon would be $\\frac{s}{2}$.\nA hexagon contains six equilateral triangles. One of these triangles would be similar to the large equilateral triangle in the ratio $1\u00a0: 4$, since the sidelength of the small equilateral triangle is half the sidelength of the large one. Thus, the area of one of the small equilateral triangles is $1$. The area of the hexagon is then $1 \\times 6 = \\boxed{\\textbf{(C)}\\ 6}$.", "Let the side length of the equilateral triangle be $s$ and the side length of the hexagon be $y$. Since the perimeters are equal, we must have $3s=6y$ which reduces to $s=2y$. Substitute this value in to the area of an equilateral triangle to yield $\\dfrac{(2y)^2\\sqrt{3}}{4}=\\dfrac{4y^2\\sqrt{3}}{4}$. \nSetting this equal to $4$ gives us $\\dfrac{4y^2\\sqrt{3}}{4}=4\\implies 4y^2\\sqrt{3}=16\\implies y^2\\sqrt{3}=4$.\nSubstitute $y^2\\sqrt{3}$ into the area of a regular hexagon to yield $\\dfrac{3(4)}{2}=6$.\nTherefore, our answer is $\\boxed{\\textbf{(C)}\\ 6}$.", "Let the side length of the triangle be $s$ and the side length of the hexagon be $t$. As explained in Solution 1, $s=2t$, or $t=\\frac{s}{2}$. The area of the triangle is $\\frac{s^2\\sqrt3}{4}=4$ and the area of the hexagon is $\\frac{t^2\\sqrt3}{4} \\cdot 6=\\frac{3t^2\\sqrt3}{2}$. Substituting $\\frac{s}{2}$ in for $t$, we get \n\\[\\frac{\\frac{3s^2\\sqrt3}{4}}{2}=\\frac{3s^2\\sqrt3}{8}.\\]\n$\\frac{s^2\\sqrt3}{4}=4 \\implies \\frac{s^2\\sqrt3}{8}=2 \\implies \\frac{3s^2\\sqrt3}{8}=\\boxed{\\textbf{(C)}\\ 6}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2008_AMC_8_Problems_19", "eval_func": null }, "input": { "text": "Eight points are spaced around at intervals of one unit around a $2 \\times 2$ square, as shown. Two of the $8$ points are chosen at random. What is the probability that the two points are one unit apart?\n[asy] size((50)); dot((5,0)); dot((5,5)); dot((0,5)); dot((-5,5)); dot((-5,0)); dot((-5,-5)); dot((0,-5)); dot((5,-5)); [/asy]\n$\\textbf{(A)}\\ \\frac{1}{4}\\qquad\\textbf{(B)}\\ \\frac{2}{7}\\qquad\\textbf{(C)}\\ \\frac{4}{11}\\qquad\\textbf{(D)}\\ \\frac{1}{2}\\qquad\\textbf{(E)}\\ \\frac{4}{7}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The two points are one unit apart at $8$ places around the edge of the square. There are $8 \\choose 2$$= 28$ ways to choose two points. The probability is\n\\[\\frac{8}{28} = \\boxed{\\textbf{(B)}\\ \\frac27}\\]", "Arbitrarily pick a point in the grid. Clearly, we see two options for the other point to be placed, so the answer is $\\boxed{\\textbf{(B)}\\ \\frac27}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1986_AJHSME_Problems_15", "eval_func": null }, "input": { "text": "Sale prices at the Ajax Outlet Store are $50\\%$ below original prices. On Saturdays an additional discount of $20\\%$ off the sale price is given. What is the Saturday price of a coat whose original price is $\\textdollar 180$?\n$\\text{(A)}$ $\\textdollar 54$\n$\\text{(B)}$ $\\textdollar 72$\n$\\text{(C)}$ $\\textdollar 90$\n$\\text{(D)}$ $\\textdollar 108$\n$\\text{(D)}$ $\\textdollar 110$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "First we need to do the first discount, which, at Ajax Outlet Store, would be there any day of the week.\n\\begin{align*} 180 \\times 50\\% &= 180 \\times \\frac{1}{2} \\\\ &= 90 \\\\ \\end{align*}\nIf we discount $20\\%$, then $80 \\%$ will be left, so after the second discount, we get\n\\begin{align*} 90 \\times 80\\% &= 90 \\times \\frac{8}{10} \\\\ &= 9 \\times 8 \\\\ &= 72 \\\\ \\end{align*}\n$\\boxed{\\text{B}}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2016_AMC_8_Problems_16", "eval_func": null }, "input": { "text": "Annie and Bonnie are running laps around a $400$-meter oval track. They started together, but Annie has pulled ahead, because she runs $25\\%$ faster than Bonnie. How many laps will Annie have run when she first passes Bonnie?\n$\\textbf{(A) }1\\dfrac{1}{4}\\qquad\\textbf{(B) }3\\dfrac{1}{3}\\qquad\\textbf{(C) }4\\qquad\\textbf{(D) }5\\qquad \\textbf{(E) }25$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Each lap Bonnie runs, Annie runs another quarter lap, so Bonnie will run four laps before she is overtaken. This means that Annie and Bonnie are equal so that Annie needs to run another lap to overtake Bonnie. That means Annie will have run $\\boxed{\\textbf{(D)}\\ 5 }$ laps.", "Call $x$ the distance Annie runs. If Annie is $25\\%$ faster than Bonnie, then Bonnie will run a distance of $\\frac{4}{5}x$. For Annie to meet Bonnie, she must run an extra $400$ meters, the length of the track. So $x-\\left(\\frac{4}{5}\\right)x=400 \\implies x=2000$, which is $\\boxed{\\textbf{(D)}\\ 5 }$ laps." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2023_AMC_10B_Problems_4", "eval_func": null }, "input": { "text": "Jackson's paintbrush makes a narrow strip with a width of $6.5$ millimeters. Jackson has enough paint to make a strip $25$ meters long. How many square centimeters of paper could Jackson cover with paint? \n$\\textbf{(A) } 162,500 \\qquad\\textbf{(B) } 162.5 \\qquad\\textbf{(C) }1,625 \\qquad\\textbf{(D) }1,625,000 \\qquad\\textbf{(E) } 16,250$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "$6.5$ millimeters is equal to $0.65$ centimeters. $25$ meters is $2500$ centimeters. The answer is $0.65 \\times 2500$, so the answer is $\\boxed{\\textbf{(C) 1,625}}$. ", "$6.5$ millimeters can be represented as $65 \\times 10^{-2}$ centimeters. $25$ meters is $25 \\times 10^{2}$ centimeters. Multiplying out these results in $(65 \\times 10^{-2}) \\times (25 \\times 10^{2})$, which is $65 \\times 25$ making the answer $\\boxed{\\textbf{(C) 1,625}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2018_AMC_10A_Problems_16", "eval_func": null }, "input": { "text": "Right triangle $ABC$ has leg lengths $AB=20$ and $BC=21$. Including $\\overline{AB}$ and $\\overline{BC}$, how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\\overline{AC}$?\n$\\textbf{(A) }5 \\qquad \\textbf{(B) }8 \\qquad \\textbf{(C) }12 \\qquad \\textbf{(D) }13 \\qquad \\textbf{(E) }15 \\qquad$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "[asy] unitsize(4); pair A, B, C, E, P; A=(-20, 0); B=origin; C=(0,21); E=(-21, 20); P=extension(B,E, A, C); draw(A--B--C--cycle); draw(B--P); dot(\"$A$\", A, SW); dot(\"$B$\", B, SE); dot(\"$C$\", C, NE); dot(\"$P$\", P, NW); [/asy]\nAs the problem has no diagram, we draw a diagram. The hypotenuse has length $29$. Let $P$ be the foot of the altitude from $B$ to $AC$. Note that $BP$ is the shortest possible length of any segment. Writing the area of the triangle in two ways, we can solve for $BP=\\dfrac{20\\cdot 21}{29}$, which is between $14$ and $15$. \nLet the line segment be $BX$, with $X$ on $AC$. As you move $X$ along the hypotenuse from $A$ to $P$, the length of $BX$ strictly decreases, hitting all the integer values from $20, 19, \\dots 15$ (IVT). Similarly, moving $X$ from $P$ to $C$ hits all the integer values from $15, 16, \\dots, 21$. This is a total of $\\boxed{(D) 13}$ distinct line segments.", "Note that if a circle with an integer radius $r$ centered at vertex $B$ intersects hypotenuse $\\overline{AB}$, the lines drawn from $B$ to the points of intersection are integer lengths. As in the previous solution, the shortest distance $14<\\overline{BP}<15$. As a result, a circle of $14$ will reach the hypotenuse and thus does not intersect it. We also know that a circle of radius $21$ intersects the hypotenuse once and a circle of radius $\\{15, 16, 17, 18, 19, 20 \\}$ intersects the hypotenuse twice. Quick graphical thinking or Euclidean construction will prove this.\n[asy] unitsize(4); pair A, B, C, E, P; A=(-20, 0); B=origin; C=(0,21); E=(-21, 20); P=extension(B,E, A, C); draw(A--B--C--cycle); draw(B--P); dot(\"$A$\", A, SW); dot(\"$B$\", B, SE); dot(\"$C$\", C, NE); dot(\"$P$\", P, S); draw(arc((0,0),21, 90, 180)); draw(arc((0,0),20, 90, 180)); draw(arc((0,0),19, 90, 180)); draw(arc((0,0),18, 90, 180)); draw(arc((0,0),17, 90, 180)); draw(arc((0,0),16, 90, 180)); draw(arc((0,0),15, 90, 180)); [/asy]\nIt follows that we can draw circles of radii $15, 16, 17, 18, 19,$ and $20,$ that each contribute integer lengths (since these circles intersect the hypotenuse twice) from $B$ to $\\overline{AC}$ and one circle of radius $21$ that contributes only one such segment. Our answer is then \\[6 \\cdot 2 + 1 = 13 \\implies \\boxed{D}\\] " ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2003_AMC_10A_Problems_2", "eval_func": null }, "input": { "text": "Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League? \n$\\mathrm{(A) \\ } 77\\qquad \\mathrm{(B) \\ } 91\\qquad \\mathrm{(C) \\ } 143\\qquad \\mathrm{(D) \\ } 182\\qquad \\mathrm{(E) \\ } 286$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Since T-shirts cost $5$ dollars more than a pair of socks, T-shirts cost $5+4=9$ dollars. \nSince each member needs $2$ pairs of socks and $2$ T-shirts, the total cost for $1$ member is $2(4+9)=26$ dollars. \nSince $2366$ dollars was the cost for the club, and $26$ was the cost per member, the number of members in the League is $2366\\div 26=\\boxed{\\mathrm{(B)}\\ 91}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2002_AMC_10A_Problems_9", "eval_func": null }, "input": { "text": "There are 3 numbers A, B, and C, such that $1001C - 2002A = 4004$, and $1001B + 3003A = 5005$. What is the average of A, B, and C?\n$\\textbf{(A)}\\ 1 \\qquad \\textbf{(B)}\\ 3 \\qquad \\textbf{(C)}\\ 6 \\qquad \\textbf{(D)}\\ 9 \\qquad \\textbf{(E) }\\text{Not uniquely determined}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Notice that we don't need to find what $A, B,$ and $C$ actually are, just their average. In other words, if we can find $A+B+C$, we will be done. \nAdding up the equations gives $1001(A+B+C)=9009=1001(9)$ so $A+B+C=9$ and the average is $\\frac{9}{3}=3$. Our answer is $\\boxed{\\textbf{(B) }3}$.", "As there are only 2 equations and 3 variables, just set one of the variables to something convenient, like 0. Setting C as 0, we get A is -2, and B is 11 by substitution. By basic arithmetic the average is 3=>B", "Start by isolating $B$ and $C$ in both of the equations, in order to represent the variables $C$ and $B$ in terms of A. Ending up with the two equations $C = 2A +4$ and $B = -3A + 5$, we have to calculate the value of the expression $\\frac{A+B+C}{3}$. Plugging in $2A + 4$ for $C$ and $-3A + 5$ for $B$, we add them up and end up with a value of 9. Dividing 9 by 3 to compute the average, we get our answer of $\\boxed{\\textbf{(B) }3}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2007_AMC_10A_Problems_24", "eval_func": null }, "input": { "text": "centered at $A$ and $B$ each have radius $2$, as shown. Point $O$ is the of $\\overline{AB}$, and $OA = 2\\sqrt {2}$. Segments $OC$ and $OD$ are tangent to the circles centered at $A$ and $B$, respectively, and $EF$ is a common . What is the area of the shaded region $ECODF$?\n[asy] size(5cm); pair A=(-2*sqrt(2),0), B = (2*sqrt(2),0), C = A+2*dir(45), D = B+2*dir(135), E = A+2*dir(90), F = B+2*dir(90); fill((0,0)--C--E--F--D--cycle,gray(0.6)); unfill(circle(A,2)); unfill(circle(B,2)); draw(circle(A,2)); draw(circle(B,2)); draw(E--F); draw(C--(0,0)--D); draw(A+2*dir(300)--A--B--B+2*dir(240)); dot((0,0)); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); label(\"$A$\",A,dir(180)); label(\"$B$\",B,dir(0)); label(\"$C$\",C,dir(240)); label(\"$D$\",D,dir(300)); label(\"$E$\",E,dir(90)); label(\"$F$\",F,dir(90)); label(\"$O$\",(0,0),dir(270)); label(\"$2$\",A+dir(300),dir(210)); label(\"$2$\",B+dir(240),dir(330)); [/asy]\n$\\text{(A)}\\ \\frac {8\\sqrt {2}}{3} \\qquad \\text{(B)}\\ 8\\sqrt {2} - 4 - \\pi \\qquad \\text{(C)}\\ 4\\sqrt {2} \\qquad \\text{(D)}\\ 4\\sqrt {2} + \\frac {\\pi}{8} \\qquad \\text{(E)}\\ 8\\sqrt {2} - 2 - \\frac {\\pi}{2}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The area we are trying to find is simply $ABFE-(\\overarc{AEC}+\\triangle{ACO}+\\triangle{BDO}+\\overarc{BFD})$.\nObviously, $\\overline{EF}\\parallel\\overline{AB}$. Thus, $ABFE$ is a rectangle, and so its area is $b\\times{h}=2\\times{(AO+OB)}=2\\times{2(2\\sqrt{2})}=8\\sqrt{2}$.\nSince $\\overline{OC}$ is tangent to circle $A$, $\\triangle{ACO}$ is a right triangle. We know $AO=2\\sqrt{2}$ and $AC=2$, so $\\triangle{ACO}$ is an isosceles right triangle, and has $\\overline{CO}$ with length $2$. The area of $\\triangle{ACO}=\\frac{1}{2}bh=2$. By symmetry, $\\triangle{ACO}\\cong\\triangle{BDO}$, and so the area of $\\triangle{BDO}$ is also $2$.\n$\\overarc{AEC}$ (or $\\overarc{BFD}$, for that matter) is $\\frac{1}{8}$ the area of its circle since $\\angle{OAC}$ is 45 degrees and $\\angle{OAE}$ forms a right . Thus $\\overarc{AEC}$ and $\\overarc{BFD}$ both have an area of $\\frac{\\pi}{2}$.\nPlugging all of these areas back into the original equation yields $8\\sqrt{2}-(\\frac{\\pi}{2}+2+2+\\frac{\\pi}{2})=8\\sqrt{2}-(4+\\pi)=\\boxed{8\\sqrt{2}-4-\\pi}\\ \\mathrm{(B)}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2017_AMC_10A_Problems_21", "eval_func": null }, "input": { "text": "A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed in another right triangle with sides of length $3$, $4$, and $5$ so that one side of the square lies on the hypotenuse of the triangle. What is $\\dfrac{x}{y}$?\n$\\textbf{(A) } \\dfrac{12}{13} \\qquad \\textbf{(B) } \\dfrac{35}{37} \\qquad \\textbf{(C) } 1 \\qquad \\textbf{(D) } \\dfrac{37}{35} \\qquad \\textbf{(E) } \\dfrac{13}{12}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Analyze the first right triangle.\n[asy] pair A,B,C; pair D, e, F; A = (0,0); B = (4,0); C = (0,3); D = (0, 12/7); e = (12/7 , 12/7); F = (12/7, 0); draw(A--B--C--cycle); draw(D--e--F); label(\"$x$\", D/2, W); label(\"$A$\", A, SW); label(\"$B$\", B, SE); label(\"$C$\", C, N); label(\"$D$\", D, W); label(\"$E$\", e, NE); label(\"$F$\", F, S); [/asy]\nNote that $\\triangle ABC$ and $\\triangle FBE$ are similar, so $\\frac{BF}{FE} = \\frac{AB}{AC}$. This can be written as $\\frac{4-x}{x}=\\frac{4}{3}$. Solving, $x = \\frac{12}{7}$.\nNow we analyze the second triangle.\n\n[asy] pair A,B,C; pair q, R, S, T; A = (0,0); B = (4,0); C = (0,3); q = (1.297, 0); R = (2.27 , 1.297); S = (0.973, 2.27); T = (0, 0.973); draw(A--B--C--cycle); draw(q--R--S--T--cycle); label(\"$y$\", (q+R)/2, NW); label(\"$A'$\", A, SW); label(\"$B'$\", B, SE); label(\"$C'$\", C, N); label(\"$Q$\", (q-(0,0.3))); label(\"$R$\", R, NE); label(\"$S$\", S, NE); label(\"$T$\", T, W); [/asy]\nSimilarly, $\\triangle A'B'C'$ and $\\triangle RB'Q$ are similar, so $RB' = \\frac{4}{3}y$, and $C'S = \\frac{3}{4}y$. Thus, $C'B' = C'S + SR + RB' = \\frac{4}{3}y + y + \\frac{3}{4}y = 5$. Solving for $y$, we get $y = \\frac{60}{37}$. Thus, $\\frac{x}{y} = \\boxed{\\textbf{(D)}\\:\\frac{37}{35}}$.", "Set the right-angle vertex of the triangle as $(0,0)$. Notice that the hypotenuse of the triangle, as depicted in solution one, can be described by $y = 3 - \\frac{3}{4}x$, while $AE$ can be describe by $y=x$. Hence, we may solve for $x$ by solving $3- \\frac{3}{4}x = x$, which yields $\\frac{12}{7}$.\nProceed by finding the value of y via the method described in solution 1, and we will get $y = \\frac{60}{37}$. Thus, $\\frac{x}{y} = \\boxed{\\textbf{(D)}\\:\\frac{37}{35}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2001_AMC_10_Problems_20", "eval_func": null }, "input": { "text": "A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length $2000$. What is the length of each side of the octagon?\n$\\textbf{(A) } \\frac{1}{3}(2000) \\qquad \\textbf{(B) } {2000(\\sqrt{2}-1)} \\qquad \\textbf{(C) } {2000(2-\\sqrt{2})} \\qquad \\textbf{(D) } {1000} \\qquad \\textbf{(E) } {1000\\sqrt{2}}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "First, realize that each triangle is congruent, a right triangle and that the two legs are equal. Also, each side of the octagon is equal, because of the definition of regular shapes. Let $s$ be the length of a leg of the isosceles right triangle. In terms of $s$, the hypotenuse of the isosceles right triangle, which is also the length of a side of the regular octagon, is $s \\sqrt{2}$. Since the length of each side of the square is 2000, the length of each side of the regular octagon is equal to the length of a side of the square ($2000$) subtracted by $2$ times the length of a leg of the isosceles right triangle ( the total length of the side is $2s+ o$, $o$ being the length of a side of the regular octagon), which is the same as $2s$. As an expression, this is $2000-2s$, which we can equate to $s \\sqrt{2}$, ( since the octagon is regular, meaning all of the side's lengths are congruent) giving us the following equation:$2000-2s = s \\sqrt{2}$. By isolating the variable and simplifying the right side, we get the following: $2000 = s(2 + \\sqrt{2})$. Dividing both sides by $(2 + \\sqrt{2})$, we arrive with $\\frac{2000}{2 + \\sqrt{2}} = s$, now, to find the length of the side of the octagon, we can plug in $s$ and use the equation $2000-2s = o$, $o$ being the length of a side of the octagon, to derive the value of a side of the octagon. After plugging in the values, we derive $2000-2(\\frac{2000}{2 + \\sqrt{2}})$, which is the same as $2000-(\\frac{4000}{2 + \\sqrt{2}})$, factoring out a $2000$, we derive the following: $2000(1-(\\frac{2}{2 + \\sqrt{2}}))$, by rationalizing the denominator of $\\frac{2}{2 + \\sqrt{2}}$, we get $2000(1-(2 - \\sqrt{2}))$, after expanding, finally, we get $\\boxed{\\textbf{(B) }2000(\\sqrt{2} -1)}$\u00a0!(not a factorial symbol, just an exclamation point)" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2005_AMC_10B_Problems_16", "eval_func": null }, "input": { "text": "The quadratic equation $x^2+mx+n$ has roots twice those of $x^2+px+m$, and none of $m,n,$ and $p$ is zero. What is the value of $n/p$?\n$\\textbf{(A) }\\ {{{1}}} \\qquad \\textbf{(B) }\\ {{{2}}} \\qquad \\textbf{(C) }\\ {{{4}}} \\qquad \\textbf{(D) }\\ {{{8}}} \\qquad \\textbf{(E) }\\ {{{16}}}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $x^2 + px + m = 0$ have roots $a$ and $b$. Then \n\\[x^2 + px + m = (x-a)(x-b) = x^2 - (a+b)x + ab,\\]\nso $p = -(a+b)$ and $m = ab$. Also, $x^2 + mx + n = 0$ has roots $2a$ and $2b$, so\n\\[x^2 + mx + n = (x-2a)(x-2b) = x^2 - 2(a+b)x + 4ab,\\]\nand $m = -2(a+b)$ and $n = 4ab$. Thus $\\frac{n}{p} = \\frac{4ab}{-(a+b)} = \\frac{4m}{\\frac{m}{2}} = \\boxed{\\textbf{(D) }8}$. \nIndeed, consider the quadratics $x^2 + 8x + 16 = 0,\\ x^2 + 16x + 64 = 0$.", "If the roots of $x^2 + mx + n = 0$ are $2a$ and $2b$ and the roots of $x^2 + px + m = 0$ are $a$ and $b$, then using Vieta's formulas,\n\\[2a + 2b = -m\\]\n\\[a + b = -p\\]\n\\[2a(2b) = n\\]\n\\[a(b) = m\\]\nTherefore, substituting the second equation into the first equation gives \n\\[m = 2(p)\\]\nand substituting the fourth equation into the third equation gives\n\\[n = 4(m)\\]\nTherefore, $n = 8p$, so $\\frac{n}{p}= \\boxed{\\textbf{(D) }8}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2011_AMC_10A_Problems_24", "eval_func": null }, "input": { "text": "Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra?\n$\\text{(A)}\\,\\frac{1}{12} \\qquad\\text{(B)}\\,\\frac{\\sqrt2}{12} \\qquad\\text{(C)}\\,\\frac{\\sqrt3}{12} \\qquad\\text{(D)}\\,\\frac{1}{6} \\qquad\\text{(E)}\\,\\frac{\\sqrt2}{6}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "A regular unit tetrahedron can be split into eight tetrahedra that have lengths of $\\frac{1}{2}$. \nThe volume of a regular tetrahedron can be found using base area and height:\nFor a tetrahedron of side length 1, its base area is $\\frac{\\sqrt{3}}{4}$, and its height can be found using Pythagoras' Theorem. Its height is $\\sqrt{1^2-\\left(\\frac{\\sqrt3}{3}\\right)^2}=\\frac{\\sqrt2}{\\sqrt3}$. Its volume is $\\frac13\\times\\frac{\\sqrt{3}}{4}\\times\\frac{\\sqrt2}{\\sqrt3}=\\frac{\\sqrt{2}}{12}$.\nThe tetrahedron actually has side length $\\sqrt2$, so the actual volume is $\\frac{\\sqrt{2}}{12}\\times\\sqrt2^3=\\frac13$.\nOn the eight small tetrahedra, the four tetrahedra on the corners of the large tetrahedra are not inside the other large tetrahedra. Thus, $\\frac{4}{8}=\\frac{1}{2}$ of the large tetrahedra will not be inside the other large tetrahedra.\nThe intersection of the two tetrahedra is thus $\\frac12\\times\\frac13=\\frac{1}{6}=\\boxed{\\text{(D)}}$.", "The intersection of the two tetrahedra is an octahedron that has points touching the center of each face of the original cube. We can split the octahedron into two square pyramids, and from there we can find the volume of each pyramid. The sides of the square face of the pyramid will have lengths of $\\frac{\\sqrt2}{2}$, so the volume of both pyramids (the whole octahedra) will be $\\left (\\frac{\\sqrt2}{2}\\right )^2\\times2\\times\\frac{1}{2}\\times\\frac{1}{3}=1/6=\\boxed{\\text{(D)}}$.", "Notice that the intersection of the tetrahedra is one of the larger tetrahedra with the four smaller tetrahedra on its points cut off. Using the tetrahedron volume formula ($\\frac{\\text{x}^3}{6\\sqrt2}$, where $\\text{x}$ is the length of a side), and using $\\sqrt2$ and $\\frac{\\sqrt2}{2}$ as the side lengths, you get the volume of the intersection as $\\frac{1}{3}-4\\times\\frac{1}{24}=\\frac{1}{3}-\\frac{1}{6}=\\frac{1}{6}=\\boxed{\\text{(D)}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2016_AMC_10A_Problems_16", "eval_func": null }, "input": { "text": "A triangle with vertices $A(0, 2)$, $B(-3, 2)$, and $C(-3, 0)$ is reflected about the $x$-axis, then the image $\\triangle A'B'C'$ is rotated counterclockwise about the origin by $90^{\\circ}$ to produce $\\triangle A''B''C''$. Which of the following transformations will return $\\triangle A''B''C''$ to $\\triangle ABC$?\n$\\textbf{(A)}$ counterclockwise rotation about the origin by $90^{\\circ}$. \n$\\textbf{(B)}$ clockwise rotation about the origin by $90^{\\circ}$. \n$\\textbf{(C)}$ reflection about the $x$-axis \n$\\textbf{(D)}$ reflection about the line $y = x$\n$\\textbf{(E)}$ reflection about the $y$-axis.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Consider a point $(x, y)$. Reflecting it about the $x$-axis will map it to $(x, -y)$, and rotating it counterclockwise about the origin by $90^{\\circ}$ will map it to $(y, x)$. The operation that undoes this is a reflection about the $y = x$, so the answer is $\\boxed{\\textbf{(D)}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2013_AMC_10A_Problems_10", "eval_func": null }, "input": { "text": "A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?\n\n$\\textbf{(A)}\\ 15 \\qquad\\textbf{(B)}\\ 30 \\qquad\\textbf{(C)}\\ 40 \\qquad\\textbf{(D)}\\ 60 \\qquad\\textbf{(E)}\\ 70$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let the total amount of flowers be $x$. Thus, the number of pink flowers is $0.6x$, and the number of red flowers is $0.4x$. The number of pink carnations is $\\frac{2}{3}(0.6x) = 0.4x$ and the number of red carnations is $\\frac{3}{4}(0.4x) = 0.3x$. Summing these, the total number of carnations is $0.4x+0.3x=0.7x$. Dividing, we see that $\\frac{0.7x}{x} = 0.7 = \\boxed{\\textbf{(E) }70\\%}$", "We have $\\dfrac15$ for pink roses,$1-\\dfrac6{10}=\\dfrac4{10}=\\dfrac25$ red flowers, $\\dfrac6{10}-\\dfrac15=\\dfrac35-\\dfrac15=\\dfrac25$ pink carnations, $\\dfrac25\\cdot \\dfrac34=\\dfrac6{20}=\\dfrac3{10}$ red carnations we add them up to get $\\dfrac25+\\dfrac3{10}=\\dfrac7{10}=70\\%$ so our final answer is or $\\boxed{(E)70\\%}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2014_AMC_10B_Problems_2", "eval_func": null }, "input": { "text": "What is $\\frac{2^3 + 2^3}{2^{-3} + 2^{-3}}$?\n$\\textbf {(A) } 16 \\qquad \\textbf {(B) } 24 \\qquad \\textbf {(C) } 32 \\qquad \\textbf {(D) } 48 \\qquad \\textbf {(E) } 64$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We can synchronously multiply ${2^3}$ to the expresions both above and below the fraction bar. Thus, \n\\[\\frac{2^3+2^3}{2^{-3}+2^{-3}}\\\\=\\frac{2^6+2^6}{1+1}\\\\={2^6}.\\]\nHence, the fraction equals to $\\boxed{{\\textbf{(E) }64}}$.", "We have \n\\[\\frac{2^3+2^3}{2^{-3}+2^{-3}} = \\frac{8 + 8}{\\frac{1}{8} + \\frac{1}{8}} = \\frac{16}{\\frac{1}{4}} = 16 \\cdot 4 = 64,\\] so our answer is $\\boxed{{(\\textbf{E) }64}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2007_AMC_10B_Problems_20", "eval_func": null }, "input": { "text": "A set of $25$ square blocks is arranged into a $5 \\times 5$ square. How many different combinations of $3$ blocks can be selected from that set so that no two are in the same row or column?\n$\\textbf{(A) } 100 \\qquad\\textbf{(B) } 125 \\qquad\\textbf{(C) } 600 \\qquad\\textbf{(D) } 2300 \\qquad\\textbf{(E) } 3600$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "There are $25$ ways to choose the first square. The four remaining squares in its row and column and the square you chose exclude nine squares from being chosen next time.\nThere are $16$ remaining blocks to be chosen for the second square. The three remaining spaces in its row and column and the square you chose must be excluded from being chosen next time.\nFinally, the last square has $9$ remaining choices.\nThe number of ways to choose $3$ squares is $25 \\cdot 16 \\cdot 9,$ but the order in which you chose the squares does not matter as the blocks are indistinguishable, so we divide by $3!$.\n\\[\\frac{25 \\cdot 16 \\cdot 9}{3 \\cdot 2 \\cdot 1} = 25 \\cdot 8 \\cdot 3 = 100 \\cdot 6 = \\boxed{\\mathrm{(C) \\ } 600}\\]", "Once we choose our three squares, we will have occupied three separate columns $(A, B, C)$ and three separate rows. There are ${5 \\choose 3} \\times {5 \\choose 3}$ ways to choose these rows and columns. \nThere are $3$ ways to assign the square in column $A$ to a row, $2$ ways to assign the square in column $B$ to one of the remaining two rows, and poor square in column $C$ doesn't get to choose. $:($\nIn total, we have \\[{5 \\choose 3} \\times {5 \\choose 3} \\times 3!\\] which totals out to $\\boxed{\\mathrm{(C) \\ } 600}$.", "We know that there are $\\binom{25}{3}=2300$ ways to choose three blocks. However, the restriction clearly limits the number of ways we can choose our blocks. Hence, only $\\text{(A)}$, $\\text{(B)}$, or $\\text{(C)}$ could be the correct answer. Clearly, there are more than $125$ ways, thus yielding $\\boxed{\\text{(C)} 600}$ ways." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2006_AMC_10B_Problems_3", "eval_func": null }, "input": { "text": "A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of $34$ points, and the Cougars won by a margin of $14$ points. How many points did the Panthers score? \n$\\textbf{(A) } 10\\qquad \\textbf{(B) } 14\\qquad \\textbf{(C) } 17\\qquad \\textbf{(D) } 20\\qquad \\textbf{(E) } 24$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $x$ be the number of points scored by the Cougars, and $y$ be the number of points scored by the Panthers. The problem is asking for the value of $y$. \n\\begin{align*} x+y &= 34 \\\\ x-y &= 14 \\\\ 2x &= 48 \\\\ x &= 24 \\\\ y &= \\boxed{\\textbf{(A) }10} \\\\ \\end{align*}" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2014_AMC_10B_Problems_3", "eval_func": null }, "input": { "text": "Randy drove the first third of his trip on a gravel road, the next $20$ miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip?\n$\\textbf {(A) } 30 \\qquad \\textbf {(B) } \\frac{400}{11} \\qquad \\textbf {(C) } \\frac{75}{2} \\qquad \\textbf {(D) } 40 \\qquad \\textbf {(E) } \\frac{300}{7}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let the total distance be $x$. We have $\\dfrac{x}{3} + 20 + \\dfrac{x}{5} = x$, or $\\dfrac{8x}{15} + 20 = x$. Subtracting $\\dfrac{8x}{15}$ from both sides gives us $20 = \\dfrac{7x}{15}$. Multiplying by $\\dfrac{15}{7}$ gives us $x = \\boxed{{\\textbf{(E) }\\dfrac{300}{7}}}$", "The first third of his distance added to the last one-fifth of his distance equals $\\frac{8}{15}$. Therefore, $\\frac{7}{15}$ of his distance is $20$. Let $x$ be his total distance, and solve for $x$. Therefore, $x$ is equal to $\\frac{300}{7}$, or $E$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2012_AMC_10A_Problems_21", "eval_func": null }, "input": { "text": "Let points $A$ = $(0 ,0 ,0)$, $B$ = $(1, 0, 0)$, $C$ = $(0, 2, 0)$, and $D$ = $(0, 0, 3)$. Points $E$, $F$, $G$, and $H$ are midpoints of line segments $\\overline{BD},\\text{ } \\overline{AB}, \\text{ } \\overline {AC},$ and $\\overline{DC}$ respectively. What is the area of $EFGH$?\n$\\textbf{(A)}\\ \\sqrt{2}\\qquad\\textbf{(B)}\\ \\frac{2\\sqrt{5}}{3}\\qquad\\textbf{(C)}\\ \\frac{3\\sqrt{5}}{4}\\qquad\\textbf{(D)}\\ \\sqrt{3}\\qquad\\textbf{(E)}\\ \\frac{2\\sqrt{7}}{3}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Consider a tetrahedron with vertices at $A,B,C,D$ on the $xyz$-plane. The length of $EF$ is just one-half of $AD$ because it is the midsegment of $\\triangle ABD.$ The same concept applies to the other side lengths. $AD=3$ and $BC=\\sqrt{1^2+2^2}=\\sqrt{5}$. Then $EF=HG=\\frac32$ and $EH=FG=\\frac{\\sqrt{5}}{2}$. The line segments lie on perpendicular planes so quadrilateral $EFGH$ is a rectangle. The area is\n\\[EF \\cdot FG = \\frac 32 \\cdot \\frac{\\sqrt{5}}{2} = \\frac{3\\sqrt 5} 4\\implies \\boxed{\\textbf C}.\\]\n[asy] import three; draw((0,0,0)--(1,0,0)--(0,0,3)--cycle); draw((0,0,0)--(0,2,0)); draw((0,2,0)--(0,0,3)); //EFGH draw((0.5,0,1.5)--(0.5,0,0)--(0,1,0)--(0,1,1.5)--(0.5,0,1.5),red); //Points label(\"$E$\",(0.5,0,1.5),NW); label(\"$F$\",(0.5,0,0),S); label(\"$G$\",(0,1,0),S); label(\"$H$\",(0,1,1.5),NE); label(\"$A$\",(0,0,0),NE); label(\"$B$\",(1,0,0),S); label(\"$C$\",(0,2,0),S); label(\"$D$\",(0,0,3),N); [/asy]", "Computing the points of $EFGH$ gives $E(0.5, 0, 1.5), F(0.5, 0, 0), G(0,1,0), H(0,1,1.5)$. The vector $EF$ is $(0,0,-1.5)$, while the vector $HG$ is also $(0,0,-1.5)$, meaning the two sides $EF$ and $GH$ are parallel. Similarly, the vector $FG$ is $(-0.5, 1, 0)$, while the vector $EH$ is also $(-0.5, 1, 0)$. Again, these are equal in both magnitude and direction, so $FG$ and $EH$ are parallel. Thus, figure $EFGH$ is a parallelogram.\nComputation of vectors $EF$ and $HG$ is sufficient evidence that the figure is a parallelogram, since the vectors are not only point in the same direction, but are of the same magnitude, but the other vector $FG$ is needed to find the angle between the sides.\nTaking the dot product of vector $EF$ and vector $FG$ gives $0 \\cdot -0.5 + 0 \\cdot 1 + -1.5 \\cdot 0 = 0$, which means the two vectors are perpendicular. (Alternately, as above, note that vector $EF$ goes directly down on the z-axis, while vector $FG$ has no z-component and lie completely in the xy plane.) Thus, the figure is a parallelogram with a right angle, which makes it a rectangle. With the distance formula in three dimensions, we find that $EF = \\frac{3}{2}$ and $FG = \\frac{\\sqrt{5}}{2}$, giving an area of $\\frac32 \\cdot \\frac{\\sqrt{5}}{2} = \\boxed{\\textbf{(C)}\\ \\frac{3\\sqrt{5}}{4}}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2008_AMC_10A_Problems_9", "eval_func": null }, "input": { "text": "Suppose that\n\\[\\frac{2x}{3}-\\frac{x}{6}\\]\nis an integer. Which of the following statements must be true about $x$?\n$\\mathrm{(A)}\\ \\text{It is negative.}\\\\\\mathrm{(B)}\\ \\text{It is even, but not necessarily a multiple of 3.}\\\\\\mathrm{(C)}\\ \\text{It is a multiple of 3, but not necessarily even.}\\\\\\mathrm{(D)}\\ \\text{It is a multiple of 6, but not necessarily a multiple of 12.}\\\\\\mathrm{(E)}\\ \\text{It is a multiple of 12.}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "\\[\\frac{2x}{3}-\\frac{x}{6}\\quad\\Longrightarrow\\quad\\frac{4x}{6}-\\frac{x}{6}\\quad\\Longrightarrow\\quad\\frac{3x}{6}\\quad\\Longrightarrow\\quad\\frac{x}{2}\\]\nFor $\\frac{x}{2}$ to be an integer, $x$ must be even, but not necessarily divisible by $3$. Thus, the answer is $\\mathrm{(B)}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2008_AMC_10B_Problems_16", "eval_func": null }, "input": { "text": "Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, the sum is 0.)\n$\\mathrm{(A)}\\ {{{\\frac{3} {8}}}} \\qquad \\mathrm{(B)}\\ {{{\\frac{1} {2}}}} \\qquad \\mathrm{(C)}\\ {{{\\frac{43} {72}}}} \\qquad \\mathrm{(D)}\\ {{{\\frac{5} {8}}}} \\qquad \\mathrm{(E)}\\ {{{\\frac{2} {3}}}}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We consider 3 cases based on the outcome of the coin:\nCase 1, 0 heads:\nThe probability of this occurring on the coin flip is $\\frac{1} {4}$. The probability that 0 rolls of a die will result in an odd sum is $0$. \nCase 2, 1 head:\nThe probability of this case occurring is $\\frac{1} {2} \\cdot \\frac {1} {2} \\cdot 2 = \\frac {1} {2}.$ The probability that 1 die results in an odd number is $\\frac{1} {2}$.\nCase 3, 2 heads:\nThe probability of this occurring is $\\frac{1} {4}$. The probability that 2 dice result in an odd sum is $\\frac{1} {2}$, because regardless of what we throw on the first die, we have $\\frac{1} {2}$ probability that the second die will have the opposite parity.\nThus, the probability of having an odd sum rolled is $\\frac{1} {4} \\cdot 0 + \\frac{1} {2} \\cdot \\frac{1} {2} + \\frac{1} {4} \\cdot \\frac{1} {2}=\\frac{3} {8}\\Rightarrow \\boxed{A}$", "We use complementary counting or subtracting $P(\\text{Even})$ from $1$. We use casework now.\nCase $1$: $2$ Tails. $2$ tails occur with probability $\\frac{1}{4}$, but we will always get an even number, so the overall probability to get an even sum is $\\frac{1}{4}$.\nCase $2$: $1$ Tail: This event occurs with probability $\\frac{1}{2}$ and the probability we get an even is $\\frac{1}{2}$, so the overall probability to get an even, in this case, is also $\\frac{1}{4}$.\nWe know $P\\text{(Even)}$ is greater than $\\frac{1}{2}$, so $P\\text{(Odd)}$ is less than $\\frac{1}{2}$.\nOnly $\\boxed{\\text{(A)}\\frac{3}{8}}$ is less than $\\frac{1}{2}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2014_AMC_10B_Problems_18", "eval_func": null }, "input": { "text": "A list of $11$ positive integers has a mean of $10$, a median of $9$, and a unique mode of $8$. What is the largest possible value of an integer in the list?\n$\\textbf {(A) } 24 \\qquad \\textbf {(B) } 30 \\qquad \\textbf {(C) } 31\\qquad \\textbf {(D) } 33 \\qquad \\textbf {(E) } 35$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We start off with the fact that the median is $9$, so we must have $a, b, c, d, e, 9, f, g, h, i, j$, listed in ascending order. Note that the integers do not have to be distinct. \nSince the mode is $8$, we have to have at least $2$ occurrences of $8$ in the list. If there are $2$ occurrences of $8$ in the list, we will have $a, b, c, 8, 8, 9, f, g, h, i, j$. In this case, since $8$ is the unique mode, the rest of the integers have to be distinct. So we minimize $a,b,c,f,g,h,i$ in order to maximize $j$. If we let the list be $1,2,3,8,8,9,10,11,12,13,j$, then $j = 11 \\times 10 - (1+2+3+8+8+9+10+11+12+13) = 33$. \nNext, consider the case where there are $3$ occurrences of $8$ in the list. Now, we can have two occurrences of another integer in the list. We try $1,1,8,8,8,9,9,10,10,11,j$. Following the same process as above, we get $j = 11 \\times 10 - (1+1+8+8+8+9+9+10+10+11) = 35$. As this is the highest choice in the list, we know this is our answer. Therefore, the answer is $\\boxed{\\textbf{(E) }35}$", "Note that $x_1 + \\ldots + x_{11} = 110$ let $x_6 = 9$ so $x_1 + \\ldots + x_5 + x_7 + \\ldots + x_{11} = 101$. To maximize the value of $x_i$ where $i$ ranges from $1$ to $11$, we let any $7$ elements be $1,2,\\ldots,7$ so $x_1 + x_2 + x_3 = 57$. Now we have to let one of above $3$ values = $8$ hence $x_1 + x_2 = 49$ now let $x_1 = 35$, $x_2 = 14$ hence $\\boxed{\\textbf{(E) }35}$ is the answer." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2015_AMC_10B_Problems_3", "eval_func": null }, "input": { "text": "Isaac has written down one integer two times and another integer three times. The sum of the five numbers is $100$, and one of the numbers is $28.$ What is the other number?\n$\\textbf{(A) }8\\qquad\\textbf{(B) }11\\qquad\\textbf{(C) }14\\qquad\\textbf{(D) }15\\qquad\\textbf{(E) }18$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let the first number be $x$ and the second be $y$. We have $2x+3y=100$. We are given one of the numbers is $28$. If $x$ were to be $28$, $y$ would not be an integer, thus $y=28$. $2x+3(28)=100$, which gives $x=\\boxed{\\textbf{(A) }8}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2008_AMC_10A_Problems_13", "eval_func": null }, "input": { "text": "Doug can paint a room in $5$ hours. Dave can paint the same room in $7$ hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let $t$ be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by $t$?\n$\\mathrm{(A)}\\ \\left(\\frac{1}{5}+\\frac{1}{7}\\right)\\left(t+1\\right)=1\\qquad\\mathrm{(B)}\\ \\left(\\frac{1}{5}+\\frac{1}{7}\\right)t+1=1\\qquad\\mathrm{(C)}\\ \\left(\\frac{1}{5}+\\frac{1}{7}\\right)t=1\\\\\\mathrm{(D)}\\ \\left(\\frac{1}{5}+\\frac{1}{7}\\right)\\left(t-1\\right)=1\\qquad\\mathrm{(E)}\\ \\left(5+7\\right)t=1$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Doug can paint $\\frac{1}{5}$ of a room per hour, Dave can paint $\\frac{1}{7}$ of a room per hour, and the time they spend working together is $t-1$.\nSince rate multiplied by time gives output, $\\left(\\frac{1}{5}+\\frac{1}{7}\\right)\\left(t-1\\right)=1 \\Rightarrow \\mathrm{(D)}$", "If one person does a job in $a$ hours and another person does a job in $b$ hours, the time it takes to do the job together is $\\frac{ab}{a+b}$ hours.\nSince Doug paints a room in 5 hours and Dave paints a room in 7 hours, they both paint in $\\frac{5\\times7}{5+7} = \\frac{35}{12}$ hours. They also take 1 hour for lunch, so the total time $t = \\frac{35}{12} + 1$ hours.\nLooking at the answer choices, $(D)$ is the only one satisfied by $t = \\frac{35}{12} + 1$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2009_AMC_10B_Problems_3", "eval_func": null }, "input": { "text": "Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for 25 rooms. How many cans of paint did she use for the 25 rooms?\n$\\mathrm{(A)}\\ 10\\qquad \\mathrm{(B)}\\ 12\\qquad \\mathrm{(C)}\\ 15\\qquad \\mathrm{(D)}\\ 18\\qquad \\mathrm{(E)}\\ 25$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Losing three cans of paint corresponds to being able to paint five fewer rooms. So $\\frac 35 \\cdot 25 = \\boxed{15}$. The answer is $\\mathrm{(C)}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2002_AMC_10A_Problems_21", "eval_func": null }, "input": { "text": "The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is \n$\\text{(A) }11 \\qquad \\text{(B) }12 \\qquad \\text{(C) }13 \\qquad \\text{(D) }14 \\qquad \\text{(E) }15$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "As the unique mode is $8$, there are at least two $8$s.\nAs the range is $8$ and one of the numbers is $8$, the largest one can be at most $16$.\nIf the largest one is $16$, then the smallest one is $8$, and thus the mean is strictly larger than $8$, which is a contradiction.\nIf we have 2 8's we can add find the numbers 4, 6, 7, 8, 8, 9, 10, 12.\nThis is a possible solution but has not reached the maximum.\nIf we have 4 8's we can find the numbers 6, 6, 6, 8, 8, 8, 8, 14.\nWe can also see that they satisfy the need for the mode, median, and range to be 8. This means that the answer will be\n$\\boxed{\\text{(D)}\\ 14 }$. ", "We could express this collection as integers $a\\textsubscript{1}$ through $a\\textsubscript{8}$, with $a\\textsubscript{1}$ being the smallest and $a\\textsubscript{8}$ being the largest.\nSince the mean is $8$, we know that $a\\textsubscript{4}$ and $a\\textsubscript{5}$ must also be $8$. If they were not, the other numbers, which are lesser and greater than $a\\textsubscript{4}$ and $a\\textsubscript{5}$ respectively, would not be able to satisfy the condition that $8$ is the mode.\nThere are $8$ terms and the mean is $8$. This tells us that the sum of all the numbers is $64$.\nWe want to maximize the value of $a\\textsubscript{8}$, so we set $a\\textsubscript{6}$ and $a\\textsubscript{7}$ to $8$ as well.\nKnowing that we want to minimize numbers and that the range is $8$ , we set $a\\textsubscript{1}$, $a\\textsubscript{2}$, and $a\\textsubscript{3}$ equal to $a\\textsubscript{8} - 8$.\n{$a\\textsubscript{1}$, $a\\textsubscript{2}$, $a\\textsubscript{3}$, $a\\textsubscript{4}$, $a\\textsubscript{5}$, $a\\textsubscript{6}$, $a\\textsubscript{7}$, $a\\textsubscript{8}$} $=$ {$a\\textsubscript{8} - 8$, $a\\textsubscript{8} - 8$, $a\\textsubscript{8} - 8$, $8$, $8$, $8$, $8$, $a\\textsubscript{8}$}\nLetting the sum of all the numbers be $64$, we find that $32 + 4a_8 - 24 = 64$, which simplifies to $4a_8 = 56$. Solving, we get $\\boxed{\\text{(D)}\\ 14 }$. " ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2007_AMC_10A_Problems_22", "eval_func": null }, "input": { "text": "A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term 824. Let $S$ be the sum of all the terms in the sequence. What is the largest prime factor that always divides $S$? \n$\\mathrm{(A)}\\ 3\\qquad \\mathrm{(B)}\\ 7\\qquad \\mathrm{(C)}\\ 13\\qquad \\mathrm{(D)}\\ 37\\qquad \\mathrm{(E)}\\ 43$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "A given digit appears as the hundreds digit, the tens digit, and the units digit of a term the same number of times. Let $k$ be the sum of the units digits in all the terms. Then $S=111k=3 \\cdot 37k$, so $S$ must be divisible by $37\\ \\mathrm{(D)}$. To see that it need not be divisible by any larger prime, the sequence $123, 231, 312$ gives $S=666=2 \\cdot 3^2 \\cdot 37\\Rightarrow \\mathrm{\\boxed{(D)}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2014_AMC_10A_Problems_3", "eval_func": null }, "input": { "text": "Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for $\\textdollar 2.50$ each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs $\\textdollar 0.75$ for her to make. In dollars, what is her profit for the day?\n$\\textbf{(A)}\\ 24\\qquad\\textbf{(B)}\\ 36\\qquad\\textbf{(C)}\\ 44\\qquad\\textbf{(D)}\\ 48\\qquad\\textbf{(E)}\\ 52$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "She first sells one-half of her $48$ loaves, or $\\frac{48}{2}=24$ loaves. Each loaf sells for $\\textdollar 2.50$, so her total earnings in the morning is equal to \\[24\\cdot \\textdollar 2.50 = \\textdollar 60\\]\nThis leaves 24 loaves left, and Bridget will sell $\\dfrac{2}{3}\\times 24=16$ of them for a price of $\\textdollar\\frac{2.50}{2}=\\textdollar 1.25$. Thus, her total earnings for the afternoon is \\[16\\cdot \\textdollar 1.25 = \\textdollar 20\\]\nFinally, Bridget will sell the remaining $24-16=8$ loaves for a dollar each. This is a total of $\\textdollar 1\\cdot 8 = \\textdollar 8$\nThe total amount of money she makes is equal to $60+20+8=\\textdollar 88$.\nHowever, since Bridget spends $\\textdollar 0.75$ making each loaf of bread, the total cost to make the bread is equal to $\\textdollar 0.75\\cdot48=\\textdollar 36$.\nHer total profit is the amount of money she spent subtracted from the amount of money she made, which is \\[88-36=52\\implies\\boxed{\\textbf{(E)} \\ 52}\\]" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2003_AMC_10A_Problems_8", "eval_func": null }, "input": { "text": "What is the probability that a randomly drawn positive factor of $60$ is less than $7$?\n$\\mathrm{(A) \\ } \\frac{1}{10}\\qquad \\mathrm{(B) \\ } \\frac{1}{6}\\qquad \\mathrm{(C) \\ } \\frac{1}{4}\\qquad \\mathrm{(D) \\ } \\frac{1}{3}\\qquad \\mathrm{(E) \\ } \\frac{1}{2}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "For a positive number $n$ which is not a perfect square, exactly half of the positive factors will be less than $\\sqrt{n}$. \nSince $60$ is not a perfect square, half of the positive factors of $60$ will be less than $\\sqrt{60}\\approx 7.746$. \nClearly, there are no positive factors of $60$ between $7$ and $\\sqrt{60}$. \nTherefore half of the positive factors will be less than $7$. \nSo the answer is $\\boxed{\\mathrm{(E)}\\ \\frac{1}{2}}$.", "Testing all numbers less than $7$, numbers $1, 2, 3, 4, 5$, and $6$ divide $60$. The prime factorization of $60$ is $2^2\\cdot 3 \\cdot 5$. Using the formula for the number of divisors, the total number of divisors of $60$ is $(3)(2)(2) = 12$. Therefore, our desired probability is $\\frac{6}{12} = \\boxed{\\mathrm{(E)}\\ \\frac{1}{2}}$", "This is not too bad with casework. Notice that $1*60=2*30=3*20=4*15=5*12=6*10=60$. Hence, $60$ has $12$ factors, of which $6$ are less than $7$. Thus, the answer is $\\frac{6}{12} = \\boxed{\\mathrm{(E)}\\ \\frac{1}{2}}$.", "This following solution isn't recommended, but just list out all the divisors of $60$ to find $12$ divisors. $6$ of these are less than $7,$ so the answer is $\\frac{6}{12} = \\boxed{\\mathrm{(E)}\\ \\frac{1}{2}}$. " ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2022_AMC_10A_Problems_9", "eval_func": null }, "input": { "text": "A rectangle is partitioned into $5$ regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?\n[asy] size(5.5cm); draw((0,0)--(0,2)--(2,2)--(2,0)--cycle); draw((2,0)--(8,0)--(8,2)--(2,2)--cycle); draw((8,0)--(12,0)--(12,2)--(8,2)--cycle); draw((0,2)--(6,2)--(6,4)--(0,4)--cycle); draw((6,2)--(12,2)--(12,4)--(6,4)--cycle); [/asy]\n$\\textbf{(A) }120\\qquad\\textbf{(B) }270\\qquad\\textbf{(C) }360\\qquad\\textbf{(D) }540\\qquad\\textbf{(E) }720$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The top left rectangle can be $5$ possible colors. Then the bottom left region can only be $4$ possible colors, and the bottom middle can only be $3$ colors since it is next to the top left and bottom left. Similarly, we have $3$ choices for the top right and $3$ choices for the bottom right, which gives us a total of $5\\cdot4\\cdot3\\cdot3\\cdot3=\\boxed{\\textbf{(D) }540}$.", "Case 1: All the rectangles are different colors. It would be $5! = 120$ choices. \nCase 2: Two rectangles that are the same color. Grouping these two rectangles as one gives us $5\\cdot4\\cdot3\\cdot2 = 120$. But, you need to multiply this number by three because the same-colored rectangles can be chosen at the top left and bottom right, the top right and bottom left, or the bottom right and bottom left, which gives us a grand total of $360$.\nCase 3: We have two sets of rectangles chosen from these choices (top right & bottom left, top left & bottom right) that have the same color. However, the choice of the bottom left and bottom right does not work for this case, as the second pair would be chosen from two touching rectangles. Again, grouping the same-colored rectangles gives us $5\\cdot4\\cdot3 = 60$. \nTherefore, we have $120 + 360 + 60 = \\boxed{\\textbf{(D) }540}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2008_AMC_10A_Problems_6", "eval_func": null }, "input": { "text": "A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of 3 kilometers per hour, bikes at a rate of 20 kilometers per hour, and runs at a rate of 10 kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race?\n$\\mathrm{(A)}\\ 3\\qquad\\mathrm{(B)}\\ 4\\qquad\\mathrm{(C)}\\ 5\\qquad\\mathrm{(D)}\\ 6\\qquad\\mathrm{(E)}\\ 7$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $d$ be the length of one segment of the race.\nAverage speed is total distance divided by total time. The total distance is $3d$, and the total time is $\\frac{d}{3}+\\frac{d}{20}+\\frac{d}{10}=\\frac{29d}{60}$.\nThus, the average speed is $3d\\div\\left(\\frac{29d}{60}\\right)=\\frac{180}{29}$. This is closest to $6$, so the answer is $\\mathrm{(D)}$.", "Since the three segments are all the same length, the triathlete's average speed is the harmonic mean of the three given rates. Therefore, the average speed is \\[\\frac{3}{\\frac{1}{3}+\\frac{1}{20}+\\frac{1}{10}}=\\frac{3}{\\frac{29}{60}}=\\frac{180}{29}\\approx6\\Rightarrow\\boxed{\\mathrm{(D)}\\ 6}\\]." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2020_AMC_10A_Problems_1", "eval_func": null }, "input": { "text": "What value of $x$ satisfies\n\\[x- \\frac{3}{4} = \\frac{5}{12} - \\frac{1}{3}?\\]\n$\\textbf{(A)}\\ {-}\\frac{2}{3}\\qquad\\textbf{(B)}\\ \\frac{7}{36}\\qquad\\textbf{(C)}\\ \\frac{7}{12}\\qquad\\textbf{(D)}\\ \\frac{2}{3}\\qquad\\textbf{(E)}\\ \\frac{5}{6}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Adding $\\frac{3}{4}$ to both sides, $x= \\frac{5}{12} - \\frac{1}{3} + \\frac{3}{4} = \\frac{5}{12} - \\frac{4}{12} + \\frac{9}{12}=\\boxed{\\textbf{(E) }\\frac{5}{6}}$.", "Multiplying $12$ on both sides gets us $12x-9=1 \\Rightarrow 12x=10$, therefore $\\boxed{x=\\textbf{(E)}~\\frac{5}{6}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2008_AMC_10A_Problems_5", "eval_func": null }, "input": { "text": "Which of the following is equal to the product \n\\[\\frac{8}{4}\\cdot\\frac{12}{8}\\cdot\\frac{16}{12}\\cdot\\cdots\\cdot\\frac{4n+4}{4n}\\cdot\\cdots\\cdot\\frac{2008}{2004}?\\]\n$\\textbf{(A)}\\ 251\\qquad\\textbf{(B)}\\ 502\\qquad\\textbf{(C)}\\ 1004\\qquad\\textbf{(D)}\\ 2008\\qquad\\textbf{(E)}\\ 4016$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "$\\frac {8}{4}\\cdot\\frac {12}{8}\\cdot\\frac {16}{12}\\cdots\\frac {4n + 4}{4n}\\cdots\\frac {2008}{2004} = \\frac {1}{4}\\cdot\\left(\\frac {8}{8}\\cdot\\frac {12}{12}\\cdots\\frac {4n}{4n}\\cdots\\frac {2004}{2004}\\right)\\cdot 2008 = \\frac{2008}{4} =$ $502 \\Rightarrow B$.", "Notice that everything cancels out except for $2008$ in the numerator and $4$ in the denominator.\nThus, the product is $\\frac{2008}{4}=502$, and the answer is $\\textbf{(B)}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2001_AMC_10_Problems_19", "eval_func": null }, "input": { "text": "Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible?\n$\\textbf{(A)}\\ 6 \\qquad \\textbf{(B)}\\ 9 \\qquad \\textbf{(C)}\\ 12 \\qquad \\textbf{(D)}\\ 15 \\qquad \\textbf{(E)}\\ 18$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let's use stars and bars.\nLet the donuts be represented by $O$s. We wish to find all possible combinations of glazed, chocolate, and powdered donuts that give us $4$ in all. The four donuts we want can be represented as $OOOO$. Notice that we can add two \"dividers\" to divide the group of donuts into three different kinds; the first will be glazed, second will be chocolate, and the third will be powdered. For example, $O|OO|O$ represents one glazed, two chocolate, and one powdered. We have six objects in all, and we wish to turn two into dividers, which can be done in $\\binom{6}{2}=15$ ways. Our answer is hence $\\boxed{\\textbf{(D)}\\ 15}$. Notice that this can be generalized to get the stars and bars (balls and urns) identity.", "Similar to Solution 1, we can generalize the problem by using the stars and bars formula, yielding an answer of $\\dbinom{4+3-1}{3-1}=\\dbinom{6}{2}=\\boxed{\\textbf{(D)}\\ 15}$.", "Simple casework works here as well:\nSet up the following ratios:\n\\[4:0:0\\]\n\\[3:1:0\\]\n\\[2:2:0\\]\n\\[2:1:1\\]\nIn three of these cases we see that there are two of the same ratios (so like two boxes would have $0$), and so if we swapped those two donuts, we would have the same case. Thus we get $\\frac{3!}{2!1!}$ for those $3$ (You can also set it up and logically symmetry applies). For the other case where each ratio of donuts is different, we get the normal $\\binom{4}{3}=6.$ Thus, our answer is $3 \\cdot 3+6 = \\boxed{\\textbf{(D) }15}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2005_AMC_10B_Problems_19", "eval_func": null }, "input": { "text": "On a certain math exam, $10\\%$ of the students got $70$ points, $25\\%$ got $80$ points, $20\\%$ got $85$ points, $15\\%$ got $90$ points, and the rest got $95$ points. What is the difference between the mean and the median score on this exam?\n$\\textbf{(A) }\\ 0 \\qquad \\textbf{(B) }\\ 1 \\qquad \\textbf{(C) }\\ 2 \\qquad \\textbf{(D) }\\ 4 \\qquad \\textbf{(E) }\\ 5$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "To begin, we see that the remaining $30\\%$ of the students got $95$ points. Assume that there are $20$ students; we see that $2$ students got $70$ points, $5$ students got $80$ points, $4$ students got $85$ points, $3$ students got $90$ points, and $6$ students got $95$ points. The median is $85$, since the $10^{\\text{th}}$ and $11^{\\text{th}}$ terms are both $85$. The mean is $\\dfrac{70\\,(2)+80\\,(5)+85\\,(4)+90\\,(3)+95\\,(6)}{20}=\\dfrac{1720}{20}=86$. The difference between the mean and median, therefore, is $\\boxed{\\textbf{(B)}\\ 1}$.", "The remaining $30\\%$ of the students got $95$ points.\nThe mean is equal to $10\\%\\cdot70 + 25\\%\\cdot80 + 20\\%\\cdot85 + 15\\%\\cdot90 + 30\\%\\cdot95 = 86$.\nThe score greater than or equal to $50\\%$ of other scores is the median. Since $35\\%$ scored $80$ or lower and the next $20\\%$ scored $85$, the median is $85$. The difference between the mean and the median is $\\boxed{\\textbf{(B)}\\ 1}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2015_AMC_10B_Problems_13", "eval_func": null }, "input": { "text": "The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?\n$\\textbf{(A) } 20 \\qquad\\textbf{(B) } \\dfrac{360}{17} \\qquad\\textbf{(C) } \\dfrac{107}{5} \\qquad\\textbf{(D) } \\dfrac{43}{2} \\qquad\\textbf{(E) } \\dfrac{281}{13}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We find the $x$-intercepts and the $y$-intercepts to find the intersections of the axes and the line. If $x=0$, then $y=12$. If $y$ is $0$, then $x=5$. Our three vertices are $(0,0)$, $(5,0)$, and $(0,12)$. Two of our altitudes are $5$ and $12$, and since it is a $5$-$12$-$13$ right triangle, the hypotenuse is $13$. Since the area of the triangle is $30$, so our final altitude is $\\frac{30(2)}{13}=\\frac{60}{13}$. The sum of our altitudes is $\\frac{60+156+65}{13}=\\boxed{\\textbf{(E)} \\dfrac{281}{13}}$. Note that there is no need to calculate the final answer after we know that the third altitude has length $\\frac{60}{13}$ since $E$ is the only choice with a denominator of $13$ and $13$ is relatively prime to $5$ and $12$.", "Noticing that the line has coefficients $12$ and $5$, we can suspect that we have a $5$-$12$-$13$ triangle on our hands. Because we want the sum of the altitudes, the third altitude that is not an axis must have a denominator of $13$. Since the other altitudes are integers, we choose the option with $13$ as the denominator, namely $E$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2022_AMC_10B_Problems_11", "eval_func": null }, "input": { "text": "All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices below is logically equivalent to the statement \"No school bigger than Euclid HS sold more T-shirts than Euclid HS\"?\n$\\textbf{(A) }$ All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS.\n$\\textbf{(B) }$ No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.\n$\\textbf{(C) }$ All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS.\n$\\textbf{(D) }$ All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS.\n$\\textbf{(E) }$ All schools smaller than Euclid HS sold more T-shirts than Euclid HS.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $B$ denote a school that is bigger than Euclid HS, and $M$ denote a school that sold more T-shirts than Euclid HS.\nIt follows that $\\neg B$ denotes a school that is Euclid HS, and $\\neg M$ denotes a school that T-shirts than Euclid HS.\nConverting everything to conditional statements (if-then form), the given statement becomes \\[B\\implies\\neg M.\\]\nIts contrapositive is $M\\implies\\neg B,$ which is $\\boxed{\\textbf{(B)}}.$\nNote that \"not bigger than\" does not mean \"smaller than\", and \"not selling more\" does not mean \"selling fewer\". There is an equality case. Therefore, none of the other answer choices is equivalent to $B\\implies\\neg M.$", "Suppose we have five schools: Euclid HS with $50$ students and $10$ T-shirts sold, school $A$ with $51$ students and $10$ T-shirts sold, school $B$ with $49$ students and $10$ T-shirts sold, school $C$ with $49$ students and $9$ T-shirts sold, and school $D$ with $51$ students and $9$ T-shirts sold (This configuration is legal.). Then, school $B$ rules out $\\textbf{(A)}$, school $A$ rules out $\\textbf{(C)}$, school $D$ rules out $\\textbf{(D)}$, and school $C$ rules out $\\textbf{(E)}$, leaving us with $\\boxed{\\textbf{(B)}}$ as the correct answer." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2004_AMC_10A_Problems_14", "eval_func": null }, "input": { "text": "The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is $20$ cents. If she had one more quarter, the average value would be $21$ cents. How many dimes does she have in her purse?\n$\\text {(A)}\\ 0 \\qquad \\text {(B)}\\ 1 \\qquad \\text {(C)}\\ 2 \\qquad \\text {(D)}\\ 3\\qquad \\text {(E)}\\ 4$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let the total value, in cents, of the coins Paula has originally be $v$, and the number of coins she has be $n$. Then $\\frac{v}{n}=20\\Longrightarrow v=20n$ and $\\frac{v+25}{n+1}=21$. Substituting yields: $20n+25=21(n+1),$ so $n=4$, $v = 80.$ Then, we see that the only way Paula can satisfy this rule is if she had $3$ quarters and $1$ nickel in her purse. Thus, she has $\\boxed{\\mathrm{(A)}\\ 0}$ dimes.", "If the new coin was worth $20$ cents, adding it would not change the mean. The additional $5$ cents raise the mean by $1$, thus the new number of coins must be $5$. Therefore there were $4$ coins worth a total of $4\\times20=80$ cents. As in the previous solution, we conclude that the only way to get $80$ cents using $4$ coins is $25+25+25+5$. Thus, having three quarters, one nickel, and no dimes $\\boxed{\\mathrm{(A)}\\ 0}.$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2021_AMC_10A_Problems_2", "eval_func": null }, "input": { "text": "Portia's high school has $3$ times as many students as Lara's high school. The two high schools have a total of $2600$ students. How many students does Portia's high school have?\n$\\textbf{(A)} ~600 \\qquad\\textbf{(B)} ~650 \\qquad\\textbf{(C)} ~1950 \\qquad\\textbf{(D)} ~2000\\qquad\\textbf{(E)} ~2050$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The following system of equations can be formed with $P$ representing the number of students in Portia's high school and $L$ representing the number of students in Lara's high school:\n\\begin{align*} P&=3L, \\\\ P+L&=2600. \\end{align*}\nSubstituting $P=3L$ gives $4L=2600.$ Solving for $L$ gives $L=650.$ Since we need to find $P,$ we multiply $650$ by $3$ to get $P=\\boxed{\\textbf{(C)} ~1950}.$", "Suppose Lara's high school has $x$ students, so Portia's high school has $3x$ students. We have $x+3x=2600,$ or $4x=2600.$ The answer is \\[3x=2600\\cdot\\frac 34=650\\cdot3=\\boxed{\\textbf{(C)} ~1950}.\\]", "Clearly, $2600$ is $4$ times the number of students in Lara's high school. Therefore, Lara's high school has $2600\\div4=650$ students, and Portia's high school has $650\\cdot3=\\boxed{\\textbf{(C)} ~1950}$ students.", "The number of students in Portia's high school must be a multiple of $3.$ This eliminates $\\textbf{(B)},\\textbf{(D)},$ and $\\textbf{(E)}.$ Since $\\textbf{(A)}$ is too small (as it is clear that $600+\\frac{600}{3}<2600$), we are left with $\\boxed{\\textbf{(C)} ~1950}.$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2008_AMC_10B_Problems_3", "eval_func": null }, "input": { "text": "Assume that $x$ is a positive real number . Which is equivalent to $\\sqrt[3]{x\\sqrt{x}}$?\n$\\mathrm{(A)}\\ x^{1/6}\\qquad\\mathrm{(B)}\\ x^{1/4}\\qquad\\mathrm{(C)}\\ x^{3/8}\\qquad\\mathrm{(D)}\\ x^{1/2}\\qquad\\mathrm{(E)}\\ x$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "$\\sqrt[3]{x\\sqrt{x}}=\\sqrt[3]{\\sqrt{x^3}}=\\sqrt[6]{x^3}=x^{3/6}=x^{1/2}\\ \\boxed{(D)}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2007_AMC_10A_Problems_10", "eval_func": null }, "input": { "text": "The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is $20$, the father is $48$ years old, and the average age of the mother and children is $16$. How many children are in the family?\n$\\text{(A)}\\ 2 \\qquad \\text{(B)}\\ 3 \\qquad \\text{(C)}\\ 4 \\qquad \\text{(D)}\\ 5 \\qquad \\text{(E)}\\ 6$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $n$ be the number of children. Then the total ages of the family is $48 + 16(n+1)$, and the total number of people in the family is $n+2$. So\n\\[20 = \\frac{48 + 16(n+1)}{n+2} \\Longrightarrow 20n + 40 = 16n + 64 \\Longrightarrow n = 6\\ \\mathrm{(E)}.\\]", "Let $x$ be the number of the children and the mom. The father, who is $48$, plus the sum of the ages of the kids and mom divided by the number of kids and mom plus $1$ (for the dad) = $20$. This is because the average age of the entire family is $20.$\nThis statement, written as an equation, is: \\[\\frac{48+16x}{x+1}=20\\]\n\\[48+16x=20x+20\\]\n\\[4x=28\\]\n\\[x=7\\]\n$7$ people - $1$ mom = $6$ children.\nTherefore, the answer is $\\boxed{E}$", "Let $m$ be the Mom's age.\nLet the number of children be $x$ and their average age be $y$. Their age totaled up is simply $xy$. \nWe have the following two equations:\n$\\frac{m+48+xy}{2+x}=20$, where $m+48+xy$ is the family's total age and $2+x$ is the total number of people in the family.\n$\\frac{m+48+xy}{2+x}=20$\n$m+48+xy=40+20x$\nThe next equation is $\\frac{m+xy}{1+x}=16$, where $m+xy$ is the total age of the Mom and the children, and $1+x$ is the number of children along with the Mom.\n$\\frac{m+xy}{1+x}=16$\n$m+xy=16+16x$.\nWe know the value for $m+xy$, so we substitute the value back in the first equation.\n$m+48+xy=40+20x$\n$(16+16x)+48=40+20x$.\n$x=6$. \nEarlier, we set $x$ to be the number of children. Therefore, there are $\\boxed{\\text{(E)} 6}$ children." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2004_AMC_10A_Problems_6", "eval_func": null }, "input": { "text": "Bertha has 6 daughters and no sons. Some of her daughters have 6 daughters, and the rest have none. Bertha has a total of 30 daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters?\n$\\mathrm{(A) \\ } 22 \\qquad \\mathrm{(B) \\ } 23 \\qquad \\mathrm{(C) \\ } 24 \\qquad \\mathrm{(D) \\ } 25 \\qquad \\mathrm{(E) \\ } 26$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Since Bertha has $6$ daughters, she has $30-6=24$ granddaughters, of which none have daughters. Of Bertha's daughters, $\\frac{24}6=4$ have daughters, so $6-4=2$ do not have daughters. Therefore, of Bertha's daughters and granddaughters, $24+2=26$ do not have daughters. $\\boxed{\\mathrm{(E)}\\ 26}$", "Bertha has $30 - 6 = 24$ granddaughters, none of whom have any daughters. The granddaughters are the children of $24/6 = 4$ of Bertha's daughters, so the number of women having no daughters is $30 - 4 = \\boxed{26}$.", "Draw a tree diagram and see that the answer can be found in the sum of $6+6$ granddaughters, $5+5$ daughters, and $4$ more daughters. Adding them together gives the answer of $\\boxed{\\mathrm{(E)}\\ 26}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2005_AMC_10A_Problems_11", "eval_func": null }, "input": { "text": "A wooden cube $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$?\n$\\textbf{(A) } 3\\qquad \\textbf{(B) } 4\\qquad \\textbf{(C) } 5\\qquad \\textbf{(D) } 6\\qquad \\textbf{(E) } 7$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Since there are $n^2$ little faces on each face of the big wooden cube, there are $6n^2$ little faces painted red. \nSince each unit cube has $6$ faces, there are $6n^3$ little faces total. \nSince one-fourth of the little faces are painted red, \n$\\frac{6n^2}{6n^3}=\\frac{1}{4}$\n$\\frac{1}{n}=\\frac{1}{4}$\n$n=\\boxed{\\textbf{(B) }4}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2002_AMC_10B_Problems_12", "eval_func": null }, "input": { "text": "For which of the following values of $k$ does the equation $\\frac{x-1}{x-2} = \\frac{x-k}{x-6}$ have no solution for $x$?\n$\\textbf{(A) } 1 \\qquad \\textbf{(B) } 2 \\qquad \\textbf{(C) } 3 \\qquad \\textbf{(D) } 4 \\qquad \\textbf{(E) } 5$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The domain over which we solve the equation is $\\mathbb{R} \\setminus \\{2,6\\}$.\nWe can now cross-multiply to get rid of the fractions, we get $(x-1)(x-6)=(x-k)(x-2)$.\nSimplifying that, we get $7x-6 = (k+2)x - 2k$. Clearly for $k=5$\u00a0we get the equation $-6=-10$\u00a0which is never true. The answer is $\\boxed{\\mathrm{ (E)}\\ 5}$\nFor other $k$, one can solve for $x$: $x(5-k) = 6-2k$, hence $x=\\frac {6-2k}{5-k}$. We can easily verify that for none of the other 4 possible values of $k$ is this equal to $2$ or $6$, hence there is a solution for $x$ in each of the other cases." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2004_AMC_10A_Problems_19", "eval_func": null }, "input": { "text": "A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?\n[asy] size(250);defaultpen(linewidth(0.8)); draw(ellipse(origin, 3, 1)); fill((3,0)--(3,2)--(-3,2)--(-3,0)--cycle, white); draw((3,0)--(3,16)^^(-3,0)--(-3,16)); draw((0, 15)--(3, 12)^^(0, 16)--(3, 13)); filldraw(ellipse((0, 16), 3, 1), white, black); draw((-3,11)--(3, 5)^^(-3,10)--(3, 4)); draw((-3,2)--(0,-1)^^(-3,1)--(-1,-0.89)); draw((0,-1)--(0,15), dashed); draw((3,-2)--(3,-4)^^(-3,-2)--(-3,-4)); draw((-7,0)--(-5,0)^^(-7,16)--(-5,16)); draw((3,-3)--(-3,-3), Arrows(6)); draw((-6,0)--(-6,16), Arrows(6)); draw((-2,9)--(-1,9), Arrows(3)); label(\"$3$\", (-1.375,9.05), dir(260), fontsize(7)); label(\"$A$\", (0,15), N); label(\"$B$\", (0,-1), NE); label(\"$30$\", (0, -3), S); label(\"$80$\", (-6, 8), W);[/asy]\n$\\mathrm{(A) \\ } 120 \\qquad \\mathrm{(B) \\ } 180 \\qquad \\mathrm{(C) \\ } 240 \\qquad \\mathrm{(D) \\ } 360 \\qquad \\mathrm{(E) \\ } 480$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The cylinder can be \"unwrapped\" into a rectangle, and we see that the stripe is a parallelogram with base $3$ and height $80$. Thus, we get $3\\times80=240\\Rightarrow\\boxed{\\mathrm{(C)}\\ 240}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2020_AMC_10B_Problems_20", "eval_func": null }, "input": { "text": "Let $B$ be a right rectangular prism (box) with edges lengths $1,$ $3,$ and $4$, together with its interior. For real $r\\geq0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point in $B$. The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$, where $a,$ $b,$ $c,$ and $d$ are positive real numbers. What is $\\frac{bc}{ad}?$\n$\\textbf{(A) } 6 \\qquad\\textbf{(B) } 19 \\qquad\\textbf{(C) } 24 \\qquad\\textbf{(D) } 26 \\qquad\\textbf{(E) } 38$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Split $S(r)$ into 4 regions: \n1. The rectangular prism itself\n2. The extensions of the faces of $B$\n3. The quarter cylinders at each edge of $B$\n4. The one-eighth spheres at each corner of $B$\n\nRegion 1: The volume of $B$ is $1 \\cdot 3 \\cdot 4 = 12$, so $d=12$.\nRegion 2: This volume is equal to the surface area of $B$ times $r$ (these \"extensions\" are just more boxes). The volume is then $\\text{SA} \\cdot r=2(1 \\cdot 3 + 1 \\cdot 4 + 3 \\cdot 4)r=38r$ to get $c=38$.\nRegion 3: We see that there are 12 quarter-cylinders, 4 of each type. We have 4 quarter-cylinders of height 1, 4 quarter-cylinders of height 3, 4 quarter-cylinders of height 4. Since 4 quarter-cylinders make a full cylinder, the total volume is $4 \\cdot \\dfrac{1\\pi r^2}{4} + 4 \\cdot \\dfrac{3\\pi r^2}{4} + 4 \\cdot \\dfrac{4 \\pi r^2}{4}=8 \\pi r^2$. Therefore, $b=8\\pi$. \nRegion 4: There is an eighth-sphere of radius $r$ at each corner of $B$. Since there are 8 corners, these eighth-spheres add up to 1 full sphere of radius $r$. The volume of this sphere is then $\\frac{4}{3}\\pi \\cdot r^3$, so $a=\\frac{4\\pi}{3}$.\nUsing these values, $\\dfrac{bc}{ad}=\\frac{(8\\pi)(38)}{(4\\pi/3)(12)} = \\boxed{\\textbf{(B) }19}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2002_AMC_10A_Problems_2", "eval_func": null }, "input": { "text": "Given that a, b, and c are non-zero real numbers, define $(a, b, c) = \\frac{a}{b} + \\frac{b}{c} + \\frac{c}{a}$, find $(2, 12, 9)$.\n$\\textbf{(A)}\\ 4 \\qquad \\textbf{(B)}\\ 5 \\qquad \\textbf{(C)}\\ 6 \\qquad \\textbf{(D)}\\ 7 \\qquad \\textbf{(E)}\\ 8$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "$(2, 12, 9)=\\frac{2}{12}+\\frac{12}{9}+\\frac{9}{2}=\\frac{1}{6}+\\frac{4}{3}+\\frac{9}{2}=\\frac{1}{6}+\\frac{8}{6}+\\frac{27}{6}=\\frac{36}{6}=6$. Our answer is then $\\boxed{\\textbf{(C) }6}$.", "Without computing the answer exactly, we see that $2/12=\\text{a little}$, $12/9=\\text{more than }1$, and $9/2=4.5$.\nThe sum is $4.5 + (\\text{more than }1) + (\\text{a little}) = (\\text{more than }5.5) + (\\text{a little})$, and as all the options are integers, the correct one is $\\boxed{\\textbf{(C) }6}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2009_AMC_10B_Problems_14", "eval_func": null }, "input": { "text": "On Monday, Millie puts a quart of seeds, $25\\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $25\\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?\n$\\textbf{(A)}\\ \\text{Tuesday}\\qquad \\textbf{(B)}\\ \\text{Wednesday}\\qquad \\textbf{(C)}\\ \\text{Thursday}\\qquad \\textbf{(D)}\\ \\text{Friday}\\qquad \\textbf{(E)}\\ \\text{Saturday}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "On Monday, day 1, the birds find $\\frac 14$ quart of millet in the feeder. On Tuesday they find\n\\[\\frac 14 + \\frac 34 \\cdot \\frac 14\\]\nquarts of millet. On Wednesday, day 3, they find\n\\[\\frac 14 + \\frac 34 \\cdot \\frac 14 + \\left(\\frac34\\right)^2 \\cdot \\frac 14\\]\nquarts of millet. The number of quarts of millet they find on day $n$ is\n\\[\\frac 14 + \\frac 34 \\cdot \\frac 14 + \\left(\\frac34\\right)^2 \\cdot \\frac 14 + \\cdots + \\left(\\frac 34\\right)^{n-1} \\cdot \\frac 14 = \\frac {(\\frac 14)(1 - (\\frac 34)^n)}{1 - \\frac 34} = 1 - \\left(\\frac 34\\right)^n .\\]\nThe birds always find $\\frac 34$ quart of other seeds, so more than half the seeds are millet if $1 - \\left(\\frac 34\\right)^n > \\frac 34$, that is, when $\\left(\\frac 34\\right)^n < \\frac 14$. Because $\\left(\\frac 34\\right)^4 = \\frac {81}{256} > \\frac 14$ and $\\left(\\frac 34\\right)^5 = \\frac {243}{1024} < \\frac 14$, this will first occur on day $5$ which is $\\boxed {\\text{Friday}}$. The answer is $\\mathrm{(D)}$.", "Notice that, every day, there is always $\\frac34$ quarts of other seeds in the feeder, so we simply need to calculate when the quantity of millet seeds is greater than $\\frac34$ quarts. \nEvery day that elapses, a fourth of the millet is taken away (eaten by birds), and then a fourth of a quart is added (by Millie). Thus, if $n$ was the quantity of millet seeds the previous day, then on the next day, the quantity is $\\dfrac34n+\\dfrac14=\\dfrac14\\left(3n+1\\right).$\nMonday: $\\dfrac14\\quad$\nTuesday: $\\dfrac14\\left(3\\cdot\\dfrac14+1\\right)=\\dfrac{7}{16}$\nWednesday: $\\dfrac14\\left(3\\cdot\\dfrac{7}{16}+1\\right)=\\dfrac{37}{64}$\nThursday: $\\dfrac14\\left(3\\cdot\\dfrac{37}{64}+1\\right)=\\dfrac{175}{256}$\nFriday: $\\dfrac14\\left(3\\cdot\\dfrac{175}{256}+1\\right)=\\dfrac{781}{1024}>\\dfrac{768}{1024}=\\dfrac34.$\nBrute forcing, we see that it takes $4$ days. Thus, the answer is $\\boxed{\\text{Friday}}$, or $\\boxed{\\text{(D)}}$. \nNote: Yes there is a pattern to it, $1-\\left(\\frac{3}{4}\\right)^x$ where $x$ is the number of days, as shown in Solution 1. This solution is only if you didn't notice the pattern. " ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2003_AMC_10B_Problems_11", "eval_func": null }, "input": { "text": "A line with slope $3$ intersects a line with slope $5$ at point $(10,15)$. What is the distance between the $x$-intercepts of these two lines?\n$\\textbf{(A) } 2 \\qquad\\textbf{(B) } 5 \\qquad\\textbf{(C) } 7 \\qquad\\textbf{(D) } 12 \\qquad\\textbf{(E) } 20$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Using the point-slope form, the equation of each line is\n\\[y-15=3(x-10) \\longrightarrow y=3x-15\\]\n\\[y-15=5(x-10) \\longrightarrow y=5x-35\\]\nSubstitute in $y=0$ to find the $x$-intercepts.\n\\[0=3x-15\\longrightarrow x=5\\]\n\\[0=5x-35\\longrightarrow x=7\\]\nThe difference between them is $7-5=\\boxed{\\textbf{(A) \\ } 2}$.", "The $x$-intercepts of a line is where the line's $y$-coordinate is $=0$. The slope is defined as $\\frac{\\text{change in } y\\text{-coordinate}}{\\text{change in } x\\text{-coordinate}}$. Therefore, the line with slope $3$ can be expressed as\n\\[\\frac{\\text{change in }y\\text{-coordinate}}{\\text{change in }x\\text{-coordinate}}=3\\]\nFrom $(10,15)$ to the $x$-intercept, the line must move $-15$ units to the $x$-axis. Therefore, \\[\\frac{\\text{change in }y\\text{-coordinate}}{\\text{change in }x\\text{-coordinate}}=\\frac{-15}{x_1}=3\\implies x_1=-5\\].\nTherefore, the $x$-intercept of the line with slope $3$ is $(10-5,0)=(5,0)$.Note that some sources may state \"the $x$-intercept is $5$\" instead of \"the $x$-intercept is $(5,0)$.\nIf an idea functions great for one part of the problem but does not find the solution to the problem, we want to use the idea again. Using the idea for the line with slope $5$, we find\n\\[\\frac{\\text{change in }y\\text{-coordinate}}{\\text{change in }x\\text{-coordinate}}=\\frac{-15}{x_2}=5\\implies x_2=-3\\].\nTherefore, the $x$-intercept of the line with slope $5$ is $(10-3,0)=(7,0)$. The distance between $\\boxed{(5,0)--(7,0)=2\\implies A}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2002_AMC_10A_Problems_5", "eval_func": null }, "input": { "text": "Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.\n[asy] unitsize(.3cm); path c=Circle((0,2),1); filldraw(Circle((0,0),3),grey,black); filldraw(Circle((0,0),1),white,black); filldraw(c,white,black); filldraw(rotate(60)*c,white,black); filldraw(rotate(120)*c,white,black); filldraw(rotate(180)*c,white,black); filldraw(rotate(240)*c,white,black); filldraw(rotate(300)*c,white,black); [/asy]\n$\\textbf{(A)}\\ \\pi \\qquad \\textbf{(B)}\\ 1.5\\pi \\qquad \\textbf{(C)}\\ 2\\pi \\qquad \\textbf{(D)}\\ 3\\pi \\qquad \\textbf{(E)}\\ 3.5\\pi$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The outer circle has radius $1+1+1=3$, and thus area $9\\pi$. The little circles have area $\\pi$ each; since there are 7, their total area is $7\\pi$. Thus, our answer is $9\\pi-7\\pi=\\boxed{2\\pi\\Rightarrow \\textbf{(C)}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2002_AMC_10B_Problems_4", "eval_func": null }, "input": { "text": "What is the value of $(3x - 2)(4x + 1) - (3x - 2)4x + 1$ when $x=4$?\n$\\mathrm{(A)}\\ 0 \\qquad\\mathrm{(B)}\\ 1 \\qquad\\mathrm{(C)}\\ 10 \\qquad\\mathrm{(D)}\\ 11 \\qquad\\mathrm{(E)}\\ 12$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "By the distributive property, \n\\[(3x-2)[(4x+1)-4x] + 1 = 3x-2 + 1 = 3x-1 = 3(4) - 1 = \\boxed{\\mathrm{(D)}\\ 11}\\].", "Inputting 4 into $x$ in the original equation,\n\\[[3(4)-2][4(4)+1]-[3(4)-2][4(4)]+1 = 170-160+1 = \\boxed{\\mathrm{(D)}\\ 11}\\]" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2009_AMC_10B_Problems_6", "eval_func": null }, "input": { "text": "Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages?\n$\\mathrm{(A)}\\ 10\\qquad \\mathrm{(B)}\\ 12\\qquad \\mathrm{(C)}\\ 16\\qquad \\mathrm{(D)}\\ 18\\qquad \\mathrm{(E)}\\ 24$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The age of each person is a factor of $128 = 2^7$. So the twins could be $2^0 = 1, 2^1 = 2, 2^2 = 4, 2^3 = 8$ years of age and, consequently Kiana could be $128$, $32$, $8$, or $2$ years old, respectively. Because Kiana is younger than her brothers, she must be $2$ years old. So the sum of their ages is $2 + 8 + 8 = \\boxed{18}$. The answer is $\\mathrm{(D)}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2004_AMC_10B_Problems_10", "eval_func": null }, "input": { "text": "A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains $100$ cans, how many rows does it contain?\n$\\mathrm{(A)\\ }5\\qquad\\mathrm{(B)\\ }8\\qquad\\mathrm{(C)\\ }9\\qquad\\mathrm{(D)\\ }10\\qquad\\mathrm{(E)\\ }11$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The sum of the first $n$ odd numbers is $n^2$. As in our case $n^2=100$, we have $n=\\boxed{\\mathrm{(D)\\ }10}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2023_AMC_12B_Problems_4", "eval_func": null }, "input": { "text": "Jackson's paintbrush makes a narrow strip with a width of $6.5$ millimeters. Jackson has enough paint to make a strip $25$ meters long. How many square centimeters of paper could Jackson cover with paint? \n$\\textbf{(A) } 162,500 \\qquad\\textbf{(B) } 162.5 \\qquad\\textbf{(C) }1,625 \\qquad\\textbf{(D) }1,625,000 \\qquad\\textbf{(E) } 16,250$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "$6.5$ millimeters is equal to $0.65$ centimeters. $25$ meters is $2500$ centimeters. The answer is $0.65 \\times 2500$, so the answer is $\\boxed{\\textbf{(C) 1,625}}$. ", "$6.5$ millimeters can be represented as $65 \\times 10^{-2}$ centimeters. $25$ meters is $25 \\times 10^{2}$ centimeters. Multiplying out these results in $(65 \\times 10^{-2}) \\times (25 \\times 10^{2})$, which is $65 \\times 25$ making the answer $\\boxed{\\textbf{(C) 1,625}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2002_AMC_12P_Problems_2", "eval_func": null }, "input": { "text": "The function $f$ is given by the table\n\\[\\begin{tabular}{|c||c|c|c|c|c|} \\hline x & 1 & 2 & 3 & 4 & 5 \\\\ \\hline f(x) & 4 & 1 & 3 & 5 & 2 \\\\ \\hline \\end{tabular}\\]\nIf $u_0=4$ and $u_{n+1} = f(u_n)$ for $n \\ge 0$, find $u_{2002}$\n$\\text{(A) }1 \\qquad \\text{(B) }2 \\qquad \\text{(C) }3 \\qquad \\text{(D) }4 \\qquad \\text{(E) }5$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We can guess that the series given by the problem is periodic in some way. Starting off, $u_0=4$ is given. $u_1=u_{0+1}=f(u_0)=f(4)=5,$ so $u_1=5.$ $u_2=u_{1+1}=f(u_1)=f(5)=2,$ so $u_2=2.$ $u_3=u_{2+1}=f(u_2)=f(2)=1,$ so $u_3=1.$ $u_4=u_{3+1}=f(u_3)=f(1)=4,$ so $u_4=4.$ Plugging in $4$ will give us $5$ as found before, and plugging in $5$ will give $2$ and so on. This means that our original guess of the series being periodic was correct. Summing up our findings in a nice table,\n\\[\\begin{tabular}{|c||c|c|c|c|c|c|} \\hline n & 0 & 1 & 2 & 3 & 4 & ...\\\\ \\hline un & 4 & 5 & 2 & 1 & 4 & ...\\\\ \\hline \\end{tabular}\\]\nin which the next $u_n$ is found by simply plugging in the number from the last box into $f(x).$ The function is periodic every $4$ terms. $2002 \\equiv 2\\pmod{4}$, and counting $4$ starting from $u_1$ will give us our answer of $\\boxed{\\textbf{(B) } 2}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2009_AMC_12B_Problems_15", "eval_func": null }, "input": { "text": "Assume $0 < r < 3$. Below are five equations for $x$. Which equation has the largest solution $x$?\n$\\textbf{(A)}\\ 3(1 + r)^x = 7\\qquad \\textbf{(B)}\\ 3(1 + r/10)^x = 7\\qquad \\textbf{(C)}\\ 3(1 + 2r)^x = 7$\n$\\textbf{(D)}\\ 3(1 + \\sqrt {r})^x = 7\\qquad \\textbf{(E)}\\ 3(1 + 1/r)^x = 7$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Intuitively, $x$ will be largest for that option for which the value in the parentheses is smallest.\nFormally, first note that each of the values in parentheses is larger than $1$. Now, each of the options is of the form $3f(r)^x = 7$. This can be rewritten as $x\\log f(r) = \\log\\frac 73$. As $f(r)>1$, we have $\\log f(r)>0$. Thus $x$ is the largest for the option for which $\\log f(r)$ is smallest. And as $\\log f(r)$ is an increasing function, this is the option for which $f(r)$ is smallest.\nWe now get the following easier problem: Given that $00$.\nThe sum of digits of this number is $a+b+c$, hence we get the equation $100a+10b+c = 6(a+b+c)$. This simplifies to $94a + 4b - 5c = 0$. Clearly for $a>0$ there are no solutions, hence $a=0$ and we get the equation $4b=5c$. This obviously has only one valid solution $(b,c)=(5,4)$, hence the only solution is the number $54$.", "The sum of the digits is at most $9+9+9=27$. Therefore the number is at most $6\\cdot 27 = 162$. Since the number is $6$ times the sum of its digits, it must be divisible by $6$, therefore also by $3$, therefore the sum of its digits must be divisible by $3$. With this in mind we can conclude that the number must be divisible by $18$, not just by $6$. Since the number is divisible by $18$, it is also divisible by $9$, therefore the sum of its digits is divisible by $9$, therefore the number is divisible by $54$, which leaves us with $54$, $108$ and $162$. Only $54$ is $6$ times its digits, hence the answer is $\\boxed{1}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2018_AMC_12A_Problems_6", "eval_func": null }, "input": { "text": "For positive integers $m$ and $n$ such that $m+101:~ a_n = a_{n-1}a_{n+1} - 1$. This can be rewritten as $a_{n+1} = \\frac{a_n +1}{a_{n-1}}$. We have $a_1=x$ and $a_2=2000$, and we compute:\n\\begin{align*} a_3 & = \\frac{a_2+1}{a_1} = \\frac{2001}x \\\\ a_4 & = \\frac{a_3+1}{a_2} = \\frac{ \\dfrac{2001}x + 1 }{ 2000 } = \\frac{2001 + x}{2000x} \\\\ a_5 & = \\frac{a_4+1}{a_3} = \\frac{ \\frac{2001 + x}{2000x} + 1 }{ \\frac{2001}x } = \\frac{ \\frac{2001 + 2001x}{2000x} }{ \\frac{2001}x } = \\frac{1+x}{2000} \\\\ a_6 & = \\frac{a_5+1}{a_4} = \\frac{ \\frac{1+x}{2000} + 1 }{ \\frac{2001 + x}{2000x} } = \\frac{ \\frac{2001+x}{2000} }{ \\frac{2001 + x}{2000x} } = x \\\\ a_7 & = \\frac{a_6+1}{a_5} = \\frac{ x+1 }{ \\frac{1+x}{2000} } = 2000 \\end{align*}\nAt this point we see that the sequence will become periodic: we have $a_6=a_1$, $a_7=a_2$, and each subsequent term is uniquely determined by the previous two.\nHence if $2001$ appears, it has to be one of $a_1$ to $a_5$. As $a_2=2000$, we only have four possibilities left. Clearly $a_1=2001$ for $x=2001$, and $a_3=2001$ for $x=1$. The equation $a_4=2001$ solves to $x = \\frac{2001}{2000\\cdot 2001 - 1}$, and the equation $a_5=2001$ to $x=2000\\cdot 2001 - 1$.\nNo two values of $x$ we just computed are equal, and therefore there are $\\boxed{4}$ different values of $x$ for which the sequence contains the value $2001$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2017_AMC_12B_Problems_25", "eval_func": null }, "input": { "text": "A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$-player teams, but no two teams may have exactly the same $5$ members. The site statistics show a curious fact: The average, over all subsets of size $9$ of the set of $n$ participants, of the number of complete teams whose members are among those $9$ people is equal to the reciprocal of the average, over all subsets of size $8$ of the set of $n$ participants, of the number of complete teams whose members are among those $8$ people. How many values $n$, $9\\leq n\\leq 2017$, can be the number of participants?\n$\\textbf{(A) } 477 \\qquad \\textbf{(B) } 482 \\qquad \\textbf{(C) } 487 \\qquad \\textbf{(D) } 557 \\qquad \\textbf{(E) } 562$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let there be $T$ teams. For each team, there are ${n-5\\choose 4}$ different subsets of $9$ players that includes a given full team, so the total number of team-(group of 9) pairs is \n\\[T{n-5\\choose 4}.\\]\nThus, the expected value of the number of full teams in a random set of $9$ players is\n\\[\\frac{T{n-5\\choose 4}}{{n\\choose 9}}.\\]\nSimilarly, the expected value of the number of full teams in a random set of $8$ players is\n\\[\\frac{T{n-5\\choose 3}}{{n\\choose 8}}.\\]\nThe condition is thus equivalent to the existence of a positive integer $T$ such that\n\\[\\frac{T{n-5\\choose 4}}{{n\\choose 9}}\\frac{T{n-5\\choose 3}}{{n\\choose 8}} = 1.\\]\n\\[T^2\\frac{(n-5)!(n-5)!8!9!(n-8)!(n-9)!}{n!n!(n-8)!(n-9)!3!4!} = 1\\]\n\\[T^2 = \\big((n)(n-1)(n-2)(n-3)(n-4)\\big)^2 \\frac{3!4!}{8!9!}\\]\n\\[T^2 = \\big((n)(n-1)(n-2)(n-3)(n-4)\\big)^2 \\frac{144}{7!7!8\\cdot8\\cdot9}\\]\n\\[T^2 = \\big((n)(n-1)(n-2)(n-3)(n-4)\\big)^2 \\frac{1}{4\\cdot7!7!}\\]\n\\[T = \\frac{(n)(n-1)(n-2)(n-3)(n-4)}{2^5\\cdot3^2\\cdot5\\cdot7}\\]\nNote that this is always less than ${n\\choose 5}$, so as long as $T$ is integral, $n$ is a possibility. Thus, we have that this is equivalent to \n\\[2^5\\cdot3^2\\cdot5\\cdot7\\big|(n)(n-1)(n-2)(n-3)(n-4).\\]\nIt is obvious that $5$ divides the RHS, and that $7$ does iff $n\\equiv 0,1,2,3,4\\mod 7$. Also, $3^2$ divides it iff $n\\not\\equiv 5,8\\mod 9$. One can also bash out that $2^5$ divides it in $16$ out of the $32$ possible residues $\\mod 32$. \nNote that $2016 = 7*9*32$ so by using all numbers from $2$ to $2017$, inclusive, it is clear that each possible residue $\\mod 7,9,32$ is reached an equal number of times, so the total number of working $n$ in that range is $5\\cdot 7\\cdot 16 = 560$. However, we must subtract the number of \"working\" $2\\leq n\\leq 8$, which is $3$. Thus, the answer is $\\boxed{\\textbf{(D) } 557}$.\nAlternatively, it is enough to approximate by finding the floor of $2017 \\cdot \\frac57 \\cdot \\frac79 \\cdot \\frac12 - 3$ to get $\\boxed{\\textbf{(D) } 557}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2011_AMC_12A_Problems_24", "eval_func": null }, "input": { "text": "Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, and $DA=12$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?\n$\\textbf{(A)}\\ \\sqrt{15} \\qquad \\textbf{(B)}\\ \\sqrt{21} \\qquad \\textbf{(C)}\\ 2\\sqrt{6} \\qquad \\textbf{(D)}\\ 5 \\qquad \\textbf{(E)}\\ 2\\sqrt{7}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Note as above that ABCD must be tangential to obtain the circle with maximal radius. Let $E$, $F$, $G$, and $H$ be the points on $AB$, $BC$, $CD$, and $DA$ respectively where the circle is tangent. Let $\\theta=\\angle BAD$ and $\\alpha=\\angle ADC$. Since the quadrilateral is cyclic(because we want to maximize the circle, so we set the quadrilateral to be cyclic), $\\angle ABC=180^{\\circ}-\\alpha$ and $\\angle BCD=180^{\\circ}-\\theta$. Let the circle have center $O$ and radius $r$. Note that $OHD$, $OGC$, $OFB$, and $OEA$ are right angles. \nHence $FOG=\\theta$, $GOH=180^{\\circ}-\\alpha$, $EOH=180^{\\circ}-\\theta$, and $FOE=\\alpha$. \nTherefore, $AEOH\\sim OFCG$ and $EBFO\\sim HOGD$. \nLet $x=CG$. Then $CF=x$, $BF=BE=9-x$, $GD=DH=7-x$, and $AH=AE=x+5$. Using $AEOH\\sim OFCG$ and $EBFO\\sim HOGD$ we have $r/(x+5)=x/r$, and $(9-x)/r=r/(7-x)$. By equating the value of $r^2$ from each, $x(x+5)=(7-x)(9-x)$. Solving we obtain $x=3$ so that $r=\\boxed{\\textbf{(C)}\\ 2\\sqrt{6}}$.", "To maximize the radius of the circle, we also need to maximize its area. To maximize the area of the circle, the quadrilateral must be tangential (have an incircle). In a tangential quadrilateral, the sum of opposite sides is equal to the semiperimeter of the quadrilateral. $14+7=12+9$, so this particular quadrilateral has an incircle. By definition, given $4$ side lengths, a cyclic quadrilateral has the maximum area of any quadrilateral with those side lengths. Therefore, to maximize the area of the quadrilateral and thus the incircle, we assume that this quadrilateral is cyclic. \nFor cyclic quadrilaterals, Brahmagupta's formula gives the area as $\\sqrt{(s-a)(s-b)(s-c)(s-d)}$ where $s$ is the semiperimeter and $a, b, c,$ and $d$ are the side lengths. Breaking it up into $4$ triangles, we see the area of a tangential quadrilateral is also equal to $r*s$. Equate these two equations.\nSubstituting $s$, the semiperimeter, and $A$, the area and solving for $r$,we get $\\boxed{\\textbf{(C)}\\ 2\\sqrt{6}}$.", "By Pitot's Theorem, since $AB+CD=AD+BC$, there exists a circle tangent to all four sides of quadrilateral $ABCD$. Thus, we need to find the radius of this circle.\nLet the circle be tangent to $AB$, $BC$, $CD$, and $AD$ at points $P$, $Q$, $R$, and $S$, respectively. Also, let $AP=x$. Then $AS$ also equals $x$. Let the center of the circle be $O$. Observe that $AO$ bisects angle $\\angle A$, so $\\cot\\frac{A}{2}=\\frac{x}{r}$. Moreover, $\\cot\\frac{C}{2}=\\frac{9-(14-x)}{r}=\\frac{x-5}{r}$. But $\\angle \\frac{C}{2}=90^\\circ-\\angle \\frac{A}{2}$, so $\\cot\\frac{C}{2}=\\tan\\frac{A}{2}$, and we find that $\\frac{r}{x}=\\frac{x-5}{r}$. Hence, $r^2=x(x-5)$.\nSimilarly, $\\cot\\frac{B}{2}=\\frac{14-x}{r}$, and $\\cot\\frac{D}{2}=\\frac{12-x}{r}=\\tan\\frac{B}{2}=\\frac{r}{14-x}$. Therefore, $r^2=\\frac{12-x}{14-x}$. Putting this together with the above equation yields \\[(12-x)(14-x)=x(x-5)\\]\\[\\Longrightarrow x^2-26x+168=x^2-5x\\]\\[\\Longrightarrow 21x=168\\]\\[\\Longrightarrow x=8.\\] Thus, $r^2=8(8-5)=8(3)=24$, and $r=\\boxed{\\textbf{(C)}\\ 2\\sqrt{6}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2016_AMC_12A_Problems_16", "eval_func": null }, "input": { "text": "The graphs of $y=\\log_3 x, y=\\log_x 3, y=\\log_\\frac{1}{3} x,$ and $y=\\log_x \\dfrac{1}{3}$ are plotted on the same set of axes. How many points in the plane with positive $x$-coordinates lie on two or more of the graphs? \n$\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ 3\\qquad\\textbf{(C)}\\ 4\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ 6$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Setting the first two equations equal to each other, $\\log_3 x = \\log_x 3$.\nSolving this, we get $\\left(3, 1\\right)$ and $\\left(\\frac{1}{3}, -1\\right)$.\nSimilarly with the last two equations, we get $\\left(3, -1\\right)$ and $\\left(\\frac{1}{3}, 1\\right)$.\nNow, by setting the first and third equations equal to each other, we get $\\left(1, 0\\right)$.\nPairing the first and fourth or second and third equations won't work because then $\\log x \\leq 0$.\nPairing the second and fourth equations will yield $x = 1$, but since you can't divide by $\\log 1 = 0$, it doesn't work.\nAfter trying all pairs, we have a total of $5$ solutions $\\rightarrow \\boxed{\\textbf{(D)} 5}$", "Note that $\\log_b a =\\log_c a / \\log_c b$.\nThen $\\log_b a = \\log_a a / \\log_a b = 1/ \\log_a b$\n$\\log_\\frac{1}{a} b = \\log_a \\frac{1}{a} / \\log_a b = -1/ \\log_a b$\n$\\log_\\frac{1}{b} a = -\\log_a b$\nTherefore, the system of equations can be simplified to:\n$y = t$\n$y = -t$\n$y = \\frac{1}{t}$\n$y = -\\frac{1}{t}$\nwhere $t = \\log_3 x$. Note that all values of $t$ correspond to exactly one positive $x$ value, so all $(t,y)$ intersections will correspond to exactly one $(x,y)$ intersection in the positive-x area.\nGraphing this system of functions will generate a total of $5$ solutions $\\rightarrow \\boxed{\\textbf{(D)} 5}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2017_AMC_12B_Problems_14", "eval_func": null }, "input": { "text": "An ice-cream novelty item consists of a cup in the shape of a 4-inch-tall frustum of a right circular cone, with a 2-inch-diameter base at the bottom and a 4-inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height 4 inches, whose base, at the bottom, is the top base of the frustum. What is the total volume of the ice cream, in cubic inches?\n$\\textbf{(A)}\\ 8\\pi \\qquad \\textbf{(B)}\\ \\frac{28\\pi}{3} \\qquad \\textbf{(C)}\\ 12\\pi \\qquad \\textbf{(D)}\\ 14\\pi \\qquad \\textbf{(E)}\\ \\frac{44\\pi}{3}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The top cone has radius 2 and height 4 so it has volume $\\dfrac{1}{3} \\pi (2)^2 \\times 4$. \nThe frustum is made up by taking away a small cone of radius 1, height 4 from a large cone of radius 2, height 8, so it has volume\n$\\dfrac{1}{3} \\pi (2)^2 \\times 8 - \\dfrac{1}{3} \\pi (1)^2 \\times 4$.\nAdding, we get $\\dfrac{1}{3} \\pi (16+32-4) = \\boxed{ \\textbf{E} \\ \\dfrac{44\\pi}{3}}$. ", "Find the volume of the cone with the method in Solution 1. The volume of the frustrum is $\\pi\\int_{0}^{4} \\left(-\\frac{1}{4}x+2\\right)^2 dx=\\frac{28\\pi}{3}$\nAdding, we get $\\dfrac{44\\pi}{3}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2009_AMC_12A_Problems_24", "eval_func": null }, "input": { "text": "The tower function of twos is defined recursively as follows: $T(1) = 2$ and $T(n + 1) = 2^{T(n)}$ for $n\\ge1$. Let $A = (T(2009))^{T(2009)}$ and $B = (T(2009))^A$. What is the largest integer $k$ for which \\[\\underbrace{\\log_2\\log_2\\log_2\\ldots\\log_2B}_{k\\text{ times}}\\] is defined?\n$\\textbf{(A)}\\ 2009\\qquad \\textbf{(B)}\\ 2010\\qquad \\textbf{(C)}\\ 2011\\qquad \\textbf{(D)}\\ 2012\\qquad \\textbf{(E)}\\ 2013$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Testing the first two (or three) positive integers instead of 2009, $k$ seems to always be 4 more. Put E and go on to tackle ", "We just look at the last three logarithms for the moment, and use the fact that $\\log_2 T(k) = T(k - 1)$. We wish to find:\n\\begin{align*} \\log_2\\log_2\\log_2\\left({T(2009)^{\\left({T(2009)}^{T(2009)}\\right)}}\\right) &= \\log_2(T(2009)\\log_2(T(2009)\\log_2 T(2009)))\\\\ &= \\log_2(T(2009)\\log_2(T(2009)T(2008)))\\\\ &= \\log_2 \\{ T(2009) [ T(2008) + T(2007) ] \\} \\\\ \\end{align*}\n\nNow we realize that $T(n - 1)$ is much smaller than $T(n)$. So we approximate this, remembering we have rounded down, as:\nWe have used $3$ logarithms so far. Applying $2007$ more to the left of our expression, we get $T(1) = 2$. Then we can apply the logarithm $2$ more times, until we get to $0$. So our answer is approximately $3 + 2007 + 2 = 2012$. But we rounded down, so that means that after $2012$ logarithms we get a number slightly greater than $0$, so we can apply logarithms one more time. We can be sure it is small enough so that the logarithm can only be applied $1$ more time since $2012 + 1 = 2013$ is the largest answer choice. So the answer is $\\mathbf{(E)}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2008_AMC_12A_Problems_6", "eval_func": null }, "input": { "text": "Heather compares the price of a new computer at two different stores. Store $A$ offers $15\\%$ off the sticker price followed by a $$90$ rebate, and store $B$ offers $25\\%$ off the same sticker price with no rebate. Heather saves $$15$ by buying the computer at store $A$ instead of store $B$. What is the sticker price of the computer, in dollars?\n$\\mathrm{(A)}\\ 750\\qquad\\mathrm{(B)}\\ 900\\qquad\\mathrm{(C)}\\ 1000\\qquad\\mathrm{(D)}\\ 1050\\qquad\\mathrm{(E)}\\ 1500$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let the sticker price be $x$.\nThe price of the computer is $0.85x-90$ at store $A$ and $0.75x$ at store $B$.\nHeather saves $$15$ at store $A$, so $0.85x-90+15=0.75x$.\nSolving, we find $x=750$, and thus the answer is $\\mathrm{(A)}$.", "The $$ 90$ at store $A$ is $$ 15$ greater than the additional $25\\%-15\\% = 10\\%$ off at store $B$.\nThus the $10\\%$ off is equal to $$ 90$ $-$ $$ 15$ $=$ $$ 75$, and therefore the sticker price is $$ 750$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2014_AMC_12B_Problems_2", "eval_func": null }, "input": { "text": "Orvin went to the store with just enough money to buy $30$ balloons. When he arrived he discovered that the store had a special sale on balloons: buy $1$ balloon at the regular price and get a second at $\\frac{1}{3}$ off the regular price. What is the greatest number of balloons Orvin could buy?\n$\\textbf{(A)}\\ 33\\qquad\\textbf{(B)}\\ 34\\qquad\\textbf{(C)}\\ 36\\qquad\\textbf{(D)}\\ 38\\qquad\\textbf{(E)}\\ 39$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "If every balloon costs $n$ dollars, then Orvin has $30n$ dollars. For every balloon he buys for $n$ dollars, he can buy another for $\\frac{2n}{3}$ dollars. This means it costs him $\\frac{5n}{3}$ dollars to buy a bundle of $2$ balloons. With $30n$ dollars, he can buy $\\frac{30n}{\\frac{5n}{3}} = 18$ sets of two balloons, so the total number of balloons he can buy is $18\\times2 \\implies \\boxed{\\textbf{(C)}\\ 36 }$", "Similar to solution 1, but quicker. \nWe see the fraction $\\frac13,$ so we let each balloon cost $$3$ WLOG. Thus, Orvin has $$3\\cdot30=$90.$ \nEach pair of balloons (one full-priced and one a third off) costs $$3+\\left(1-\\frac13\\right)\\cdot$3=$3+$2=$5.$ Therefore, Orvin can buy $\\frac{$90}{$5}=18$ pairs of balloons, at maximum. $18\\cdot2=36$ balloons. " ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2002_AMC_12P_Problems_10", "eval_func": null }, "input": { "text": "Let $f_n (x) = \\text{sin}^n x + \\text{cos}^n x.$ For how many $x$ in $[0,\\pi]$ is it true that\n\\[6f_{4}(x)-4f_{6}(x)=2f_{2}(x)?\\]\n$\\text{(A) }2 \\qquad \\text{(B) }4 \\qquad \\text{(C) }6 \\qquad \\text{(D) }8 \\qquad \\text{(E) more than }8$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Divide by 2 on both sides to get \\[3f_{4}(x)-2f_{6}(x)=f_{2}(x)\\]\nSubstituting the definitions of $f_{2}(x)$, $f_{4}(x)$, and $f_{6}(x)$, we may rewrite the expression as\n\\[3(\\text{sin}^4{x} + \\text{cos}^4{x}) - 2(\\text{sin}^6{x} + \\text{cos}^6{x}) = 1\\]\nWe now simplify each term separately using some algebraic manipulation and the Pythagorean identity.\nWe can rewrite $3(\\text{sin}^4 x + \\text{cos}^4 x)$ as\n$3(\\text{sin}^2 x + \\text{cos}^2 x)^2 - 6\\text{sin}^2 x \\text{cos}^2 x$, which is equivalent to \n$3 - 6\\text{sin}^2 x \\text{cos}^2 x$.\nAs for $2(\\text{sin}^6 x + \\text{cos}^6 x)$, we may factor it as \n$2(\\text{sin}^2 x + \\text{cos}^2 x)(\\text{sin}^4 x + \\text{cos}^4 x - \\text{sin}^2 x \\text{cos}^2 x)$ which can be rewritten as\n$2(\\text{sin}^4 x + \\text{cos}^4 x - \\text{sin}^2 x \\text{cos}^2 x)$, and then as\n$2(\\text{sin}^2 x + \\text{cos}^2)^2) - 6\\text{sin}^2 x \\text{cos}^2 x$, which is equivalent to \n$2 - 6\\text{sin}^2 x \\text{cos}^2 x$.\nPutting everything together, we have $(3 - 6\\text{sin}^2 {x} \\text{cos}^2 {x}) - (2 - 6\\text{sin}^2 {x} \\text{cos}^2 {x}) = 1$ or $1 = 1$. Therefore, the given equation $3f_{4}(x)-2f_{6}(x)=f_{2}(x)$ is true for all real $x$, meaning that there are more than 8 values of $x$ that satisfy the given equation and so the answer is $\\boxed{\\text{(E) more than }8}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2006_AMC_12A_Problems_3", "eval_func": null }, "input": { "text": "The ratio of Mary's age to Alice's age is $3:5$. Alice is $30$ years old. How old is Mary?\n$\\textbf{(A)}\\ 15\\qquad\\textbf{(B)}\\ 18\\qquad\\textbf{(C)}\\ 20\\qquad\\textbf{(D)}\\ 24\\qquad\\textbf{(E)}\\ 50$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $m$ be Mary's age. Then $\\frac{m}{30}=\\frac{3}{5}$. Solving for $m$, we obtain $m=\\boxed{\\textbf{(B) }18}.$", "We can see this is a combined ratio of $8$, $(5+3)$. We can equalize by doing $30\\div5=6$, and $6\\cdot3=\\boxed{\\textbf{(B) }18}$. With the common ratio of $8$ and difference ratio of $6$, we see $6\\cdot8=30+18$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2014_AMC_12B_Problems_3", "eval_func": null }, "input": { "text": "Randy drove the first third of his trip on a gravel road, the next $20$ miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip?\n$\\textbf{(A)}\\ 30\\qquad\\textbf{(B)}\\ \\frac{400}{11}\\qquad\\textbf{(C)}\\ \\frac{75}{2}\\qquad\\textbf{(D)}\\ 40\\qquad\\textbf{(E)}\\ \\frac{300}{7}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "If the first and last legs of his trip account for $\\frac{1}{3}$ and $\\frac{1}{5}$ of his trip, then the middle leg accounts for $1 - \\frac{1}{3} - \\frac{1}{5} = \\frac{7}{15}$ths of his trip. This is equal to $20$ miles. Letting the length of the entire trip equal $x$, we have \n\\[\\frac{7}{15}x = 20 \\implies x=\\boxed{\\textbf{(E)}\\ \\frac{300}{7}}\\]" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2012_AMC_12A_Problems_21", "eval_func": null }, "input": { "text": "Let $a$, $b$, and $c$ be positive integers with $a\\ge$ $b\\ge$ $c$ such that\n$a^2-b^2-c^2+ab=2011$ and\n$a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$.\nWhat is $a$?\n$\\textbf{(A)}\\ 249\\qquad\\textbf{(B)}\\ 250\\qquad\\textbf{(C)}\\ 251\\qquad\\textbf{(D)}\\ 252\\qquad\\textbf{(E)}\\ 253$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Add the two equations.\n$2a^2 + 2b^2 + 2c^2 - 2ab - 2ac - 2bc = 14$.\nNow, this can be rearranged and factored.\n$(a^2 - 2ab + b^2) + (a^2 - 2ac + c^2) + (b^2 - 2bc + c^2) = 14$\n$(a - b)^2 + (a - c)^2 + (b - c)^2 = 14$\n$a$, $b$, and $c$ are all integers, so the three terms on the left side of the equation must all be perfect squares.\nWe see that the only is possibility is $14 = 9 + 4 + 1$. \n$(a-c)^2 = 9 \\Rightarrow a-c = 3$, since $a-c$ is the biggest difference. It is impossible to determine by inspection whether $a-b = 1$ or $2$, or whether $b-c = 1$ or $2$.\nWe want to solve for $a$, so take the two cases and solve them each for an expression in terms of $a$. Our two cases are $(a, b, c) = (a, a-1, a-3)$ or $(a, a-2, a-3)$. Plug these values into one of the original equations to see if we can get an integer for $a$.\n$a^2 - (a-1)^2 - (a-3)^2 + a(a-1) = 2011$, after some algebra, simplifies to\n$7a = 2021$. $2021$ is not divisible by $7$, so $a$ is not an integer. \nThe other case gives $a^2 - (a-2)^2 - (a-3)^2 + a(a-2) = 2011$, which simplifies to $8a = 2024$. Thus, $a = 253$ and the answer is $\\boxed{\\textbf{(E)}\\ 253}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2007_AMC_12B_Problems_9", "eval_func": null }, "input": { "text": "A function $f$ has the property that $f(3x-1)=x^2+x+1$ for all real numbers $x$. What is $f(5)$?\n$\\mathrm{(A)}\\ 7 \\qquad \\mathrm{(B)}\\ 13 \\qquad \\mathrm{(C)}\\ 31 \\qquad \\mathrm{(D)}\\ 111 \\qquad \\mathrm{(E)}\\ 211$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "$3x-1 =5 \\implies x= 2$\n$f(3(2)-1) = 2^2+2+1=7 \\implies (A)$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2012_AMC_12B_Problems_24", "eval_func": null }, "input": { "text": "Define the function $f_1$ on the positive integers by setting $f_1(1)=1$ and if $n=p_1^{e_1}p_2^{e_2}\\cdots p_k^{e_k}$ is the prime factorization of $n>1$, then \\[f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\\cdots (p_k+1)^{e_k-1}.\\]\nFor every $m\\ge 2$, let $f_m(n)=f_1(f_{m-1}(n))$. For how many $N$s in the range $1\\le N\\le 400$ is the sequence $(f_1(N),f_2(N),f_3(N),\\dots )$ unbounded?\n A sequence of positive numbers is unbounded if for every integer $B$, there is a member of the sequence greater than $B$.\n$\\textbf{(A)}\\ 15\\qquad\\textbf{(B)}\\ 16\\qquad\\textbf{(C)}\\ 17\\qquad\\textbf{(D)}\\ 18\\qquad\\textbf{(E)}\\ 19$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "First of all, notice that for any odd prime $p$, the largest prime that divides $p+1$ is no larger than $\\frac{p+1}{2}$, therefore eventually the factorization of $f_k(N)$ does not contain any prime larger than $3$. Also, note that $f_2(2^m) = f_1(3^{m-1})=2^{2m-4}$, when $m=4$ it stays the same but when $m\\geq 5$ it grows indefinitely. Therefore any number $N$ that is divisible by $2^5$ or any number $N$ such that $f_k(N)$ is divisible by $2^5$ makes the sequence $(f_1(N),f_2(N),f_3(N),\\dots )$ unbounded. There are $12$ multiples of $2^5$ within $400$. $2^4 5^2=400$ also works: $f_2(2^4 5^2) = f_1(3^4 \\cdot 2) = 2^6$.\nNow let's look at the other cases. Any first power of prime in a prime factorization will not contribute the unboundedness because $f_1(p^1)=(p+1)^0=1$. At least a square of prime is to contribute. So we test primes that are less than $\\sqrt{400}=20$:\n$f_1(3^4)=4^3=2^6$ works, therefore any number $\\leq 400$ that are divisible by $3^4$ works: there are $4$ of them.\n$3^3 \\cdot Q^2$ could also work if $Q^2$ satisfies $2~|~f_1(Q^2)$, but $3^3 \\cdot 5^2 > 400$.\n$f_1(5^3)=6^2 = 2^2 3^2$ does not work.\n$f_1(7^3)=8^2=2^6$ works. There are no other multiples of $7^3$ within $400$.\n$7^2 \\cdot Q^2$ could also work if $4~|~f_1(Q^2)$, but $7^2 \\cdot 3^2 > 400$ already.\nFor number that are only divisible by $p=11, 13, 17, 19$, they don't work because none of these primes are such that $p+1$ could be a multiple of $2^5$ nor a multiple of $3^4$.\nIn conclusion, there are $12+1+4+1=18$ number of $N$'s ... $\\framebox{D}$.", "Say a number $n$ is $\\textit{boring}$ if the sequence $\\{f_k(n)\\}$ is bounded; otherwise $n$ is $\\textit{interesting}$. \nIt is clear that $n$ is interesting iff $f_1(n)$ is. For a prime, $p$, $f_1(p)=1$. Thus, for a prime, $p_k$, in the prime factorization of $n$; if $e_k=1$ then $f_1(n)=f_1(n/p_k)$, so $n$ is interesting iff $n/p_k$ is. Continuing in this manner, we can divide $n$ by all such primes $p_k$ for which $e_k=1$; and $n$ is interesting iff each of these resulting numbers are. Finally we will end up with a number whose prime factorization contains only exponents $\\ge 2$. Let $S$ be the set of such numbers. It suffices to find all interesting numbers in $S$; all other interesting numbers will be multiples of these. Note that $f_1(n)\\mid f_1(kn)$ for all $k\\in\\mathbb{N}$; so by induction a multiple of an interesting number is also interesting.\nThe set $S$ contains either powers of primes $\\le 19$, or a product of two such powers.\nWe have $f_2(2^a)=f_1(3^{a-1}) = 2^{2(a-2)}$; since $2(a-2)> a$ iff $a\\ge 5$, so $2^5$ and its multiples are interesting; there are $12$ such.\nWe have $f_2(3^b)=f_1(2^{2(b-1)}) = 3^{2b-3}$; since $2b-3> b$ iff $b\\ge 4$, so $3^4$ and its multiples are interesting; there are $4$ such.\nAmong the remaining prime powers only $7^3 \\textbf{ is interesting}$ but it has no other multiples in $[400]$.\nWe are now left with numbers $n\\in S$ that are products of two prime powers, i.e $n=p^aq^b$, $a,b \\ge 2$. \nWe have $f_2(2^a3^b)= 2^{2(a-2)}3^{2b-3}$, so a number of this form is interesting iff $a\\ge 5$ or $b\\ge 4$; they are already counted above.\nIf $5^2\\mid n$ then $n=2^2\\cdot 5^2$ or $2^3\\cdot 5^2$ or $3^2\\cdot 5^2$ (all boring) or $n=2^4\\cdot 5^2=400$, $\\textbf{which is interesting}$, but has no other multiples. \nIf $7^2\\mid n$ then $n$ is $2^2\\cdot 7^2$ or $2^3\\cdot 7^2$ and both are boring. \nNo larger prime can divide $n$ because even $11^2\\cdot 2^2 > 400$. \nWe have found all the interesting numbers: $12$ multiples of $2^5$, $4$ multiples of $3^4$, $2^4\\cdot 5^2$, and $7^3$ for a total of $12+4+1+1= 18$ ... $\\framebox{D}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2014_AMC_12B_Problems_18", "eval_func": null }, "input": { "text": "The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is $bad$ if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?\n$\\textbf {(A) } 1 \\qquad \\textbf {(B) } 2 \\qquad \\textbf {(C) } 3 \\qquad \\textbf {(D) } 4 \\qquad \\textbf {(E) } 5$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We see that there are $5!$ total ways to arrange the numbers. However, we can always rotate these numbers so that, for example, the number 1 is always at the top of the circle. Thus, there are only $4!$ ways under rotation, which is not difficult to list out. We systematically list out all $24$ cases. \nNow, we must examine if they satisfy the conditions. We can see that by choosing one number at a time, we can always obtain subsets with sums 1, 2, 3, 4, and 5. By choosing the full circle, we can obtain 15. By choosing everything except for 1, 2, 3, 4, and 5, we can obtain subsets with sums of 10, 11, 12, 13, and 14. \nThis means that we now only need to check for 6, 7, 8, and 9. However, once we have found a set summing to 6, we can choose everything else and obtain a set summing to 9, and similarly for 7 and 8. Thus, we only need to check each case for whether or not we can obtain 6 or 7. \nWe find that there are only 4 arrangements that satisfy these conditions. However, each of these is a reflection of another. We divide by 2 for these reflections to obtain a final answer of $\\boxed{\\textbf {(B) }2}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2015_AMC_12B_Problems_3", "eval_func": null }, "input": { "text": "Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number?\n$\\textbf{(A)}\\; 8 \\qquad\\textbf{(B)}\\; 11 \\qquad\\textbf{(C)}\\; 14 \\qquad\\textbf{(D)}\\; 15 \\qquad\\textbf{(E)}\\; 18$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $a$ be the number written two times, and $b$ the number written three times. Then $2a + 3b = 100$. Plugging in $a = 28$ doesn't yield an integer for $b$, so it must be that $b = 28$, and we get $2a + 84 = 100$. Solving for $a$, we obtain $a = \\boxed{\\textbf{(A)}\\; 8}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2007_AMC_12B_Problems_13", "eval_func": null }, "input": { "text": "A traffic light runs repeatedly through the following cycle: green for $30$ seconds, then yellow for $3$ seconds, and then red for $30$ seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching?\n$\\mathrm{(A)}\\ \\frac{1}{63} \\qquad \\mathrm{(B)}\\ \\frac{1}{21} \\qquad \\mathrm{(C)}\\ \\frac{1}{10} \\qquad \\mathrm{(D)}\\ \\frac{1}{7} \\qquad \\mathrm{(E)}\\ \\frac{1}{3}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The traffic light runs through a $63$ second cycle.\nLetting $t=0$ reference the moment it turns green, the light changes at three different times: $t=30$, $t=33$, and $t=63$\nThis means that the light will change if the beginning of Leah's interval lies in\n$[27,30]$, $[30,33]$ or $[60,63]$\nThis gives a total of $9$ seconds out of $63$\n$\\frac{9}{63} = \\frac{1}{7} \\Rightarrow \\mathrm{(D)}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2009_AMC_12B_Problems_3", "eval_func": null }, "input": { "text": "Twenty percent less than 60 is one-third more than what number?\n$\\mathrm{(A)}\\ 16\\qquad \\mathrm{(B)}\\ 30\\qquad \\mathrm{(C)}\\ 32\\qquad \\mathrm{(D)}\\ 36\\qquad \\mathrm{(E)}\\ 48$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Twenty percent less than 60 is $\\frac 45 \\cdot 60 = 48$. One-third more than a number n is $\\frac 43n$. Therefore $\\frac 43n = 48$ and the number is $\\boxed {36}$. The answer is $\\mathrm{(D)}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2002_AMC_12A_Problems_21", "eval_func": null }, "input": { "text": "Consider the sequence of numbers: $4,7,1,8,9,7,6,\\dots$ For $n>2$, the $n$-th term of the sequence is the units digit of the sum of the two previous terms. Let $S_n$ denote the sum of the first $n$ terms of this sequence. The smallest value of $n$ for which $S_n>10,000$ is: \n$\\text{(A) }1992 \\qquad \\text{(B) }1999 \\qquad \\text{(C) }2001 \\qquad \\text{(D) }2002 \\qquad \\text{(E) }2004$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The sequence is infinite. As there are only $100$ pairs of digits, sooner or later a pair of consecutive digits will occur for the second time. As each next digit only depends on the previous two, from this point on the sequence will be periodic.\n(Additionally, as every two consecutive digits uniquely determine the previous one as well, the first pair of digits that will occur twice must be the first pair $4,7$.)\nHence it is a good idea to find the period. Writing down more terms of the sequence, we get:\n\\[4,7,1,8,9,7,6,3,9,2,1,3,4,7,\\dots\\]\nand we found the period. The length of the period is $12$, and its sum is $4+7+\\cdots+1+3 = 60$. Hence for each $k$ we have $S_{12k} = 60k$.\nWe have $\\lfloor 10000/60 \\rfloor = 166$ and $166\\cdot 12 = 1992$, therefore $S_{1992} = 60\\cdot 166 = 9960$.\nThe rest can now be computed by hand, we get $S_{1998} = 9960+4+7+1+8+9+7= 9996$, and $S_{1999}=9996 + 6 = 10002$, thus the answer is $\\boxed{\\text{(B) }1999}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2006_AMC_12B_Problems_22", "eval_func": null }, "input": { "text": "Suppose $a$, $b$ and $c$ are positive integers with $a+b+c=2006$, and $a!b!c!=m\\cdot 10^n$, where $m$ and $n$ are integers and $m$ is not divisible by $10$. What is the smallest possible value of $n$?\n$\\mathrm{(A)}\\ 489 \\qquad \\mathrm{(B)}\\ 492 \\qquad \\mathrm{(C)}\\ 495 \\qquad \\mathrm{(D)}\\ 498 \\qquad \\mathrm{(E)}\\ 501$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The power of $10$ for any factorial is given by the well-known algorithm \n\\[\\left\\lfloor \\frac n{5}\\right\\rfloor + \\left\\lfloor \\frac n{25}\\right\\rfloor + \\left\\lfloor \\frac n{125}\\right\\rfloor + \\cdots\\]\nIt is rational to guess numbers right before powers of $5$ because we won't have any extra numbers from higher powers of $5$. As we list out the powers of 5, it is clear that $5^{4}=625$ is less than 2006 and $5^{5}=3125$ is greater. Therefore, set $a$ and $b$ to be 624. Thus, c is $2006-(624\\cdot 2)=758$. Applying the algorithm, we see that our answer is $152+152+188= \\boxed{492}$.", "Clearly, the power of $2$ that divides $n!$ is larger or equal than the power of $5$ which divides\nit. Hence we are trying to minimize the power of $5$ that will divide $a!b!c!$.\nConsider $n! = 1\\cdot 2 \\cdot \\dots \\cdot n$. Each fifth term is divisible by $5$, each $25$-th one\nby $25$, and so on. Hence the total power of $5$ that divides $n$ is $\\left\\lfloor \\frac n{5}\\right\\rfloor + \\left\\lfloor \\frac n{25}\\right\\rfloor + \\cdots$. (For any $n$ only finitely many terms in the sum\nare\nnon-zero.)\nIn our case we have $a<2006$, so the largest power of $5$ that will be less than $a$ is at most\n$5^4 = 625$. Therefore the power of $5$ that divides $a!$ is equal to $\\left\\lfloor \\frac a{5}\\right\\rfloor + \\left\\lfloor \\frac a{25}\\right\\rfloor + \\left\\lfloor \\frac a{125}\\right\\rfloor + \\left\\lfloor \\frac a{625}\\right\\rfloor$. The same\nis true for $b$ and $c$.\nIntuition may now try to lure us to split $2006$ into $a+b+c$ as evenly as possible, giving\n$a=b=669$ and $c=668$. However, this solution is not optimal.\nTo see how we can do better, let's rearrange the terms as follows:\n\\begin{align*} \\text{result} & = \\Big\\lfloor \\frac a{5}\\Big\\rfloor + \\Big\\lfloor \\frac b{5}\\Big\\rfloor + \\Big\\lfloor \\frac c{5}\\Big\\rfloor \\\\ & + \\Big\\lfloor \\frac a{25}\\Big\\rfloor + \\Big\\lfloor \\frac b{25}\\Big\\rfloor + \\Big\\lfloor \\frac c{25}\\Big\\rfloor \\\\ & + \\Big\\lfloor \\frac a{125}\\Big\\rfloor + \\Big\\lfloor \\frac b{125}\\Big\\rfloor + \\Big\\lfloor \\frac c{125}\\Big\\rfloor \\\\ & + \\Big\\lfloor \\frac a{625}\\Big\\rfloor + \\Big\\lfloor \\frac b{625}\\Big\\rfloor + \\Big\\lfloor \\frac c{625}\\Big\\rfloor \\end{align*}\nThe idea is that the rows of the above equation are roughly equal to $\\left\\lfloor \\frac n{5}\\right\\rfloor$, $\\left\\lfloor \\frac n{25}\\right\\rfloor$, etc.\nMore precisely, we can now notice that for any positive integers $a,b,c,k$ we can write $a,b,c$ in the form $a=a_0k + a_1$, $b=b_0k+b_1$, $c=c_0k + c_1$, where all $a_i,b_i,c_i$ are integers and $0\\leq a_1,b_1,c_1 b$. Representing this as an inequality, we have\n\n$27 \\le 27(c-b) \\le 243$.\n\nSubstituting $2k+1$, we have\n\n$27 \\le 2k+1 \\le 243$\n$\\Rightarrow 13 \\le k \\le 121$.\n\nHowever, we know that $abc$ is a $3$-digit number, and since $k^2$ is less than $abc$, $k^2$ must be at most $961$, or $31^2$. Therefore $k \\le 31$. Plugging this back into our inequality, we have\n\n$13 \\le k \\le 31$\n$\\Rightarrow 27 \\le 2k+1 \\le63$\n$\\Rightarrow 27 \\le 27(c-b) \\le 63$\n$\\Rightarrow 1 \\le (c-b) \\le \\frac{7}{3}$.\n\nBut (c-b) must be an integer, so now we have\n\n$1 \\le (c-b) \\le 2$\n$\\Rightarrow 27 \\le 27(c-b) \\le 54$\n$\\Rightarrow 27 \\le 2k+1 \\le 54$\n$\\Rightarrow 13 \\le k \\le\\frac{53}{2}$\n\n$k$ is also an integer, so now we have\n\n$\\Rightarrow 13 \\le k \\le 26$\n$\\Rightarrow 27 \\le 2k+1 \\le 53$\n$\\Rightarrow 27 \\le 27(c-b) \\le 53$\n$\\Rightarrow 1 \\le (c-b) \\le \\frac{53}{27}$.\n\nOnce again, $(c-b)$ must be an integer, so we have\n$1 \\le (c-b) \\le 1$\n$\\Rightarrow (c-b) = 1$\n$\\Rightarrow 27(c-b) = 27$\n$\\Rightarrow 2k+1 = 27$\n$\\Rightarrow k = 13$\n\nThe two squares are $13^2$ and $14^2$, or $169$ and $196$. A third of the distance between them is $9$, and $169 +9 = 178$. $1 + 7 + 8 = 16 \\Rightarrow \\boxed{\\text{C}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2020_AMC_12B_Problems_14", "eval_func": null }, "input": { "text": "Bela and Jenn play the following game on the closed interval $[0, n]$ of the real number line, where $n$ is a fixed integer greater than $4$. They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval $[0, n]$. Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?\n$\\textbf{(A)} \\text{ Bela will always win.} \\qquad \\textbf{(B)} \\text{ Jenn will always win.} \\qquad \\textbf{(C)} \\text{ Bela will win if and only if }n \\text{ is odd.} \\\\ \\textbf{(D)} \\text{ Jenn will win if and only if }n \\text{ is odd.} \\qquad \\textbf{(E)} \\text { Jenn will win if and only if } n>8.$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "If Bela selects the middle number in the range $[0, n]$ and then mirror whatever number Jenn selects, then if Jenn can select a number within the range, so can Bela. Jenn will always be the first person to run out of a number to choose, so the answer is $\\boxed{\\textbf{(A)} \\text{ Bela will always win.}}$", "First of all, realize that the parity of $n$ likely has no effect on the strategy, since Bela obviously wins when $n=1$, and it's not hard to find a winning strategy for $n=2$ or $n=3$ (start with $n/2$). Similarly, there is no reason the strategy would change when $n>8.$\nSo we are left with $\\textbf{(A)}$ and $\\textbf{(B)}.$ From here it is best to try out random numbers and try to find the strategy that will let Bela win, but if you can't find it, realize that it is more likely the answer is $\\boxed{\\textbf{(A)} \\text{ Bela will always win}}$ since Bela has the first move and thus has more control.", "We can just play the game. We can draw a number line to $9$ and assume Bela and Jenn will only play the integer values. From now after playing one round of the game, Bela will win so answers $\\textbf{(B)},\\textbf{(D)},$ and $\\textbf{(E)}$ are eliminated. Now we want to test if Bela will win if $n$ is even so we can draw another number line this time up to $10$. Now after playing the game, we can find that Bela won yet again so the answer is $\\boxed{\\textbf{(A)} \\text{ Bela will always win}}.$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2023_AMC_12A_Problems_23", "eval_func": null }, "input": { "text": "How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation\n\\[(1+2a)(2+2b)(2a+b) = 32ab?\\]\n$\\textbf{(A) }0\\qquad\\textbf{(B) }1\\qquad\\textbf{(C) }2\\qquad\\textbf{(D) }3\\qquad\\textbf{(E) }\\text{an infinite number}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Using AM-GM on the two terms in each factor on the left, we get\n\\[(1+2a)(2+2b)(2a+b) \\ge 8\\sqrt{2a \\cdot 4b \\cdot 2ab}= 32ab,\\]\nmeaning the equality condition must be satisfied. This means $1 = 2a = b$, so we only have $\\boxed{1}$ solution.", "Equation $(1+2a)(2+2b)(2a+b)=32ab$ is equivalent to\n\\[b(2a-1)^2+2a(b-1)^2+(2a-b)^2=0,\\]\nwhere $a$, $b>0$. Therefore $2a-1=b-1=2a-b=0$, so $(a,b)=\\left(\\tfrac12,1\\right)$. Hence the answer is $\\boxed{\\textbf{(B) }1}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2021_AMC_12A_Problems_2", "eval_func": null }, "input": { "text": "Under what conditions is $\\sqrt{a^2+b^2}=a+b$ true, where $a$ and $b$ are real numbers?\n$\\textbf{(A) }$ It is never true.\n$\\textbf{(B) }$ It is true if and only if $ab=0$.\n$\\textbf{(C) }$ It is true if and only if $a+b\\ge 0$.\n$\\textbf{(D) }$ It is true if and only if $ab=0$ and $a+b\\ge 0$.\n$\\textbf{(E) }$ It is always true.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "One can square both sides to get $a^{2}+b^{2}=a^{2}+2ab+b^{2}$. Then, $0=2ab\\implies ab=0$. Also, it is clear that both sides of the equation must be nonnegative. The answer is $\\boxed{\\textbf{(D)}}$.", "Complete the square of the left side by rewriting the radical to be \\[\\sqrt{(a+b)^{2}-2ab}.\\]\nFrom there it is evident for the square root of the left to be equal to the right, $ab$ must be equal to zero.\nAlso, we know that the equivalency of square root values only holds true for nonnegative values of $a+b$, making the correct answer $\\boxed{\\textbf{(D)}}.$", "The left side of the original equation is the arithmetic square root, which is always nonnegative. So, we need $a+b\\ge 0,$ which refutes $\\textbf{(B)}$ and $\\textbf{(E)}.$ Next, picking $(a,b)=(0,0)$ refutes $\\textbf{(A)},$ and picking $(a,b)=(1,2)$ refutes $\\textbf{(C)}.$ By POE (Process of Elimination), the answer is $\\boxed{\\textbf{(D)}}.$", "If we graph $\\sqrt{x^2+y^2}=x+y,$ then we get the union of:\nTherefore, the answer is $\\boxed{\\textbf{(D)}}.$\nThe graph of $\\sqrt{x^2+y^2}=x+y$ is shown below.\n[asy] /* Made by MRENTHUSIASM */ size(200); int xMin = -5; int xMax = 5; int yMin = -5; int yMax = 5; draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5)); draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5)); label(\"$x$\",(xMax,0),(2,0)); label(\"$y$\",(0,yMax),(0,2)); draw((xMax,0)--(0,0)--(0,yMax),red+linewidth(1.5)); [/asy]" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2008_AMC_12A_Problems_3", "eval_func": null }, "input": { "text": "Suppose that $\\tfrac{2}{3}$ of $10$ bananas are worth as much as $8$ oranges. How many oranges are worth as much as $\\tfrac{1}{2}$ of $5$ bananas?\n$\\mathrm{(A)}\\ 2\\qquad\\mathrm{(B)}\\ \\frac{5}{2}\\qquad\\mathrm{(C)}\\ 3\\qquad\\mathrm{(D)}\\ \\frac{7}{2}\\qquad\\mathrm{(E)}\\ 4$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "If $\\frac{2}{3}\\cdot10\\ \\text{bananas}=8\\ \\text{oranges}$, then $\\frac{1}{2}\\cdot5\\ \\text{bananas}=\\left(\\frac{1}{2}\\cdot 5\\ \\text{bananas}\\right)\\cdot\\left(\\frac{8\\ \\text{oranges}}{\\frac{2}{3}\\cdot10\\ \\text{bananas}}\\right)=3\\ \\text{oranges}\\Longrightarrow\\mathrm{(C)}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2006_AMC_12B_Problems_10", "eval_func": null }, "input": { "text": "In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?\n$\\text {(A) } 43 \\qquad \\text {(B) } 44 \\qquad \\text {(C) } 45 \\qquad \\text {(D) } 46 \\qquad \\text {(E) } 47$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "If the second size has length x, then the first side has length 3x, and we have the third side which has length 15. By the triangle inequality, we have: \\[\\\\ x+15>3x \\Rightarrow 2x<15 \\Rightarrow x<7.5\\] Now, since we want the greatest perimeter, we want the greatest integer x, and if $x<7.5$ then $x=7$. Then, the first side has length $3*7=21$, the second side has length $7$, the third side has length $15$, and so the perimeter is $21+7+15=43 \\Rightarrow \\boxed{\\text {(A)}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2003_AMC_12B_Problems_6", "eval_func": null }, "input": { "text": "The second and fourth terms of a geometric sequence are $2$ and $6$. Which of the following is a possible first term?\n$\\textbf{(A) } -\\sqrt{3} \\qquad\\textbf{(B) } -\\frac{2\\sqrt{3}}{3} \\qquad\\textbf{(C) } -\\frac{\\sqrt{3}}{3} \\qquad\\textbf{(D) } \\sqrt{3} \\qquad\\textbf{(E) } 3$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let the first term be $a$ and the common ratio be $r$. Therefore, \n\\[ar=2\\ \\ (1) \\qquad \\text{and} \\qquad ar^3=6\\ \\ (2)\\]\nDividing $(2)$ by $(1)$ eliminates the $a$, yielding $r^2=3$, so $r=\\pm\\sqrt{3}$.\nNow, since $ar=2$, $a=\\frac{2}{r}$, so $a=\\frac{2}{\\pm\\sqrt{3}}=\\pm\\frac{2\\sqrt{3}}{3}$.\nWe therefore see that $\\boxed{\\textbf{(B)}\\ -\\frac{2\\sqrt{3}}{3}}$ is a possible first term." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2004_AMC_12B_Problems_11", "eval_func": null }, "input": { "text": "All the students in an algebra class took a $100$-point test. Five students scored $100$, each student scored at least $60$, and the mean score was $76$. What is the smallest possible number of students in the class?\n$\\mathrm{(A)}\\ 10 \\qquad \\mathrm{(B)}\\ 11 \\qquad \\mathrm{(C)}\\ 12 \\qquad \\mathrm{(D)}\\ 13 \\qquad \\mathrm{(E)}\\ 14$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let the number of students be $n\\geq 5$. Then the sum of their scores is at least $5\\cdot 100 + (n-5)\\cdot 60$. At the same time, we need to achieve the mean $76$, which is equivalent to achieving the sum $76n$.\nHence we get a necessary condition on $n$: we must have $5\\cdot 100 + (n-5)\\cdot 60 \\leq 76n$.\nThis can be simplified to $200 \\leq 16n$. The smallest integer $n$ for which this is true is $n=13$.\nTo finish our solution, we now need to find one way how $13$ students could have scored on the test. We have $13\\cdot 76 = 988$ points to divide among them. The five $100$s make $500$, hence we must divide the remaining $488$ points among the other $8$ students. This can be done e.g. by giving $61$ points to each of them. \nHence the smallest possible number of students is $\\boxed{\\mathrm{(D)}\\ 13}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2003_AMC_12B_Problems_19", "eval_func": null }, "input": { "text": "Let $S$ be the set of permutations of the sequence $1,2,3,4,5$ for which the first term is not $1$. A permutation is chosen randomly from $S$. The probability that the second term is $2$, in lowest terms, is $a/b$. What is $a+b$? \n$\\mathrm{(A)}\\ 5 \\qquad\\mathrm{(B)}\\ 6 \\qquad\\mathrm{(C)}\\ 11 \\qquad\\mathrm{(D)}\\ 16 \\qquad\\mathrm{(E)}\\ 19$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "There are $4$ choices for the first element of $S$, and for each of these choices there are $4!$ ways to arrange the remaining elements. If the second element must be $2$, then there are only $3$ choices for the first element and $3!$ ways to arrange the remaining elements. Hence the answer is $\\frac{3 \\cdot 3!}{4 \\cdot 4!} = \\frac {18}{96} = \\frac{3}{16}$, and $a+b=19 \\Rightarrow \\mathrm{(E)}$.", "There is a $\\frac {1}{4}$ chance that the number $1$ is the second term. Let $x$ be the chance that $2$ will be the second term. Since $3, 4,$ and $5$ are in similar situations as $2$, this becomes $\\frac {1}{4} + 4x = 1$\nSolving for $x$, we find it equals $\\frac {3}{16}$, therefore $3 + 16 = 19 \\Rightarrow \\mathrm{(E)}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2020_AMC_12B_Problems_20", "eval_func": null }, "input": { "text": "Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance?\n$\\textbf{(A)}\\ \\frac{9}{64} \\qquad\\textbf{(B)}\\ \\frac{289}{2048} \\qquad\\textbf{(C)}\\ \\frac{73}{512} \\qquad\\textbf{(D)}\\ \\frac{147}{1024} \\qquad\\textbf{(E)}\\ \\frac{589}{4096}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Define two ways of painting to be in the same $class$ if one can be rotated to form the other.\nWe can count the number of ways of painting for each specific $class$.\nCase 1: Black-white color distribution is 0-6 (out of 6 total faces)\nTrivially $1^2 = 1$ way to paint the cubes.\nCase 2: Black-white color distribution is 1-5\nTrivially all $\\dbinom{6}{5} = 6$ ways belong to the same $class$, so $6^2$ ways to paint the cubes.\nCase 3: Black-white color distribution is 2-4\nThere are two $classes$ for this case: the $class$ where the two red faces are touching and the other $class$ where the two red faces are on opposite faces. There are $3$ members of the latter $class$ since there are $3$ unordered pairs of $2$ opposite faces of a cube. Thus, there are $\\dbinom{6}{4} - 3 = 12$ members of the former $class$. Thus, $12^2 + 3^2$ ways to paint the cubes for this case.\nCase 4: Black-white color distribution is 3-3\nBy simple intuition, there are also two $classes$ for this case, the $class$ where the three red faces meet at a single vertex, and the other class where the three red faces are in a \"straight line\" along the edges of the cube. Note that since there are $8$ vertices in a cube, there are $8$ members of the former class and $\\dbinom{6}{3} - 8 = 12$ members of the latter class. Thus, $12^2 + 8^2$ ways to paint the cubes for this case.\nNote that by symmetry (since we are only switching the colors), the number of ways to paint the cubes for black-white color distributions 4-2, 5-1, and 6-0 is 2-4, 1-5, and 0-6 (respectively).\nThus, our total answer is\\[\\frac{2(6^2 + 1^2 + 12^2 + 3^2) + 12^2 + 8^2}{2^{12}} = \\frac{588}{4096} = \\boxed{\\textbf{(D) } \\frac{147}{1024}}.\\]\n" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2002_AMC_12A_Problems_2", "eval_func": null }, "input": { "text": "Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly?\n$\\textbf{(A) } 15\\qquad \\textbf{(B) } 34\\qquad \\textbf{(C) } 43\\qquad \\textbf{(D) } 51\\qquad \\textbf{(E) } 138$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We work backwards; the number that Cindy started with is $3(43)+9=138$. Now, the correct result is $\\frac{138-3}{9}=\\frac{135}{9}=15$. Our answer is $\\boxed{\\textbf{(A) }15}$.", "Let the number be $x$. We transform the problem into an equation: $\\frac{x-9}{3}=43$. Solve for $x$ gives us $x=138$. Therefore, the correct result is $\\frac{138-3}{9}=\\frac{135}{9}=\\boxed{\\textbf{(A) }15}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2011_AMC_12B_Problems_25", "eval_func": null }, "input": { "text": "For every $m$ and $k$ integers with $k$ odd, denote by $\\left[\\frac{m}{k}\\right]$ the integer closest to $\\frac{m}{k}$. For every odd integer $k$, let $P(k)$ be the probability that\n\\[\\left[\\frac{n}{k}\\right] + \\left[\\frac{100 - n}{k}\\right] = \\left[\\frac{100}{k}\\right]\\]\nfor an integer $n$ randomly chosen from the interval $1 \\leq n \\leq 99!$. What is the minimum possible value of $P(k)$ over the odd integers $k$ in the interval $1 \\leq k \\leq 99$?\n$\\textbf{(A)}\\ \\frac{1}{2} \\qquad \\textbf{(B)}\\ \\frac{50}{99} \\qquad \\textbf{(C)}\\ \\frac{44}{87} \\qquad \\textbf{(D)}\\ \\frac{34}{67} \\qquad \\textbf{(E)}\\ \\frac{7}{13}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Answer: $(D) \\frac{34}{67}$\n\nFirst of all, you have to realize that \nif $\\left[\\frac{n}{k}\\right] + \\left[\\frac{100 - n}{k}\\right] = \\left[\\frac{100}{k}\\right]$\nthen $\\left[\\frac{n - k}{k}\\right] + \\left[\\frac{100 - (n - k)}{k}\\right] = \\left[\\frac{100}{k}\\right]$\nSo, we can consider what happen in $1\\le n \\le k$ and it will repeat. Also since range of $n$ is $1$ to $99!$, it is always a multiple of $k$. So we can just consider $P(k)$ for $1\\le n \\le k$.\n\nLet $\\text{fpart}(x)$ be the fractional part function\nThis is an AMC exam, so use the given choices wisely. With the given choices, and the previous explanation, we only need to consider $k = 99$, $87$, $67$, $13$. $1\\le n \\le k$\n\nFor $k > \\frac{200}{3}$, $\\left[\\frac{100}{k}\\right] = 1$. 3 of the $k$ that we should consider land in here.\nFor $n < \\frac{k}{2}$, $\\left[\\frac{n}{k}\\right] = 0$, then we need $\\left[\\frac{100 - n}{k}\\right] = 1$\nelse for $\\frac{k}{2}< n < k$, $\\left[\\frac{n}{k}\\right] = 1$, then we need $\\left[\\frac{100 - n}{k}\\right] = 0$\n\nFor $n < \\frac{k}{2}$, $\\left[\\frac{100 - n}{k}\\right] = \\left[\\frac{100}{k} - \\frac{n}{k}\\right]= 1$\nSo, for the condition to be true, $100 - n > \\frac{k}{2}$ . ( $k > \\frac{200}{3}$, no worry for the rounding to be $> 1$)\n$100 > k > \\frac{k}{2} + n$, so this is always true.\nFor $\\frac{k}{2}< n < k$, $\\left[\\frac{100 - n}{k}\\right] = 0$, so we want $100 - n < \\frac{k}{2}$, or $100 < \\frac{k}{2} + n$\n$100 <\\frac{k}{2} + n < \\frac{3k}{2}$\nFor k = 67, $67 > n > 100 - \\frac{67}{2} = 66.5$\nFor k = 69, $69 > n > 100 - \\frac{69}{2} = 67.5$\netc.\n\nWe can clearly see that for this case, $k = 67$ has the minimum $P(k)$, which is $\\frac{34}{67}$. Also, $\\frac{7}{13} > \\frac{34}{67}$ .\nSo for AMC purpose, answer is $\\boxed{\\textbf{(D) }\\frac{34}{67}}$.", "It suffices to consider $0\\le n < k-1.$ Now for each of these $n,$ let $f(n)=\\left[\\frac{n}{k}\\right], g(n)=\\left[\\frac{100}{k}\\right]-\\left[\\frac{100-n}{k}\\right].$ If we let $k=67,$ then the following graphs result for $f$ and $g.$\n$f:$\n[asy] size(10cm); for (int i = 0; i < 67; ++i) { \tif (i<=33) dot((i,0)); else dot((i,1)); } label(\"(0,0)\",(0,0),SW); label(\"(66,1)\",(66,1),NE); [/asy]\n$g:$\n[asy] size(10cm); for (int i = 0; i < 67; ++i) { \tdot((i,0)); } label(\"(0,0)\",(0,0),NW); label(\"(66,0)\",(66,0),SE); [/asy]\nOur probability is the number of $0\\le i67$ $87$ and $99$ do not yield better probabilities. Therefore, our answer is $\\boxed{\\frac{34}{67}}.$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2015_IMO_Problems_5", "eval_func": null }, "input": { "text": "Let $\\mathbb{R}$ be the set of real numbers. Determine all functions $f$:$\\mathbb{R}\\rightarrow\\mathbb{R}$ satisfying the equation\n$f(x+f(x+y))+f(xy) = x+f(x+y)+yf(x)$\nfor all real numbers $x$ and $y$.\n", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "$f(x+f(x+y)) + f(xy) = x + f(x+y) + yf(x)$ for all real numbers $x$ and $y$.\n(1) Put $x=y=0$ in the equation,\nWe get$f(0 + f(0)) + f(0) = 0 + f(0) + 0$\nor $f(f(0)) = 0$\nLet $f(0) = k$, then $f(k) = 0$\n(2) Put $x=0, y=k$ in the equation,\nWe get $f(0 + f(k)) + f(0) = 0 + f(k) + kf(0)$\nBut $f(k) = 0$ and $f(0) = k$\nso, $f(0) + f(0) = f(0)^2$\nor $f(0)[f(0) - 2] = 0$\nHence $f(0) = 0, 2$\nCase $1$\u00a0: $f(0) = 0$\nPut $x=0, y=x$ in the equation,\nWe get $f(0 + f(x)) + f(0) = 0 + f(x) + xf(0)$\nor, $f(f(x)) = f(x)$\nSay $f(x) = z$, we get $f(z) = z$\nSo, $f(x) = x$ is a solution -- fallacy\nCase $2$\u00a0: $f(0) = 2$\nAgain put $x=0, y=x$ in the equation,\nWe get $f(0 + f(x)) + f(0) = 0 + f(x) + xf(0)$\nor, $f(f(x)) + 2 = f(x) + 2x$\nWe observe that $f(x)$ must be a polynomial of power $1$ as any other power (for that matter, any other function) will make the $LHS$ and $RHS$ of different powers and will not have any non-trivial solutions. -- fallacy\nAlso, if we put $x=0$ in the above equation we get $f(2) = 0$\n$f(x) = 2-x$ satisfies both the above.\nHence, the solutions are $\\boxed{\\color{red}{f(x) = x}}$ and $\\boxed{\\color{red}{f(x) = 2-x}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1975_IMO_Problems_4", "eval_func": null }, "input": { "text": "When $4444^{4444}$ is written in decimal notation, the sum of its digits is $A$. Let $B$ be the sum of the digits of $A$. Find the sum of the digits of $B$. ($A$ and $B$ are written in decimal notation.)", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Note that\n\\[4444^{4444}<10000^{4444}=\\left(10^4\\right)^{4444}=10^{17776}\\]\nTherefore $4444^{4444}$ has fewer than 17776 digits. This shows that $A<9\\cdot 17776=159984$. The sum of the digits of $A$ is then maximized when $A=99999$, so $B\\leq 45$. Note that out of all of the positive integers less than or equal to 45, the maximal sum of the digits is 12.\nIt's not hard to prove that any base-10 number is equivalent to the sum of its digits modulo 9. Therefore $4444^{4444}\\equiv A\\equiv B(\\bmod{9})$. This motivates us to compute $X$, where $1\\leq X \\leq 12$, such that $4444^{4444}\\equiv X(\\bmod{9})$. The easiest way to do this is by searching for a pattern. Note that\n\\[4444^1\\equiv 7(\\bmod 9)\\\\4444^2\\equiv 4(\\bmod 9)\\\\4444^3\\equiv 1(\\bmod 9)\\]\nand since $4444=3\\times 1481+1$,\n\\[4444^{4444}\\equiv 4444^{3\\times1481+1}\\equiv \\left(4444^3\\right)^{1481}\\times 4444\\equiv 1\\times 4444\\equiv 7(\\bmod{9})\\]\nThus, $X=7$, which means that the sum of the digits of $B$ is $\\boxed{7}$.\n" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1992_IMO_Problems_3", "eval_func": null }, "input": { "text": "Consider nine points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $n$ such that whenever exactly n edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We show that for $n = 32$ we can find a coloring without a monochrome triangle.\nTake two squares $R_1R_2R_3R_4$ and $B_1B_2B_3B_4$. Leave the diagonals of each square uncolored, color the remaining edges of $R$ red and the remaining edges of $B$\nblue. Color blue all the edges from the ninth point $X$ to the red square, and red\nall the edges from $X$ to the blue square. Color $R_iB_j$ red if $i$ and $j$ have the same parity and blue otherwise.\nClearly $X$ is not the vertex of a monochrome square, because if $XY$ and $XZ$ are\nthe same color then, $YZ$ is either uncolored or the opposite color. There is no triangle within the red square or the blue square, and hence no monochrome triangle. It remains to consider triangles of the form $R_iR_jB_k$ and $B_iB_jR_k.$ But if $i$ and $j$ have the same parity, then $R_iR_j$ is uncolored (and similarly $B_iB_j$), whereas if\nthey have opposite parity, then $R_iB_k$ and $R_jB_k$ have opposite colors (and similarly $B_iR_k$ and $B_jR_k$).\nIt remains to show that for $n = 33$ we can always find a monochrome triangle.\nThere are three uncolored edges. Take a point on each of the uncolored edges.\nThe edges between the remaining $6$ points must all be colored. Take one of\nthese, $X.$ At least $3$ of the $5$ edges to $X$, say $XA$, $XB$, $XC$ must be the same color\n(say red). If $AB$ is also red, then $XAB$ is monochrome. Similarly, for $BC$ and $CA.$\nBut if $AB$, $BC$ and $CA$ are all blue, then $ABC$ is monochrome." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1965_IMO_Problems_5", "eval_func": null }, "input": { "text": "Consider $\\triangle OAB$ with acute angle $AOB$. Through a point $M \\neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range over (a) the side $AB$, (b) the interior of $\\triangle OAB$?", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $O(0,0),A(a,0),B(b,c)$.\nEquation of the line $AB: y=\\frac{c}{b-a}(x-a)$.\nPoint $M \\in AB\u00a0: M(\\lambda,\\frac{c}{b-a}(\\lambda-a))$.\nEasy, point $P(\\lambda,0)$.\nPoint $Q = OB \\cap MQ$, $MQ \\bot OB$.\nEquation of $OB\u00a0: y=\\frac{c}{b}x$, equation of $MQ\u00a0: y=-\\frac{b}{c}(x-\\lambda)+\\frac{c}{b-a}(\\lambda-a)$.\nSolving: $x_{Q}=\\frac{1}{b^{2}+c^{2}}\\left[b^{2}\\lambda+\\frac{c^{2}(\\lambda-a)b}{b-a}\\right]$.\nEquation of the first altitude: $x=\\frac{1}{b^{2}+c^{2}}\\left[b^{2}\\lambda+\\frac{c^{2}(\\lambda-a)b}{b-a}\\right] \\quad (1)$.\nEquation of the second altitude: $y=-\\frac{b}{c}(x-\\lambda)\\quad\\quad (2)$.\nEliminating $\\lambda$ from (1) and (2):\n\t\\[ac \\cdot x + (b^{2}+c^{2}-ab)y=abc\\]\na line segment $MN , M \\in OA , N \\in OB$.\nSecond question: the locus consists in the $\\triangle OMN$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1997_IMO_Problems_5", "eval_func": null }, "input": { "text": "Find all pairs $(a,b)$ of integers $a,b \\ge 1$ that satisfy the equation\n$a^{b^{2}}=b^{a}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Case 1: $(1 \\le a \\le b)$\n$(a^{b})^{b}=b^{a}$\nLooking at this expression since $b \\ge a$ then $a^{b} \\le b$.\nHere we look at subcase $a>1$ which gives $a^{b}>b$ for all $(1 < a \\le b)$. This contradicts condition $a^{b} \\le b$, and thus $a$ can't be more than one giving the solution of $a=1$ with $b \\ge 1$. So we substitute the value of $a=1$ into the original equation to get $1^{b^2}=b^{1}$ which solves to $b=1$ and our first pair $(a,b)=(1,1)$\n\nCase 2: $(1 \\le b < a)$\n$a^{b^2}=b^{a}$\nsince $a>b$, then $b^{2}1$ and $b^{a-2b^{2}}>0$. This gives $a-2b^{2}>1$\nThis implies that $a>2b^{2}$ for $b>1$\nLet $a=kb^{2}$ with $k \\in \\mathbb{Z} ^{+}$. Since $a>2b^{2}$, then $k \\ge 3$\n$(kb^{2}b^{-2})^{b^{2}}=b^{kb^{2}-2b^{2}}$\n$k^{b^{2}}=b^{(k-2)b^{2}}$\n$k=b^{k-2}$, which gives $b \\ge 2$\nsubcase $k=3$:\n$3=b^{3-2}=b$ and $a=kb^{2}=(3)(3)^{2}=27$. which provides 2nd pair $(a,b)=(27,3)$\nsubcase $k=4$:\n$4=b^{4-2}=b^{2}$, thus $b=2$ and $a=kb^{2}=(4)(2)^{2}=16$. which provides 3rd pair $(a,b)=(16,2)$\nsubcase $k \\ge 5$:\n$k=b^{k-2}$, thus $b=k^{1/(k-2)}$ which decreases with $k$ and $b \\to 1$ as $k \\to \\infty$ . From subcase $k=4$, we know that $b=2$, thus for subcase $k \\ge 5$, $10$ vertices of odd degree, then its edge set can be partitioned into $k$ paths, and if all its vertices have even degree, then it has an eulerian circuit.\nWe have a bipartite graph with bipartition $A,B$ here, constructed as follows: the horizontal lines which contain points from our set represent the vertices in $A$, the vertical lines represent the vertices in $B$, and the point represent the edges between the vertices corresponding to the rows and columns which contain them. What we must prove is that we can color the edge set of a bipartite graph in two colors s.t. the difference between the number of red edges and white edges adjacent to each vertex is $\\le 1$ in absolute value. It suffices to prove this for connected bipartite graphs, since then we can apply the result to each connected component of the graph.\nIf all the vertices have even degree, then we can find an eulerian circuit. The graph is bipartite, so this circuit has an even number of edges, and we can thus color the edges in the circuit alternatively red and white so that two edges which are consecutive in the circuit have different colors. It's clear that this coloration satisfies the requirements.\nIf, on the other hand, the graph has $2k>0$ vertices with odd degre, then we partition the edge set into $k$ paths, and in each path we color the edges alternatively red and white. Again, it's easy to verify the required properties of the coloration.\n", "We induct on number of points. The small cases are easily checked. Let there exist such a function for $n$ points. We will show there is a function for $n+1$ points.\nIf there exists a line $\\ell$, parallel to any of the coordinate axes (from the next time, any line will be parallel to either of the coordinate axes, unless otherwise mentioned ), containing odd number of points, then choose a point $P_x \\in \\ell$, and consider $S \\setminus P_x$. By inductive hypothesis there exists such a function $f\u00a0: P \\to \\left \\{ -1, 1 \\right \\}$ for $S \\setminus P_x$. Since $P \\in \\ell, P \\ne P_x$ contain even number of points, we have $\\sum_{\\substack{P \\in \\ell \\\\ P \\ne P_x}} f(P) = 0$. Let $\\ell '$ be the line $\\perp$ to $\\ell$, passing through $P_x$. Let $\\sum _{\\substack{P \\in \\ell ' \\\\ P \\ne P_x}} f(P) = t$, where $t \\in \\left \\{ -1,0 , 1 \\right \\}$. Now for $S$ (with $n+1$ points) define $g$ as $g(P) = f(P)$ for $P \\in S \\setminus P_x$, and $g(P_x) = -t$, if $t \\ne 0$, or $1$ or $-1$ if $t=0$. It indeed works as such a function for $n+1$ points.\nIf the above is not the case, i.e. if all the lines contain even number (greater than zero) of points, pick up an arbitrary point $P_y \\in S$. Let $\\ell$ and $\\ell '$ be the two lines containing $P_y$. Also $\\ell \\perp \\ell '$. Consider $S \\setminus P_y$, for this set , by the inductive hypothesis, there exists such a function $f$. In $S \\setminus P_y$ , $\\ell$ and $\\ell '$ contains odd number of points and if $L$ is a line different from $\\ell$ and $\\ell '$, then it contains even number of points. So $\\sum_{\\substack{P \\in L \\\\ L \\ne \\ell, \\ell '}} f(P) = 0$. Therefore it is seen that\n$\\sum_{\\substack{P \\in \\ell \\\\ P \\ne P_y}} f(P)$ $= - \\sum _{\\substack{P \\in S \\\\ P \\notin \\ell , \\ell ' }} f(P)$ $= \\sum_{\\substack{P \\in \\ell ' \\\\ P \\ne P_y}} f(P) = t$.\nSince $\\ell \\setminus P_y$ contain odd number of points, $t$ is either $-1$ or $1$. Wlog $t = 1$. Now for $S$, (containing $n+1$ points) define $g$ as\u00a0: $g(P) = f(P)$, for $P \\in S \\setminus P_y$ and $g(P_y) = -t$.\nSo the induction is complete, and the statement is established.\n" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1961_IMO_Problems_3", "eval_func": null }, "input": { "text": "Solve the equation \n$\\cos^n{x} - \\sin^n{x} = 1$\nwhere $n$ is a given positive integer.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Since $cos^2x + sin^2x = 1$, we cannot have solutions with $n\\ne2$ and $0<|cos(x)|,|sin(x)|<1$. Nor can we have solutions with $n=2$, because the sign is wrong. So the only solutions have $sin (x) = 0$ or $cos (x) = 0$, and these are: $x =$ multiple of $\\pi$, and $n$ even; $x$ even multiple of $\\pi$ and $n$ odd; $x$ = even multiple of $\\pi + \\frac{3\\pi}{2}$ and $n$ odd." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1990_IMO_Problems_1", "eval_func": null }, "input": { "text": "Chords $AB$ and $CD$ of a circle intersect at a point $E$ inside the circle. Let $M$ be an interior point of the segment $\\overline{EB}$. The tangent line at $E$ to the circle through $D, E$, and $M$ intersects the lines $\\overline{BC}$ and ${AC}$ at $F$ and $G$, respectively.\nIf $\\frac{AM}{AB} = t$, find $\\frac{EG}{EF}$ in terms of $t$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "With simple angle chasing, we find that triangles $CEG$ and $MDB$ are similar.\nso, $GE/EC = MD/MB$. (*)\nagain with simple angle chasing, we find that triangles $CEF$ and $AMD$ are similar.\nso, $EF/DM = CE/AM$. (**)\nso, by (*) and (**), we have $GE/EF = MA/MB = t/(t-1)$.\n", "This problem can be bashed with PoP and Ratio Lemma. Rewriting the given ratio gets $\\frac{MA}{MB}=\\frac{t}{1-t}$.\nBy Ratio Lemma, $\\frac{FB}{FC}=\\frac{BE}{CE} \\cdot \\frac{\\sin{\\angle{FEB}}}{\\sin{\\angle{FEC}}}=\\frac{DE}{AE} \\cdot \\frac{\\sin{\\angle{EDM}}}{\\sin{\\angle{DME}}}=\\frac{DE}{AE} \\cdot \\frac{EM}{DE}=\\frac{ME}{EA}$. Similarly, $\\frac{GA}{GC}=\\frac{ME}{EB}$. We can rewrite these equalities to get $\\frac{AM}{EM}=\\frac{BC}{FB}$ and $\\frac{BM}{EM}=\\frac{AC}{GC}$.\nUsing Ratio Lemma, $\\frac{GE}{\\sin{\\angle{ACD}}}=\\frac{GC}{\\sin{\\angle{GED}}}$ and $\\frac{EF}{\\sin{\\angle{BCD}}}=\\frac{FC}{\\sin{\\angle{FEC}}}$. Since $\\angle{GED}=\\angle{FEC}$, we have $\\frac{FE}{GE}=\\frac{FC}{GC} \\cdot \\frac{\\sin{\\angle{BCD}}}{\\sin{\\angle{ACD}}}$ (eq 1). Note that by Ratio Lemma, $\\frac{\\sin{\\angle{BCD}}}{\\sin{\\angle{ACD}}}=\\frac{CA}{CB} \\cdot \\frac{EB}{EA}$. Plugging this into (eq 1), we get $\\frac{EF}{GE}=\\frac{FC}{GC} \\cdot \\frac{CA}{CB} \\cdot \\frac{EB}{EA}=\\frac{\\frac{EA}{EM} \\cdot FB}{\\frac{EB}{EM} \\cdot GA} \\cdot \\frac{CA}{CB} \\cdot \\frac{EB}{EA}=\\frac{FB}{GA} \\cdot \\frac{CA}{CB}=\\frac{EM}{AM} \\cdot \\frac{MB}{EM}=\\frac{MB}{MA}=\\frac{1-t}{t}$. So $\\frac{EG}{EF}=\\boxed{\\frac{t}{1-t}}$.\n" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1979_IMO_Problems_5", "eval_func": null }, "input": { "text": "Determine all real numbers a for which there exists non-negative reals $x_{1}, \\ldots, x_{5}$ which satisfy the relations $\\sum_{k=1}^{5} kx_{k}=a,$ $\\sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $\\sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $\\Sigma_1= \\sum_{k=1}^{5} kx_{k}$, $\\Sigma_2=\\sum_{k=1}^{5} k^{3}x_{k}$ and $\\Sigma_3=\\sum_{k=1}^{5} k^{5}x_{k}$. For all pairs $i,j\\in \\mathbb{Z}$, let\\[\\Sigma(i,j)=i^2j^2\\Sigma_1-(i^2+j^2)\\Sigma_2+\\Sigma_3\\]Then we have on one hand\\[\\Sigma(i,j)=i^2j^2\\Sigma_1-(i^2+j^2)\\Sigma_2+\\Sigma_3=\\sum_{k=1}^5(i^2j^2k-(i^2+j^2)k^3+k^5)x_k =\\sum_{k=1}^5k(i^2j^2-(i^2+j^2)k^2+k^4)x_k\\]Therefore \\\\(1)\\[\\Sigma(i,j)=\\sum_{k=1}^5k(k^2-i^2)(k^2-j^2)x_k\\]and on the other hand \\\\ (2)\\[\\Sigma(i,j)=i^2j^2a-(i^2+j^2)a^2+a^3=a(a-i^2)(a-j^2)\\]Then from (1) we have\\[\\Sigma(0,5)=\\sum_{k=1}^5k^3(k^2-5^2)x_k\\leq 0\\]and from (2)\\[\\Sigma(0,5)=a^2(a-25)\\]so $a\\in [0,25]$ Besides we also have from (1)\\[\\Sigma(0,1)=\\sum_{k=1}^5k^3(k^2-1)x_k\\geq 0\\]and from (2)\\[\\Sigma(0,1)=a^2(a-1)\\geq 0 \\implies a\\notin (0,1)\\]and for $n=1,2,3,4$\\[\\Sigma(n,n+1)=\\sum_{k=1}^5k(k^2-n^2)(k^2-(n+1)^2)x_k\\]where in the right hand we have that\\[k0\\],\\[k=n,n+1 , \\implies (k^2-n^2)(k^2-(n+1)^2)=0\\]and\\[k>n \\implies (k^2-n^2)(k^2-(n+1)^2)>0\\], so\\[\\Sigma(n,n+1)\\geq 0\\]for $n=1,2,3,4$ From the latter and (2) we also have\\[\\Sigma(n,n+1)=a(a-n^2)(a-(n+1)^2))\\geq 0\\implies a\\notin (n^2,(n+1)^2)\\]So we have that\\[a\\in [0,25]-\\bigcup_{n=0}^4(n^2,(n+1^2))=\\{0,1,4,9,16,25\\}\\]\nIf $a=k^2$, $k=0,1,2,3,4,5$ take $x_k=k$, $x_j=0$ for $j\\neq k$. Then $\\Sigma_1=k^2=a$, $\\Sigma_2=k^3k=k^4=a^2$, and $\\Sigma_3=k^5k=k^6=a^3$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1968_IMO_Problems_2", "eval_func": null }, "input": { "text": "Find all natural numbers $x$ such that the product of their digits (in decimal notation) is equal to $x^2 - 10x - 22$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let the decimal expansion of $x$ be $\\overline{d_1d_2d_3\\dots d_n}$, where $d_i$ are base-10 digits. We then have that $x\\geq d_1\\cdot 10^{n-1}$. However, the product of the digits of $x$ is $d_1d_2d_3\\dots d_n\\leq d_1\\cdot 10\\cdot 10\\dots 10=d_1\\cdot 10^{n-1}$, with equality only when $x$ is a one-digit integer. Therefore the product of the digits of $x$ is always at most $x$, with equality only when $x$ is a base-10 digit. This implies that $x^2-10x-22\\leq x$, so $x^2-11x-22\\leq 0$. Every natural number from 1 to 12 satisfies this inequality, so we only need to check these possibilities. It is easy to rule out 1 through 11, since $x^2-10x-22<0$ for those values. However, $12^2-10\\cdot 12-22=2$, which is the product of the digits of 12. Therefore $\\boxed{12}$ is the only natural number with the desired properties. $\\blacksquare$", "It is pretty obvious that $x$ cannot be three digits or more, because then $x^2 - 10x - 22$ is way too big.\nWrite $x = 10a + b$ where $a$ and $b$ are digits satisfying $0 \\leq a, b < 10$. Then, we can use SFFT:\n\\[(10a + b)^2 - 10(10a + b) - 22 = ab\\]\n\\[(10a + b)^2 - 10(10a + b) - 24 = ab - 2\\]\n\\[(10a + b + 2)(10a + b - 12) = ab - 2.\\]\nWe have\n\\[(10a + b + 2)(10a + b - 12) \\geq (10a + 2)(10a - 12) = 100a^2 - 100a + 24 = 100(a^2 - a) + 24.\\]\nIt is therefore clear that $a$ must be either $0$ or $1$. We can then split into two cases:\n$\\mathbf{a = 0:}$\nWe have $(b + 2)(b - 12) = -2$ or $b^2 - 10b - 22 = 0$, which is only satisfied when $b = -2$ or $12$.\n$\\mathbf{a = 1:}$\nWe have $(b + 12)(b - 2) = b - 2$. This is only satisfied when $b = 2$, or $b + 12 = 0$. Therefore, $b = 2$, and so $x = \\boxed{12}.\\square$", "Let, \n$x^2-10x-22=y$\n$\\implies x^2-10+25-47=y$\n$\\implies (x-5)^2=47+y$\nNow note that, if $p$ is a prime such that $p|y$ then $7\\geq p$.\nThat means, $y=2^a*3^b*5^c*7^d$\nBut, $a^2 \\not\\equiv 2 (mod3), a^2 \\not\\equiv 2 (mod5), a^2 \\not\\equiv 5 (mod7)$ which means $3,5,7$ don't divivde $(x-5)^2-47=y.$\nSo, $y=2^a$ and $y+17=2^a+47=(x-5)^2$\nIt is easy to see that $a$ has one solution and that is $2.$( Prove it by contradiction)\nSo, $(x-5)^2=47+2=49$\n$\\implies x=12$\n$\\blacksquare$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1978_IMO_Problems_3", "eval_func": null }, "input": { "text": "Let $0 1)$. On the first day, one medal and $\frac{1}{7}$ of the remaining $m - 1$ medals were awarded. On the second day, two medals and $\frac{1}{7}$ of the now remaining medals were awarded; and so on. On the n-th and last day, the remaining $n$ medals were awarded. How many days did the contest last, and how many medals were awarded altogether?", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "This is not a particularly elegant solution, but if you start from 1 and go all the way in a clever method, by only guessing those that are 1 more than a multiple of 7, you arrive at the answer of 36." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1962_IMO_Problems_4", "eval_func": null }, "input": { "text": "Solve the equation $\\cos^2{x}+\\cos^2{2x}+\\cos^2{3x}=1$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "First, note that we can write the left hand side as a cubic function of $\\cos^2 x$. So there are at most $3$ distinct values of $\\cos^2 x$ that satisfy this equation. Therefore, if we find three values of $x$ that satisfy the equation and produce three different $\\cos^2 x$, then we found all solutions to this cubic equation (without expanding it, which is another viable option). Indeed, we find that $\\frac{\\pi}2$, $\\frac{\\pi}4$, and $\\frac{\\pi}6$ all satisfy the equation, and produce three different values of $\\cos^2 x$, namely $0$, $\\frac12$, and $\\frac34$. So we solve $\\cos^2 x = \\text{each of these}$. Therefore, our solutions are:\n$x = \\frac{(2k+1)\\pi}2,\\, \\frac{(2k+1)\\pi}4,\\, \\frac{(6k+1)\\pi}6,\\, \\frac{(6k+5)\\pi}6 \\quad \\forall k\\in Z$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1963_IMO_Problems_6", "eval_func": null }, "input": { "text": "Five students, $A,B,C,D,E$, took part in a contest. One prediction was that the contestants would finish in the order $ABCDE$. This prediction was very poor. In fact no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order $DAECB$. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We are given that no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so in order $ABCDE$. None of them finished in that order. Also only two of them had their actual positions in $DAECB$. After imposing these two conditions the list of possible outcomes is: \n(1)$CAEBD$,\n(2)$DCAEB$,\n(3)$DCEBA$,\n(4)$EDACB$. \nOne more condition is that two disjoint pairs of students predicted to finish consecutively actually did so. Out of the above four in the list, (1) and (2) have $AE$ as the correctly predicted consecutive finishers(but only 1 pair), (3) has no correctly predicted consecutive finishers. But (4) has 2 disjoint correctly predicted consecutive finishers who are $DA$ and $CB$.\nHence, order is $EDACB$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2023_IMO_Problems_1", "eval_func": null }, "input": { "text": "Determine all composite integers $n>1$ that satisfy the following property: if $d_1,d_2,\\dots,d_k$ are all the positive divisors of $n$ with $1=d_1q$, the divisor is larger than $q$. In the second case, all divisors divisible by $q^2$ are also larger than $q$, and otherwise are of the form $p^x \\cdot q^1$ or $p^x$ for some nonnegative integer $x$. If the divisor is of the form $p^x$, then it is a power of $p$. If it is of the form $p^x \\cdot q^1$, the smallest of these factors is $p^0 \\cdot q^1 = q$. Therefore, (in the case where $2$ or more primes divide $n$) the ordered tuple of divisors is of the form $(1,\\, p,\\, p^2 \\dots,\\, p^a,\\, q \\dots,\\, n)$ for some integer $a\\geq 1$, since after each divisor $p^x$, the next smallest divisor is either $p^{x+1}$ or simply $q$. \nIf $a\\geq 2$, the condition fails. This is because $p^{a-1} \\nmid p^a + q$, since $p^a$ is divisible by $p^{a-1}$, but $q$ is not since it is a prime different from $p$. If $a=1$, then $p^{a-1}=p^0=1$, which does divide $q$. Therefore $a$ must equal $1$ for the condition to be satisfied in this case. However, we know that the ordered list of divisors satisfies $d_i \\cdot d_{k+1-i}=n$, meaning since the first $3$ divisors are $(1, p, q)$, then the last $3$ divisors are $(\\frac{n}{q}, \\frac{n}{p}, n)$, so $(\\frac{n}{q})$ must divide $(\\frac{n}{p} + n)$. But $\\frac{n}{q}$ is divisible by $p$, so $\\frac{n}{p} + n$ must also be divisible by $p$, but since $a=1$ $\\frac{n}{p}$ is and $n$ isn't.\nWhen $n=p^x$, it is easy to verify this works for all primes $p$ and all $x\\geq 2$, since $p^y \\vert p^{y+1} + p^{y+2}$, and the divisors are ordered as ${1,\\, p,\\, p^2\u2026\\, p^x}$. ", "Similar argument as above but restated.\nLet $n = p_1^{e_1} p_2^{d_2} \\dots p_k^{e_k}$, with $p_1 < p_2 < \\dots < p_k$ primes. \nIf $k=1$, it's easy to verify that this property holds, since $p^l \\mid p^{l+1} + p^{l+2}$.\nSuppose $k \\geq 2$. All divisors of $n$ which contain a factor of $p_k$ for $k \\geq 2$ is at least as large as $p_2$. So, the only divisors which are possibly less than $p_2$ are of the form $p_1^{j}$. Also, by construction $p_1$ is less than $p_2$. Putting it all together, this says that the smallest factors are $(1, p_1, \\dots, p_1^j, p_2)$ for some $j \\geq 1$.\nIf $j \\geq 2$, then $p_1^{j-1} \\nmid p_1^j + p_2$, since the LHS has a factor of $p_1$ and the RHS does not since.\nIf $j = 1$, then the largest three factors are $(\\frac{n}{p_2}, \\frac{n}{p_1}, n)$. But $\\frac{n}{p_2}$ and $n$ have a factor of $p_1^{e_1}$, while $\\frac{n}{p_1}$ only has a factor of $p_1^{e_1 - 1}$, so $\\frac{n}{p_2} \\nmid \\frac{n}{p_1} + n$.\nHence the prime powers are the only composite numbers which satisfy the property." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2022_IMO_Problems_2", "eval_func": null }, "input": { "text": "Let $\\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f\u00a0: \\mathbb{R}^+ \\to \\mathbb{R}^+$ such that for each $x \\in \\mathbb{R}^+$, there is exactly one $y \\in \\mathbb{R}^+$ satisfying\n\\[xf (y) + yf (x) \\le 2\\].", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Answer: The unique solution is the function \\( f(x) = \\frac{1}{x} \\) for every \\( x \\in \\mathbb{R}^+ \\). This function clearly satisfies the required property since the expression \\( xf(y) + yf(x) = \\frac{x}{y} + \\frac{y}{x} \\) is greater than 2 for every \\( y \\neq x \\) (directly from AM-GM) and equal to 2 (with equality) for the unique value \\( y = x \\).\nProof: Let's consider a solution based on some ideas we encountered in the preparation classes for the Olympiad, specifically involving auxiliary sets and functions with specific properties.\nThe fact that for every \\( x \\in \\mathbb{R}^+ \\), there exists a unique \\( y \\in \\mathbb{R}^+ \\) that satisfies the equation \\( xf(y) + yf(x) \\leq 2 \\) can be equivalently expressed as follows: there exists a well-defined function \\( g: \\mathbb{R}^+ \\to \\mathbb{R}^+ \\) given by \\( g(x)\u00a0:= y \\), where \\( y \\) is the one mentioned above. The well-definedness of this function is evident due to the existence and uniqueness, and it satisfies the equation \\( P(x): \\quad xf(g(x)) + f(x)g(x) \\leq 2 \\) while applying the same property for \\( x \\mapsto g(x) \\) gives another unique \\( y\u00a0:= g(g(x)) \\) such that \\( g(x)f(y) + yf(g(x)) \\leq 2 \\). Therefore, we have \\( xf(y) + yf(x) > 2 \\) for all \\( y \\neq g(x) \\).\nSince this inequality holds for \\( y = x \\) (from \\( xf(y) + yf(x) > 2 \\)), the uniqueness assumption implies that \\( g(g(x)) = x \\), making \\( g \\) an involution (hence bijective).\nGenerally, working with an involution naturally leads us to consider its fixed points, especially since we aim to show that \\( g(x) = x \\) identically (which holds for the solution \\( f(x) = \\frac{1}{x}\\)). Let's define the set of fixed points of \\( g \\) as \\( \\mathcal{S}\u00a0:= \\{ x \\in \\mathbb{R}^+ \\mid g(x) = x \\} \\) and show that \\( \\mathcal{S} = \\mathbb{R}^+ \\) is the entire domain.\nAssume for a contradiction that some \\( x \\notin \\mathcal{S} \\) is not a fixed point, i.e., \\( x \\neq g(x) \\). Then, the inequality \\( 2xf(x) > 2 \\) (derived from \\( y \\mapsto x \\)) holds, implying \\( f(x) > \\frac{1}{x} \\). Similarly, \\( x \\notin \\mathcal{S} \\) implies \\( g(x) \\notin \\mathcal{S} \\) (otherwise \\( g(x) \\in \\mathcal{S} \\) implies \\( x = g(g(x)) = g(x) \\), a contradiction), leading to \\( f(g(x)) > \\frac{1}{g(x)} \\).\nApplying these inequalities to \\( P(x) \\) gives \\( xf(g(x)) + f(x)g(x) < 2 \\), which is clearly a contradiction as \\( \\frac{x}{g(x)} + \\frac{g(x)}{x} \\geqslant 2 \\), e.g., from the AM-GM inequality. Therefore, we must have \\( x \\in \\mathcal{S} \\) for every \\( x \\in \\mathbb{R}^+ \\), i.e., \\( g(x) = x \\).\nSubstituting this relationship into the original equation, we obtain \\( P(x): \\quad xf(x) + f(x)x \\leq 2 \\implies xf(x) \\leq 1 \\implies f(x) \\leq \\frac{1}{x} \\) for every \\( x \\in \\mathbb{R}^+ \\). Applying \\( yf(y) \\leq 1 \\) to the equation \\( xf(y) + yf(x) > 2 \\) (since \\( g(x) = x \\)) yields \\( f(x) > \\frac{2}{y} - \\frac{x}{y^2} \\), and taking the limit \\( y \\to x \\) from either side results in \\( f(x) \\geq \\frac{1}{x} \\).\nCombining the results, we have \\( f(x) \\leq \\frac{1}{x} \\) and \\( f(x) \\geq \\frac{1}{x} \\), implying \\( f(x) = \\frac{1}{x} \\) as desired. \\(\\blacksquare\\)\n." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2000_IMO_Problems_5", "eval_func": null }, "input": { "text": "Does there exist a positive integer $n$ such that $n$ has exactly 2000 prime divisors and $n$ divides $2^n + 1$?", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $N=2^n+1$. We will assume for the sake of contradiction that $n|N$.\n$2^n+1 \\equiv 0 \\pmod{n} \\Rightarrow 2^n \\equiv -1 \\pmod{n}$. So 2 does not divide $n$, and so $n$ is odd.\nSelect an arbitrary prime factor of $n$ and call it $p$. Let's represent $n$ in the form $p^am$, where $m$ is not divisible by $p$.\nNote that $p$ and $m$ are both odd since $n$ is odd. By repeated applications of Fermat's Little Theorem:\n$N = 2^n+1 = 2^{p^am} + 1 = (2^{p^{a-1}m})^p + 1 \\equiv 2^{p^{a-1}m} + 1$ (mod $p$)\nContinuing in this manner, and inducting on k from 1 to $a$,\n$2^{p^{a-k}m}+1 \\equiv (2^{p^{a-k-1}m})^p + 1$ (mod $p$) $\\equiv 2^{p^{a-k-1}m} + 1$ (mod $p$)\nSo we have $N \\equiv 2^m+1$ (mod $p$)\nSince $p$ is relatively prime to $m$, $N \\equiv 1+1$ (mod $p$) $\\equiv 2$ (mod $p$)\nSince $p$ is odd, $N$ is not divisible by $p$. Hence $N$ is not divisible by $n$. So we have a contradiction, and our original assumption was false, and therefore $N$ is still not divisible by $n$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2008_IMO_Problems_5", "eval_func": null }, "input": { "text": "Let $n$ and $k$ be positive integers with $k \\geq n$ and $k - n$ an even number. Let $2n$ lamps labelled $1$, $2$, ..., $2n$ be given, each of which can be either or . Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on).\nLet $N$ be the number of such sequences consisting of $k$ steps and resulting in the state where lamps $1$ through $n$ are all on, and lamps $n + 1$ through $2n$ are all off.\nLet $M$ be number of such sequences consisting of $k$ steps, resulting in the state where lamps $1$ through $n$ are all on, and lamps $n + 1$ through $2n$ are all off, but where none of the lamps $n + 1$ through $2n$ is ever switched on.\nDetermine $\\frac {N}{M}$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "For convenience, let $A$ denote the set $(1,2,\\ldots n)$ and $B$ the set $(n+1,n+2,\\ldots,2n)$.\nWe can describe each sequences of switching the lamps as a $k$-dimensional vector\n$(a_1, a_2, \\ldots, a_k)$, where $a_i \\in A \\cup B$ signifies which lamp was switched on the $i$-th move for $i=1,2,\\ldots k$. \nLet $\\cal{N}$ consist of those sequences that contain each of the numbers in $A$ an number of times and each of the numbers in $B$ an number of times. Similarly, let $\\cal{M}$ denote the set of those sequences that contain no numbers from $B$ and each of the numbers in $A$ an odd number of times. By definition, $M=|\\cal{M}|$ and $N=|\\cal{N}|$.\nDefine the mapping $f:\\cal{N} \\rightarrow \\cal{M}$ as\n\\[f(a_1, a_2, \\ldots, a_k) = (b_1,b_2,\\ldots b_k)\u00a0: b_i = \\begin{cases} a_i, & \\mbox{ if } a_i \\in A \\\\ a_i-n, & \\mbox{ if } a_i \\in B \\end{cases}\\]\nWhat we want to show now is that each element of $\\cal{M}$ is an image of exactly $2^{k-n}$ elements from $\\cal{N}$, which would imply $N = 2^{k-n}M$ and solve the problem.\nConsider an arbitrary element $y$ of $\\cal{M}$ and let $l_i$ be the number of appearances of the number $i$ in $y$ for $i=1,2,\\ldots n$. Now consider the set of pre-images of $y$, that is $X_y = \\{ x | f(x) = y \\}$.\nIt is easy to see that each element $x\\in X_y$ is derived from $y$ by an number of its $1$-s, $2$-s, and so on, where flipping means changing the number $j\\in A$ to $j+n\\in B$. Since each such set of flippings results in a unique $x$, all we want to count is the number of flippings. We can flip exactly $0, 2, 4,\\ldots$ of the $1$-s, so that results in\n\\[\\binom{l_1}{0} + \\binom{l_1}{2}+\\binom{l_1}{4}+\\cdots = 2^{l_1-1}\\]\nflippings. Combine each of them with the $2^{l_2-1}$, $2^{l_3-1}$, etc. ways of flipping the $2$-s, $3$-s etc. respectively to get the total number of flippings:\n\\[2^{l_1-1}2^{l_2-1}\\cdots2^{l_n-1} = 2^{l_1+l_2+\\cdots+l_n-n} = 2^{k-n}.\\] \nThis shows that $|X_y| = 2^{k-n}$ and the proof is complete." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2023_IMO_Problems_3", "eval_func": null }, "input": { "text": "For each integer $k \\geqslant 2$, determine all infinite sequences of positive integers $a_1, a_2, \\ldots$ for which there exists a polynomial $P$ of the form $P(x)=x^k+c_{k-1} x^{k-1}+\\cdots+c_1 x+c_0$, where $c_0, c_1, \\ldots, c_{k-1}$ are non-negative integers, such that\n\\[P\\left(a_n\\right)=a_{n+1} a_{n+2} \\cdots a_{n+k}\\]\nfor every integer $n \\geqslant 1$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $f(n)$ and $g(j)$ be functions of positive integers $n$ and $j$ respectively.\nLet $a_{n}=a_{1}+f(n)$, then $a_{n+1}=a_{1}+f(n+1)$, and $a_{n+k}=a_{1}+f(n+k)$\nLet $P=\\prod_{j=1}^{k}\\left ( a_{n+j} \\right ) = \\prod_{j=1}^{k}\\left ( a_{n}+g(j)) \\right )$\nIf we want the coefficients of $P(a_{n})$ to be positive, then $g(j)\\geq 0$ for all $j$ which will give the following value for $P$:\n$P=a_{n}^{k}+C_{k-1}a_{n}^{k-1}+...+C_{1}a_{n}+\\prod_{j=1}^{k} g(j) = P(a_{n})$\nThus for every $j$ and $n$ we need the following:\n$a_{n}+g(j)=a_{n+j}=a_{1}+f(n+j)$\nSolving for $g(j)$ we get:\n$g(j)=a_{1}+f(n+j)-a_{n}=a_{1}+f(n+j)-a_{1}-f(n)$\n$g(j)=f(n+j)-f(n)\\geq 0$ for all $n$ and $j$ because $g(j)$ needs to be greater than or equal to zero for all coefficients to be non-negative.\nThis means that $f(n)$ needs to be increasing with $n$, or staying constant, and also with $f(1)=0$ because $a_{1}=a_{1}+f(1)$.\nIn addition, since we need all coefficients to be integer, then all $f(n)$ and $g(j)$ must also be integers. We also need $g(j)$ to not be dependent of $n$, so in the expression $f(n+j)-f(n)$, the $n$ needs to cancel. This mean that the rate of change for $f(n)$ with respect to $n$ needs to be constant. This can only be achieved with $f(n)$ be the equation of a line with slope being either zero or positive integer.\nSo, we set $f(n)$ to be the equation of a line as $f(n)=mn+b$ with $m$ being the slope with a non-negative value and with $b$ the intercept at $n=0$. We know that $f(1)=0$ so $f(1)=m+b=0$ which means that $b=-m$ and our function becomes $f(n)=mn-m=(n-1)m$. Since $f(n)$ needs to be non-negative integer then $m\\geq 0 \\mid m \\in \\mathbb{Z}$ then $f(n)$ is increasing or constant, with $f(1)=0$\nThen, $g(j)=f(n+j)-f(n)=(n+j-1)m-(n-1)m=jm$\nThis gives:\n$\\prod_{j=1}^{k}\\left ( a_{n}+jm \\right )=P(a_{n})=a_{n}^{k}+C_{k-1}a_{n}^{k-1}+...+C_{1}a_{n}+k!m^{k}$\nwith $C_{0}=k!m^{k}$ and coefficients of polynomial $\\geq 0$\nThen, $a_{n}=a_{1}+f(n)$\nWhich provides the solution of all infinite sequences of positive integers as:\n$a_{n}=a_{1}+(n-1)m$, $\\forall m\\geq 0 \\mid m \\in \\mathbb{Z}$ and $a_{1} \\geq 1 \\mid a_{1} \\in \\mathbb{Z}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1983_IMO_Problems_1", "eval_func": null }, "input": { "text": "Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy the conditions: \n(i) $f(xf(y))=yf(x)$ for all $x,y$; \n(ii) $f(x)\\to0$ as $x\\to \\infty$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $x=y=1$ and we have $f(f(1))=f(1)$. Now, let $x=1,y=f(1)$ and we have $f(f(f(1)))=f(1)f(1)\\Rightarrow f(1)=[f(1)]^2$ since $f(1)>0$ we have $f(1)=1$. \nPlug in $y=x$ and we have $f(xf(x))=xf(x)$. If $a=1$ is the only solution to $f(a)=a$ then we have $xf(x)=1\\Rightarrow f(x)=\\frac{1}{x}$. We prove that this is the only function by showing that there does not exist any other $a$:\nSuppose there did exist such an $a\\ne1$. Then, letting $y=a$ in the functional equation yields $f(xa)=af(x)$. Then, letting $x=\\frac{1}{a}$ yields $f(\\frac{1}{a})=\\frac{1}{a}$. Notice that since $a\\ne1$, one of $a,\\frac{1}{a}$ is greater than $1$. Let $b$ equal the one that is greater than $1$. Then, we find similarly (since $f(b)=b$) that $f(xb)=bf(x)$. Putting $x=b$ into the equation, yields $f(b^2)=b^2$. Repeating this process we find that $f(b^{2^k})=b^{2^k}$ for all natural $k$. But, since $b>1$, as $k\\to \\infty$, we have that $b^{2^k}\\to\\infty$ which contradicts the fact that $f(x)\\to0$ as $x\\to \\infty$.", "Let $x=1$ so \\[f(f(y))=yf(1).\\] If $f(a)=f(b),$ then $af(1)=f(f(a))=f(f(b))=bf(1)\\implies a=b$ because $f(1)$ goes to the real positive integers, not $0.$ Hence, $f$ is injective. Let $x=y$ so \n\\[f(xf(x))=xf(x)\\] so $xf(x)$ is a fixed point of $f.$ Then, let $y=1$ so $f(xf(1))=f(x)\\implies f(1)=1$ as $x$ can't be $0$ so $1$ is a fixed point of $f.$ We claim $1$ is the only fixed point of $f.$ Suppose for the sake of contradiction that $a,b$ be fixed points of $f$ so $f(a)=a$ and $f(b)=b.$ Then, setting $x=a,y=b$ in (i) gives \\[f(ab)=f(af(b))=bf(a)=ab\\] so $ab$ is also a fixed point of $f.$ Also, let $x=\\frac{1}{a},y=a$ so \\[1=f(1)=f(\\frac{1}{a}\\cdot a)=f(\\frac{1}{a}\\cdot f(a))=af(\\frac{1}{a})\\implies f(\\frac{1}{a})=\\frac{1}{a}\\] so $\\frac{1}{a}$ is a fixed point of $f.$ If $f(a)=a$ with $a>1,$ then $f(a^n)$ is a fixed point of $f$, contradicting (ii). If $f(a)=a$ with $00$ since average of $2$ positive number is always positive.\n\nThus, assume for contradiction, $\\nexists n$, $f_n=f_{n-1}$.\nThen, $f_{n+1}\\le\\frac{f_{n}+f_{n-1}}{2}< \\max (f_{n},f_{n-1})$, $f_{n+2}\\le\\frac{f_{n+1}+f_{n}}{2}< \\max (f_{n},f_{n+1})\\le\\max (f_{n},f_{n-1})$\nThus, $\\max (f_{n},f_{n-1})> \\max (f_{n+1},f_{n+2})$ and that means that $\\max (f_{2n},f_{2n+1})$ is a strictly decreasing function and it must reach $0$ as $n\\rightarrow\\infty$, which contradict with the fact that $f_n>0$.\n proven.\n\n\n) Show that the constant is $\\gcd(a,b)$.\nFor any $f_n$ where $\\gcd(a,b)=d\\ne1$. $f_n=dg_n$ for $g_n$ with the same property except with $g_1=\\frac{a}{d}$ and $g_2=\\frac{b}{d}$.\nTherefore, if I prove that the constant for any $f_n$ with relatively prime $a$, $b$ is $1$, then I have shown that is true.\n\n) If $\\gcd(f_n,f_{n-1})=1$, then $\\gcd(f_n,f_{n+1})=1$.\nAssume for contradiction that $\\gcd(f_n,f_{n+1})=d\\ne1$, since both $f_n$ and $f_{n+1}$ are odd, $d$ is not divisible by $2$.\n$f_{n+1}=\\frac{f_n+f_{n-1}}{2^n}$ for some $n\\in \\mathbb{Z}^+$ such that $f_{n+1}$ is odd.\n$(2^n)f_{n+1}-f_n=f_{n-1}$\n$d(p+q)=f_{n-1}$, where $p$ and $q$ is another integer.\nThus, $f_{n-1}$ is divisible by $d$ which contradicts with the assumption that $\\gcd(f_n,f_{n-1})=1$.\n\n By induction, $\\gcd(f_n,f_{n-1})=1$ since $\\gcd(a,b)=\\gcd(f_1,f_2)=1$.\nSince there must exist some $n$ where $f_n=f_{n+1}$ (part 1), $\\gcd(f_n,f_{n+1})=f_n=1$.\n$\\mathbb{Q.E.D}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1993_USAMO_Problems_5", "eval_func": null }, "input": { "text": "Let $a_0, a_1, a_2,\\cdots$ be a sequence of positive real numbers satisfying $a_{i-1}a_{i+1}\\le a^2_i$\nfor $i = 1, 2, 3,\\cdots$ . (Such a sequence is said to be .) Show that for\neach $n > 1$,\n\\[\\frac{a_0+\\cdots+a_n}{n+1}\\cdot\\frac{a_1+\\cdots+a_{n-1}}{n-1}\\ge\\frac{a_0+\\cdots+a_{n-1}}{n}\\cdot\\frac{a_1+\\cdots+a_{n}}{n}.\\]", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Notice that because\n\\[(a_0 + a_1 + \\dots + a_{n-1})(a_1 + a_2 + \\dots + a_n) = (a_0 + (a_1 + \\dots + a_{n-1})(-a_0 + (a_0 + a_1 + \\dots + a_n))\\]\n\\[= -a_0^2 + a_0 ((a_0 + a_1 + \\dots + a_n) - (a_1 + a_2 + \\dots + a_{n-1})) + (a_1 + a_2 + \\dots + a_{n-1})(a_0 + a_1 + \\dots + a_n)\\]\n\\[= a_0 a_n + (a_1 + a_2 + \\dots + a_{n-1})(a_0 + a_1 + \\dots + a_n),\\]\nwe may subtract $\\dfrac{(a_1 + a_2 + \\dots + a_{n-1})(a_0 + a_1 + \\dots + a_n)}{n^2}$ from both sides of the inequality and observe that it is sufficient to prove that\n\\[\\frac{(a_1 + a_2 + \\dots + a_{n-1})(a_0 + a_1 + \\dots + a_n)}{(n^2 - 1)n^2} \\ge \\frac{a_0 a_n}{n^2},\\]\nor\n\\[(a_1 + a_2 + \\dots + a_{n-1})(a_0 + a_1 + \\dots + a_n) \\ge (n^2 - 1) a_0 a_n.\\]\nFortunately, this is an easy inequality. Indeed, from AM-GM applied on each group of terms we have\n\\[(a_1 + a_2 + \\dots + a_{n-1})(a_0 + a_1 + \\dots + a_n) \\ge (n^2 - 1) \\sqrt[n-1]{a_1 a_2 \\dots a_{n-1}} \\sqrt[n+1]{a_0 a_1 \\dots a_n},\\]\nand so it suffices to prove\n\\[\\sqrt[n-1]{a_1 a_2 \\dots a_{n-1}} \\sqrt[n+1]{a_0 a_1 \\dots a_n} \\ge a_0 a_n,\\]\nor, after taking both sides to the $(n^2 - 1)$ power, simplifying, and taking the $n$-th root of both sides, to prove\n\\[a_1^2 a_2^2 a_3^2 \\dots a_{n-1}^2 \\ge a_0^{n-1} a_n^{n-1}.\\]\nThis easily follows from the Fact that $a_0 a_n \\le a_i a_{n-i}$ for $1 \\le i \\le n-1$. Indeed, we are given that\n\\[a_0 a_2 \\le a_1^2\\]\n\\[a_1 a_3 \\le a_2^2\\]\n\\[a_2 a_4 \\le a_3^2\\]\n\\[\\dots\\]\n\\[a_{n-3} a_{n-1} \\le a_{n-2}^2\\]\n\\[a_{n-2} a_n \\le a_{n-1}^2.\\]\nMultiply all inequalities together and cancel $a_1, a_2^2, a_3^2, \\dots, a_{n-2}^2, a_{n-1}$ to give $a_0 a_n \\le a_1 a_{n-1}$. Similarly, by multiplying all inequalities except the first and the last, we deduce that $a_2 a_{n-2} \\ge a_1 a_{n-1} \\ge a_0 a_n$, and a simple induction argument proves the verity of the Fact for $1 \\le i \\le \\frac{n}{2}$, and so by the Commutative Property the Fact is true for all $1 \\le i \\le n-1$, as desired. Now multiply each inequality of the Fact for $i = 1, 2, 3, \\dots, n-1$ to give the desired result.\nNote: The Fact can be generalized into a Lemma: $a_x a_w \\le a_y a_z$ whenever $x \\le y \\le z \\le w$ and $x + w = y + z$. The proof is similar to that of the Fact and is left as an exercise to the reader." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1980_USAMO_Problems_3", "eval_func": null }, "input": { "text": "$A + B + C$ is an integral multiple of $\\pi$. $x, y,$ and $z$ are real numbers. If $x\\sin(A)+y\\sin(B)+z\\sin(C)=x^2\\sin(2A)+y^2\\sin(2B)+z^2\\sin(2C)=0$, show that $x^n\\sin(nA)+y^n \\sin(nB) +z^n \\sin(nC)=0$ for any positive integer $n$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $a=xe^{iA}$, $b=ye^{iB}$, $c=ze^{iC}$ be numbers in the complex plane. \nNote that $A+B+C=k\\pi$ implies $abc=xyz(e^{ik\\pi})=\\pm xyz$ which is real. Also note that $x\\sin(A), y\\sin(B), z\\sin(C)$ are the imaginary parts of $a, b, c$ and that $x^2\\sin(2A), y^2\\sin(2B), z^2\\sin(2C)$ are the imaginary parts of $a^2, b^2, c^2$ by de Moivre's Theorem. Therefore, $a+b+c$ and $a^2+b^2+c^2$ are real because their imaginary parts sum to zero.\nFinally, note that $\\frac{1}{2}\\left((a+b+c)^2-(a^2+b^2+c^2)\\right)=ab+bc+ac$ is real as well.\nIt suffices to show that $P_n=a^n+b^n+c^n$ is real for all positive integer $n$, which can be shown by induction.\nNewton Sums gives the following relationship between sums of the form $P_k=a^k+b^k+c^k$\n\\[P_k-S_1P_{k-1}+S_2P_{k-2}-S_3P_{k-3}=0\\]\nWhere $S_1=a+b+c$, $S_2=ab+bc+ac$, and $S_3=abc$. It is given that $P_0, P_1, P_2$ are real. Note that if $P_{k-1}, P_{k-2}, P_{k-3}$ are real, then clearly $P_k$ is real because all other parts of the above equation are real, completing the induction." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1981_USAMO_Problems_2", "eval_func": null }, "input": { "text": "What is the largest number of towns that can meet the following criteria. Each pair is directly linked by just one of air, bus or train. At least one pair is linked by air, at least one pair by bus and at least one pair by train. No town has an air link, a bus link and a train link. No three towns, $A, B, C$ are such that the links between $AB, AC$ and $BC$ are all air, all bus or all train.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Assume $AB$, $AC$, and $AD$ are all rail.\nNone of $BC$, $CD$, or $CD$ can be rail, as those cities would form a rail triangle with $A$.\nIf $BC$ is bus, then $BD$ is bus as well, as otherwise $B$ has all three types.\nHowever, $CD$ cannot be rail (as $\\triangle ACD$ would be a rail triangle), bus (as $BCD$ would be a bus triangle), or ferry (as $C$ and $D$ would have all three types).\nTherefore, no city can have three connections of the same type.\n\nAssume there are 5 towns - $A$, $B$, $C$, $D$, and $E$.\nTwo connections from $A$ must be of one type, and two of another; otherwise there would be at least three connections of the same type from $A$, which has been shown to be impossible.\nLet $AB$ and $AC$ be rail connections, and $AD$ and $AE$ be bus.\nAssume $CD$ is air.\n$BC$ cannot be rail ($\\triangle ABC$ would be a rail triangle) or bus ($C$ would have all three types), so $BC$ must be air.\n$DE$ cannot be bus ($\\triangle ADE$ would be a bus triangle) or rail ($D$ would have all three types), so $DE$ must be air.\n$BE$ cannot be rail ($E$ would have all three types) or bus ($B$ would have all three types), so $BE$ must be air.\nHowever, $BD$ cannot be rail ($D$ would have all three types), bus ($B$ would have all three types), or air ($D$ would have three air connections).\nTherefore, the assumption that $CD$ is air is false.\n$CD$ can equally be rail or bus; assume it is bus.\n$BC$ cannot be rail ($\\triangle ABC$ would be a rail triangle) or air ($C$ would have all three types), so $BC$ must be bus.\n$BD$ cannot be air ($B$ would have all three types) or bus ($D$ would have three bus connections), so $BD$ must be rail.\n$DE$ cannot be air ($D$ would have all three types) or bus ($D$ would have three bus connections), so $DE$ must be rail.\n$CE$ cannot be air ($C$ would have all three types) or bus ($E$ would have three bus connections), so $CE$ must be rail.\nThe only connection remaining is $BE$, which cannot be orange as both $B$ and $D$ would have all three types, but this means there are no air connections.\nTherefore, it is impossible with five (or more) towns.\n\nA four-town mapping is possible:\n$AB$, $BC$, $CD$, and $DA$ are connected by bus.\n$AC$ is connected by rail.\n$BD$ is connected by air.\nTherefore, the maximum number of towns is $4$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2003_USAMO_Problems_3", "eval_func": null }, "input": { "text": "Let $n \\neq 0$. For every sequence of integers\nsatisfying $0 \\le a_i \\le i$, for $i=0,\\dots,n$, define another sequence\nby setting $t(a_i)$ to be the number of terms in the sequence $A$ that precede the term $a_i$ and are different from $a_i$. Show that, starting from any sequence $A$ as above, fewer than $n$ applications of the transformation $t$ lead to a sequence $B$ such that $t(B) = B$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Consider some sequence $C = c_0, \\ldots, c_n$ as the image of $A$ after $t$ has been applied some finite number of times.\n If $t(c_k) = c_k = j$, then ${} c_j = \\cdots = c_{j+i} = \\cdots = c_k = j$ ($0 \\le i \\le k-j$).\n Since the $j$ terms $c_0, \\ldots, c_{j-1}$ are all less than $j$, no other terms that precede $c_{k}$ can be unequal to $c_k$. The lemma follows.\n If $t(c_k) = c_k = j$, then $t^2(c_k) = t(c_k)$ (where $f^2(x) = f(f(x))$).\n Since only $i$ terms precede the term $c_i$, we will have $t^m(c_i) \\le i$, for any integer $m$. This means that we will always have $t^m (c_0), \\ldots, t^m (c_{j-1}) < j$. This means that $c_j = t(c_j) = \\cdots = c_k = t(c_k)$, and the lemma follows by iteration.\nThus we may regard a term $c_k$ as if $t(c_k) = c_k$. We will call a sequence if all of its terms are stable.\n If $c_k =j$ is not stable, then $t(c_k) > c_k$.\n We have $c_0, \\ldots, c_{j-1} < j$, so we always have $t(c_k) \\ge c_k$. Equality implies that $c_k$ is stable.\nWe will now prove the problem by induction. For the base case, $n=1$, we have either 0,0 or 0,1, both of which are stable. Now, suppose that $t^{n-2}(a_0), \\ldots, t^{n-2}(a_{n-1})$ are stable. We must then have $t^{n-2}(a_n) \\in \\{ n-2, n-1, n\\}$. If $t^{n-2}(a_n) = n$, then it is already stable. If $t^{n-2}(a_n) = n-1$, then either it is already stable or $t^{n-1}(a_n) = n$, which is stable. If $t^{n-2}(a_n) = t^{n-2}(a_{n-1})$, then $t^{n-2}(a_n)$ must already be stable. The only possibilities remaining are $t^{n-2}(a_n) = n-2, t^{n-2}(a_{n-1}) = n-1$ and $t^{n-2}(a_n) = n-2, t^{n-2}(a_{n-1}) < n-2$. In the first case, we must have $t^{n-1}(a_n)$ equal to $n-1$ or $n$, both of which will make it stable. In the second case, we must have $t^{n-2}(a_{n-2}) = t^{n-2}(a_{n-1}) < n-2$, giving us $t^{n-1}(a_n) = n$, which will make it stable. Thus the induction is complete." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1983_USAMO_Problems_2", "eval_func": null }, "input": { "text": "Prove that the zeros of\n\\[x^5+ax^4+bx^3+cx^2+dx+e=0\\]\ncannot all be real if $2a^2<5b$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We prove the contrapositive: if the polynomial in question has the five real roots $x_1, x_2, x_3, x_4, x_5$, then $5b \\le 2a^2$.\nBecause $a = -(x_1 + x_2 + x_3 + x_4 + x_5)$ and $b = x_1x_2 + x_1x_3 + ... + x_4x_5$ by Vieta's Formulae, we have\n\\[2b = 2x_1x_2 + 2x_1x_3 + ... + 2x_4x_5 = (x_1 + x_2 + x_3 + x_4 + x_5)^2 - (x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2)\\]\n\\[=a^2 - \\frac{(1+1+1+1+1)(x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2)}{5}\\]\n\\[\\le a^2 - \\frac{(x_1 + x_2 + x_3 + x_4 + x_5)^2}{5}\\] (by Cauchy-Schwarz)\n\\[=\\frac{4a^2}{5},\\]\nso $5b \\le 2a^2$, as desired.", "For all real numbers $x_1,x_2,\\cdots x_5$,\n\\[2(x_1^2+x_2^2+\\cdots+x_5^2)\\ge\\]\n\\[x_1x_2+x_1x_3+\\cdots+x_4x_5\\]\nBy the trivial inequality,\n\\[x^2+y^2\\ge 2xy \\Rightarrow \\frac{x^2}{2} + \\frac{y^2}{2} \\ge xy\\]\nMaking such an inequality for all the variable pairs and summing them, we find the lemma is true.\nNow, let our roots be $x_1,x_2,\\cdots,x_5$. By Vieta's, $a=x_1+x_2+\\cdots+x_5$ and $b=x_1x_2+x_1x_3+\\cdots+x_4x_5$\nIf we show that for all real $x_1,x_2,\\cdots, x_5$ that $2a^2\\ge 5b$, then we have a contradiction and all of $x_1,x_2,\\cdots, x_5$ cannot be real. We start by rewriting $2a^2\\ge 5b$ as\n\\[2(x_1+x_2+\\cdots+x_5)^2\\ge 5(x_1x_2+x_1x_3+\\cdots+x_4x_5)\\]\nWe divide by $2$ and find\n\\[(x_1+x_2+\\cdots+x_5)^2\\ge \\frac{5}{2}(x_1x_2+x_1x_3+\\cdots+x_4x_5)\\]\nExpanding the LHS, we have\n\\[x_1^2+x_2^2+\\cdots+x_5^2+2(x_1x_2+x_1x_3+\\cdots+x_4x_5)\\ge\\frac{5}{2}(x_1x_2+x_1x_3+\\cdots+x_4x_5)\\]\nWe subtract the sum in brackets, and then multiply by $2$ to find\n\\[2x_1^2+2x_2^2+\\cdots+2x_5^2\\ge x_1x_2+x_1x_3+\\cdots+x_4x_5\\]\nwhich is true by our lemma." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1995_USAMO_Problems_2", "eval_func": null }, "input": { "text": "A calculator is broken so that the only keys that still work are the $\\, \\sin, \\; \\cos,$ $\\tan, \\; \\sin^{-1}, \\; \\cos^{-1}, \\,$ and $\\, \\tan^{-1} \\,$ buttons. The display initially shows 0. Given any positive rational number $\\, q, \\,$ show that pressing some finite sequence of buttons will yield $\\, q$. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We will prove the following, stronger statement\u00a0: If $m$ and $n$ are relatively prime nonnegative integers such that $n>0$, then the some finite sequence of buttons will yield $\\sqrt{m/n}$.\nTo prove this statement, we induct strongly on $m+n$. For our base case, $m+n=1$, we have $n=1$ and $m=0$, and $\\sqrt{m/n} = 0$, which is initially shown on the screen. For the inductive step, we consider separately the cases $m=0$, $0 1$ and suppose that $f(k)=k$. By Equation (1) we have that\n\\[f(k+1)! \\equiv k! \\mod k\\cdot k! .\\]\nThis implies $f(k+1)<2k$ (otherwise $f(k+1)!\\equiv 0 \\mod k\\cdot k!$). Also we have\n\\[(k+1)-1 \\mid f(k+1)-f(1)\\]\nso $f(k+1)\\equiv 1 \\mod k$. This gives the solutions $f(k+1)=1$ and $f(k+1)=k+1$. The first case is obviously impossible, so $f(k + 1) = k + 1$, as desired. By induction, $f(n) = n$ for all $n$. This also satisfies the requirements.\n $f(1)=f(2)=2$\nWe claim $f(n)=2$ by a similar induction. Again if $f(k)=2$, then by (1) we have\n\\[f(k+1)\\equiv 2 \\mod k\\cdot k!\\]\nand so $f(k+1)<2k$. Also note that\n\\[k+1-1 \\mid f(k+1)-2\\]\nand\n\\[k+1-2 \\mid f(k+1)-2\\]\nso $f(k+1)\\equiv 2 \\mod k(k-1)$. Then the only possible solution is $f(k+1)=2$. By induction, $f(n) = 2$ for all $n$, and this satisfies all requirements.\n\nIn summary, there are three solutions: $\\boxed{f(n)=1, f(n)=2, f(n)=n}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1994_USAMO_Problems_4", "eval_func": null }, "input": { "text": "Let $\\, a_1, a_2, a_3, \\ldots \\,$ be a sequence of positive real numbers satisfying $\\, \\sum_{j = 1}^n a_j \\geq \\sqrt {n} \\,$ for all $\\, n \\geq 1$. Prove that, for all $\\, n \\geq 1, \\,$\n\\[\\sum_{j = 1}^n a_j^2 > \\frac {1}{4} \\left( 1 + \\frac {1}{2} + \\cdots + \\frac {1}{n} \\right).\\]", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Note that if we try to minimize $(a_j)^2$, we would try to make the $a_j$ as equal as possible. However, by the condition given in the problem, this isn't possible, the $a_j$'s have to be an increasing sequence. Thinking of minimizing sequences, we realize that the optimal equation is $a_n = \\sqrt{n} - \\sqrt{n-1} = 1/(\\sqrt{n} + \\sqrt{n-1})$. Note that this is strictly greater than $1/(2\\sqrt{n})$. So it is greater than the sum of $(1/\\sqrt{4n})^2$ over all n from 1 to $\\infty$. So the sum we are looking to minimize is strictly greater than the sum of $1/4n$ over all $n$ from 1 to $\\infty$, which is what we wanted to do." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1990_USAMO_Problems_1", "eval_func": null }, "input": { "text": "A certain state issues license plates consisting of six digits (from 0 through 9). The state requires that any two plates differ in at least two places. (Thus the plates $\\boxed{027592}$ and $\\boxed{020592}$ cannot both be used.) Determine, with proof, the maximum number of distinct license plates that the state can use.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Consider license plates of $n$ digits, for some fixed $n$, issued with the same criteria.\nWe first note that by the pigeonhole principle, we may have at most $10^{n-1}$ distinct plates. Indeed, if we have more, then there must be two plates which agree on the first $n-1$ digits; these plates thus differ only on one digit, the last one.\nWe now show that it is possible to issue $10^{n-1}$ distinct license plates which satisfy the problem's criteria. Indeed, we issue plates with all $10^{n-1}$ possible combinations for the first $n-1$ digit, and for each plate, we let the last digit be the sum of the preceding digits taken mod 10. This way, if two plates agree on the first $n-1$ digits, they agree on the last digit and are thus the same plate, and if two plates differ in only one of the first $n-1$ digits, they must differ as well in the last digit.\nIt then follows that $10^{n-1}$ is the greatest number of license plates the state can issue. For $n=6$, as in the problem, this number is $10^5$. $\\blacksquare$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2006_USAMO_Problems_1", "eval_func": null }, "input": { "text": "Let $p$ be a prime number and let $s$ be an integer with $0 < s < p$. Prove that there exist integers $m$ and $n$ with $0 < m < n < p$ and\nif and only if $s$ is not a divisor of $p-1$.\nNote: For $x$ a real number, let $\\lfloor x \\rfloor$ denote the greatest integer less than or equal to $x$, and let $\\{x\\} = x - \\lfloor x \\rfloor$ denote the fractional part of $x$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We proceed by contradiction. Assume that $s|(p-1)$. Then for some positive integer $k$, $sk=p-1$. The conditions given are equivalent to stating that $sm \\bmod p < sn \\bmod p< s\\bmod p$.\nNow consider the following array modulo p:\n\\[\\begin{array}{ccccc} \\mbox{Row 1}& s& 2s&\\hdots& ks\\\\ \\mbox{Row 2}& s-1& 2s-1& \\hdots&ks-1\\\\ \\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\ \\mbox{Row }s& 1& s+1& \\hdots& (k-1)s+1 \\end{array}\\]\nObviously, there are $s$ rows and $k$ columns. The first entry of each row is simply $((r-1)k+1)s\\mod p$. Since we wish for $sm\\mod p$ and $sn\\mod p$ to both be in the first column, while also satisfying the given conditions, we can easily see that $sm$ must be in a row $m_r$ with $m_r>n_r$, where $n_r$ denotes the row $sn \\mod p$ is in. However, since the values of each entry decreases while $((r-1)k+1)s$ keeps increasing, we can see that the condition can never be satisfied and thus, $s\\not|(p-1).$\nTo prove the other direction, let $sj+r=p-1$ for positive integers $j$ and $r$ with $j$ being the largest integer such that $sjj+1$ such that $xs \\equiv s-1 \\mod p$.\nThus, choose $m=j+1$ and $n=x$ to complete the proof." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2021_USAMO_Problems_6", "eval_func": null }, "input": { "text": "Let $ABCDEF$ be a convex hexagon satisfying $\\overline{AB} \\parallel \\overline{DE}$, $\\overline{BC} \\parallel \\overline{EF}$, $\\overline{CD} \\parallel \\overline{FA}$, and\\[AB \\cdot DE = BC \\cdot EF = CD \\cdot FA.\\]Let $X$, $Y$, and $Z$ be the midpoints of $\\overline{AD}$, $\\overline{BE}$, and $\\overline{CF}$. Prove that the circumcenter of $\\triangle ACE$, the circumcenter of $\\triangle BDF$, and the orthocenter of $\\triangle XYZ$ are collinear.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $M_1$, $M_2$, and $M_3$ be the midpoints of $CE$, $AE$, $AC$ and $N_1$, $N_2$, and $N_3$ be the midpoints of $DF$, $BF$, and $BD$. Also, let $H$ be the orthocenter of $XYZ$. Note that we can use parallel sides to see that $X$, $Z$, and $M_3$ are collinear. Thus we have \\[\\text{Pow}(M_3,(XYZ)) = M_3Z \\cdot M_3X = \\frac 14 AB \\cdot DE\\] by midlines. Applying this argument cyclically, and noting the condition $AB \\cdot DE = BC \\cdot EF = CD \\cdot FA$, $M_1$, $M_2$, $M_3$, $N_1$, $N_2$, $N_3$ all lie on a circle concentric with $(XYZ)$. \nNext, realize that basic orthocenter properties imply that the circumcenter $O_1$ of $(ACE)$ is the orthocenter of $\\triangle M_1M_2M_3$, and likewise the circumcenter $O_2$ of $(BDF)$ is the orthocenter of $\\triangle N_1N_2N_3$.\nThe rest is just complex numbers; toss on the complex plane so that the circumcenter of $\\triangle XYZ$ is the origin. Then we have \\[o_1 = m_1+m_2+m_3 = (c+e)/2+(a+e)/2+(a+c)/2=a+c+e\\] \\[o_2 = n_1+n_2+n_3 = (b+d)/2+(d+f)/2+(b+f)/2=b+d+f\\] \\[h = x+y+z = (a+d)/2+(b+e)/2+(c+f)/2=(a+b+c+d+e+f)/2.\\]\nNote that from the above we have $h=\\frac{o_1+o_2}{2}$, so $H$ is the midpoint of segment $O_1O_2$. In particular, $H$, $O_1$, and $O_2$ are collinear, as required.", "We construct two equal triangles, prove that triangle $XYZ$ is the same as medial triangle of both this triangles. We use property of medial triangle and prove that circumcenters of constructed triangles coincide with given circumcenters.\nDenote $A' = C + E \u2013 D, B' = D + F \u2013 E, C' = A+ E \u2013 F,$\n$D' = F+ B \u2013 A, E' = A + C \u2013 B, F' = B+ D \u2013 C.$\nThen $A' \u2013 D' = C + E \u2013 D \u2013 ( F+ B \u2013 A) = (A + C + E ) \u2013 (B+ D + F).$\nDenote $D' \u2013 A' = 2\\vec V.$\nSimilarly we get $B' \u2013 E' = F' \u2013 C' = D' \u2013 A' \\implies$\n$\\triangle A'C'E' = \\triangle D'F'B'.$\nThe translation vector maps $\\triangle A'C'E'$ into $\\triangle D'F'B'$ is $2\\vec {V.}$\n$X = \\frac {A+D}{2} = \\frac { (A+ E \u2013 F) + (D + F \u2013 E)}{2} = \\frac {C' + B'}{2} = \\frac {E' + F'}{2},$\nso $X$ is midpoint of $AD, B'C',$ and $E'F'.$ Symilarly $Y$ is the midpoint of $BE, A'F',$ and $C'D', Z$ is the midpoint of $CF, A'B',$ and $D'E'.$\n$Z + V = \\frac {A' + B'}{2}+ \\frac {D' \u2013 A'}{2} = \\frac {B' + D'}{2} = Z'$ is the midpoint of $B'D'.$\nSimilarly $X' = X + V$ is the midpoint of $B'F',Y'= Y + V$ is the midpoint of $D'F'.$\nTherefore $\\triangle X'Y'Z'$ is the medial triangle of $\\triangle B'D'F'.$\n$\\triangle XYZ$ is $\\triangle X'Y'Z'$ translated on $\u2013 \\vec {V}.$\nIt is known (see diagram) that circumcenter of triangle coincide with orthocenter of the medial triangle. Therefore orthocenter $H$ of $\\triangle XYZ$ is circumcenter of $\\triangle B'D'F'$ translated on $\u2013 \\vec {V}.$\nIt is the midpoint of segment $OO'$ connected circumcenters of $\\triangle B'D'F'$ and $\\triangle A'C'E'.$\nAccording to the definition of points $A', C', E',$ quadrangles $ABCE', CDEA',$ and $AFEC'$ are parallelograms. Hence\n\\[AC' = FE, AE' = BC, CE' = AB, CA' = DE, EA' = CD, EC' = AF \\implies\\]\n\\[AC' \\cdot AE' = CE' \\cdot CA' = EA' \\cdot EC' = AB \\cdot DE \\implies\\] Power of points A,C, and E with respect circumcircle $\\triangle A'C'E'$ is equal, hence distances between these points and circumcenter of $\\triangle A'C'E'$ are the same. Therefore circumcenter $\\triangle ACE$ coincide with circumcenter $\\triangle A'C'E'.$\nSimilarly circumcenter of $\\triangle BDF$ coincide with circumcenter of $\\triangle B'D'F'.$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2009_USAMO_Problems_6", "eval_func": null }, "input": { "text": "Let $s_1, s_2, s_3, \\ldots$ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that $s_1 = s_2 = s_3 = \\cdots.$ Suppose that $t_1, t_2, t_3, \\ldots$ is also an infinite, nonconstant sequence of rational numbers with the property that $(s_i - s_j)(t_i - t_j)$ is an integer for all $i$ and $j$. Prove that there exists a rational number $r$ such that $(s_i - s_j)r$ and $(t_i - t_j)/r$ are integers for all $i$ and $j$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Suppose the $s_i$ can be represented as $\\frac{a_i}{b_i}$ for every $i$, and suppose $t_i$ can be represented as $\\frac{c_i}{d_i}$. Let's start with only the first two terms in the two sequences, $s_1$ and $s_2$ for sequence $s$ and $t_1$ and $t_2$ for sequence $t$. Then by the conditions of the problem, we have $(s_2 - s_1)(t_2 - t_1)$ is an integer, or $(\\frac{a_2}{b_2} - \\frac{a_1}{b_1})(\\frac{c_2}{d_2} - \\frac{c_1}{d_1})$ is an integer. Now we can set $r = \\frac{b_1 b_2}{d_1 d_2}$, because the least common denominator of $s_2 - s_1$ is $b_1 b_2$ and of $t_2 - t_1$ is $d_1 d_2$, and multiplying or dividing appropriately by $\\frac{b_1 b_2}{d_1 d_2}$ will always give an integer. \nNow suppose we kept adding $s_i$ and $t_i$ until we get to $s_m = \\frac{a_m}{b_m}$ in sequence $s$ and $t_m = \\frac{c_m}{d_m}$ in sequence $t$ so that $(t_m - t_i)(s_m - s_i)$ is an integer for all $i$ with $1 \\le i < m$, where $m$ is a positive integer. At this point, we will have $r$ = $\\frac{\\prod_{n=1}^{m}b_n}{\\prod_{n=1}^{m}d_n}$, because these are the least common denominators of the two sequences up to $m$. As we keep adding $s_i$ and $t_i$, $r$ will always have value $\\frac{\\prod_{n=1}^{m}b_n}{\\prod_{n=1}^{m}d_n}$, and we are done." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1997_USAMO_Problems_1", "eval_func": null }, "input": { "text": "Let $p_1,p_2,p_3,...$ be the prime numbers listed in increasing order, and let $x_0$ be a real number between $0$ and $1$. For positive integer $k$, define\n$x_{k}=\\begin{cases}0&\\text{ if }x_{k-1}=0\\\\ \\left\\{\\frac{p_{k}}{x_{k-1}}\\right\\}&\\text{ if }x_{k-1}\\ne0\\end{cases}$\nwhere $\\{x\\}$ denotes the fractional part of $x$. (The fractional part of $x$ is given by $x-\\lfloor{x}\\rfloor$ where $\\lfloor{x}\\rfloor$ is the greatest integer less than or equal to $x$.) Find, with proof, all $x_0$ satisfying $0 1$ have the property that there exists an infinite sequence $a_1$, $a_2$, $a_3$, $\\dots$ of nonzero integers such that the equality\n\\[a_k + 2a_{2k} + \\dots + na_{nk} = 0\\]\nholds for every positive integer $k$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "(Since Bertrand's is well known and provable using elementary techniques, I see nothing wrong with this-tigershark22)\nFor $n=2$, $|a_1| = 2 |a_2| = \\cdots = 2^m |a_{2^m}|$ implies that for any positive integer $m$, $|a_1| \\ge 2^m$, which is impossible.\nWe proceed to prove that the infinite sequence exists for all $n\\ge 3$.\nFirst, one notices that if we have $a_{xy} = a_x a_y$ for any integers $x$ and $y$, then it is suffice to define all $a_x$ for $x$ prime, and one only needs to verify the equation (*)\n\\[a_1+2a_2+\\cdots+na_n=0\\]\nfor the other equations will be automatically true.\nTo proceed with the construction, I need the following fact: for any positive integer $m>2$, there exists a prime $p$ such that $\\frac{m}{2}

1$, there exists a prime $p$ such that $n2$, if $m=2n$ with $n>1$, then there exists a prime $p$ such that $\\frac{m}{2} < p < m$, and if $m=2n-1$ with $n>1$, then there exists a prime $p$ such that $\\frac{m+1}{2} 2$, there exists a prime $p$ such that $\\frac{m}{2}

Q$. By the fact stated above, one can conclude that $2P > n$, and that $4Q = 2(2Q) \\ge 2P > n$. Let's construct $a_n$:\nLet $a_1=1$. There will be three cases: (i) $Q>\\frac{n}{2}$, (ii) $\\frac{n}{2} \\ge Q > \\frac{n}{3}$, and (iii) $\\frac{n}{3} \\ge Q > \\frac{n}{4}$.\nCase (i): $2Q>n$. Let $a_x = 1$ for all prime numbers $x n$. In this case, let $a_2=-1$, and $a_x = 1$ for all prime numbers $2 P$, just let $a_p=1$ (or any other non-zero integer).\nThis construction is correct because, for any $k> 1$, \n\\[a_k + 2 a_{2k} + \\cdots n a_{nk} = a_k (1 + 2 a_2 + \\cdots n a_n ) = 0\\]\nSince Bertrand's Theorem is not elementary, we still need to wait for a better proof." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1977_USAMO_Problems_1", "eval_func": null }, "input": { "text": "Determine all pairs of positive integers $(m,n)$ such that\n$(1+x^n+x^{2n}+\\cdots+x^{mn})$ is divisible by $(1+x+x^2+\\cdots+x^{m})$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Denote the first and larger polynomial to be $f(x)$ and the second one to be $g(x)$. In order for $f(x)$ to be divisible by $g(x)$ they must have the same roots. The roots of $g(x)$ are the (m+1)th roots of unity, except for 1. When plugging into $f(x)$, the root of unity is a root of $f(x)$ if and only if the terms $x^n, x^{2n}, x^{3n}, \\cdots x^{mn}$ all represent a different (m+1)th root of unity not equal to 1. \nNote that if $\\\\gcd(m+1,n)=1$, the numbers $n, 2n, 3n, \\cdots, mn$ represent a complete set of residues minus 0 modulo $m+1$. However, if $gcd(m+1,n)=a$ not equal to 1, then $\\frac{(m+1)(n)}{a}$ is congruent to $0 \\pmod {m+1}$ and thus a complete set is not formed. Therefore, $f(x)$ divides $g(x)$ if and only if $\\boxed{\\\\gcd(m+1,n)=1}.$ $\\blacksquare$", "We could instead consider $f(x)$ modulo $g(x)$. Notice that $x^{m+1} = 1 \\pmod {g(x)}$, and thus we can reduce the exponents of $f(x)$ to their equivalent modulo $m+1$. We want the resulting $h(x)$ with degree less than $m+1$ to be equal to $g(x)$ (of degree $m$), which implies that the exponents of $f(x)$ must be all different modulo $m+1$. This can only occur if and only if $gcd(m+1, n) = 1$, and this is our answer, as shown in Solution 1.", "Notice that $1+x^n+x^{2n}+...+x^{mn}=\\frac{x^{mn+n}-1}{x^n-1}$ and $1+x+x^2+...+x^m=\\frac{x^{m+1}-1}{x-1}$, so it remains to prove that $(x^n-1)(x^{m+1}-1) \\mid (x^{mn+n}-1)(x-1)$. It is clear that $(x^n-1) \\mid (x^{mn+n}-1)$ and $(x^{m+1}-1) \\mid (x^{mn+n}-1)$. For any $k \\in \\mathbb{N}$, we can use the fact that $x^k-1=\\prod_{d \\mid k} \\Phi _d(x)$, where $\\Phi _d(x)$ is the dth cyclotomic polynomial. If $\\gcd(m+1,n) = d > 1$, then $x^{m+1}-1$ and $x^n-1$ share a common cyclotomic polynomial; namely, $\\Phi _d(x)$. But since all the factors of $x^{mn+n}-1$ are distinct, $(x^{mn+n}-1)(x-1)$ cannot be divisible by $(x^n-1)(x^{m+1}-1)$. We find that $\\gcd(m+1,n) = 1$ must be the solution, since the only shared polynomial is $\\Phi _1(x)=x-1$, and we are done." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1987_USAMO_Problems_1", "eval_func": null }, "input": { "text": "Find all solutions to $(m^2+n)(m + n^2)= (m - n)^3$, where m and n are non-zero integers.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Expanding both sides, \\[m^3+mn+m^2n^2+n^3=m^3-3m^2n+3mn^2-n^3\\]\nNote that $m^3$ can be canceled and as $n \\neq 0$, $n$ can be factored out.\nWriting this as a quadratic equation in $n$: \\[2n^2+(m^2-3m)n+(3m^2+m)=0\\].\nThe discriminant $b^2-4ac$ equals \\[(m^2-3m)^2-8(3m^2+m)\\]\n\\[=m^4-6m^3-15m^2-8m\\], which we want to be a perfect square.\nMiraculously, this factors as $m(m-8)(m+1)^2$. This is square iff (if and only if) $m^2-8m$ is square or $m+1=0$. It can be checked that the only nonzero $m$ that work are $-1, 8, 9$. Finally, plugging this in and discarding extraneous roots gives all possible ordered pairs $(m, n)$ as \\[\\{(-1,-1),(8,-10),(9,-6),(9,-21)\\}\\]." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2004_USAMO_Problems_6", "eval_func": null }, "input": { "text": "A circle $\\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be the center of $\\omega$. Suppose that\nProve that $ABCD$ is an .", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Our proof is based on the following key Lemma.\n: If a circle $\\omega$, centered at $I$, is inscribed in a quadrilateral $ABCD$, then\n\\[BI^2 + \\frac{AI}{DI}\\cdot BI\\cdot CI = AB\\cdot BC.\\qquad\\qquad (1)\\]\n: Since circle $\\omega$ is inscribed in $ABCD$, we get $m\\angle DAI = m\\angle IAB = a$, $m\\angle ABI = m\\angle IBC = b$, $m\\angle BCI = m\\angle ICD = c$, $m\\angle CDI = m\\angle IDA = d$, and $a + b + c + d = 180^\\circ$. Construct a point $P$ outside of the quadrilateral such that $\\triangle ABP$ is similar to $\\triangle DCI$. We obtain\n\\begin{align*} m\\angle PAI + m\\angle PBI &= m\\angle PAB + m\\angle BAI + m\\angle PBA + m\\angle ABI \\\\ &= m\\angle IDC + a + m\\angle ICD + b \\\\ &= a + b + c + d = 180^\\circ, \\end{align*}\nimplying that the quadrilateral $PAIB$ is cyclic. By Ptolemy's Theorem, we have $AI\\cdot BP + BI\\cdot AP = AB\\cdot IP$, or\n\\[BP\\cdot\\frac{AI}{IP} + BI\\cdot\\frac{AP}{IP} = AB.\\qquad\\qquad (2)\\]\nBecause $PAIB$ is cyclic, it is not difficult to see that, as indicated in the figure, $m\\angle IPB = m\\angle IAB = a$, $m\\angle API = m\\angle ABI = b$, $m\\angle AIP = m\\angle ABP = c$, and $m\\angle PIB = m\\angle PAB = d$. Note that $\\triangle AIP$ and $\\triangle ICB$ are similar, implying that\n\\[\\frac{AI}{IP} = \\frac{IC}{CB}\\text{ and }\\frac{AP}{IP} = \\frac{IB}{CB}.\\]\nSubstituting the above equalities into the identity $(2)$, we arrive at\n\\[BP\\cdot\\frac{CI}{BC} + \\frac{BI^2}{BC} = AB,\\]\nor\n\\[BP\\cdot CI + BI^2 = AB\\cdot BC.\\qquad\\qquad (3)\\]\nNote also that $\\triangle BIP$ and $\\triangle IDA$ are similar, implying that $\\frac{BP}{BI} = \\frac{IA}{ID}$, or\n\\[BP = \\frac{AI}{ID}\\cdot IB.\\]\nSubstituting the above identity back into $(3)$ gives the desired relation $(1)$, establishing the Lemma. $\\blacksquare$\nNow we prove our main result. By the Lemma and symmetry, we have\n\\[CI^2 + \\frac{DI}{AI}\\cdot BI\\cdot CI = CD\\cdot BC.\\qquad\\qquad (4)\\]\nAdding the two identities $(1)$ and $(4)$ gives\n\\[BI^2 + CI^2 + \\left(\\frac{AI}{DI} + \\frac{DI}{AI}\\right)BI\\cdot CI = BC(AB + CD).\\]\nBy the AM-GM Inequality, we have $\\frac{AI}{DI} + \\frac{DI}{AI}\\geq 2$. Thus,\n\\[BC(AB + CD)\\geq IB^2 + IC^2 + 2IB\\cdot IC = (BI + CI)^2,\\]\nwhere the equality holds if and only if $AI = DI$. Likewise, we have\n\\[AD(AB + CD)\\geq (AI + DI)^2,\\]\nwhere the equality holds if and only if $BI = CI$. Adding the last two identities gives\n\\[(AI + DI)^2 + (BI + CI)^2\\leq (AD + BC)(AB + CD) = (AB + CD)^2,\\]\nbecause $AD + BC = AB + CD$. (The latter equality is true because the circle $\\omega$ is inscribed in the quadrilateral $ABCD$.)\nBy the given condition in the problem, all the equalities in the above discussion must hold, that is, $AI = DI$ and $BI = CI$. Consequently, we have $a = d$, $b = c$, and so $\\angle DAB + \\angle ABC = 2a + 2b = 180^\\circ$, implying that $AD\\parallel BC$. It is not difficult to see that $\\triangle AIB$ and $\\triangle DIC$ are congruent, implying that $AB = CD$. Thus, $ABCD$ is an isosceles trapezoid." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2008_USAMO_Problems_4", "eval_func": null }, "input": { "text": "Let $\\mathcal{P}$ be a with $n$ sides, $n\\ge3$. Any set of $n - 3$ diagonals of $\\mathcal{P}$ that do not intersect in the interior of the polygon determine a of $\\mathcal{P}$ into $n - 2$ triangles. If $\\mathcal{P}$ is regular and there is a triangulation of $\\mathcal{P}$ consisting of only isosceles triangles, find all the possible values of $n$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We label the vertices of $\\mathcal{P}$ as $P_0, P_1, P_2, \\ldots, P_n$. Consider a diagonal $d = \\overline{P_a\\,P_{a+k}},\\,k \\le n/2$ in the triangulation. We show that $k$ must have the form $2^{m}$ for some nonnegative integer $m$.\nThis diagonal $\\mathcal{P}$ into two regions $\\mathcal{Q},\\, \\mathcal{R}$, and is the side of an isosceles triangle in both regions. suppose the area of $Q$ is less than the area of $R$ (so the of $P$ does not lie in the interior of $Q$); it follows that the lengths of the edges and diagonals in $Q$ are all smaller than $d$. Thus $d$ must the be the base of the isosceles triangle in $Q$, from which it follows that the isosceles triangle is $\\triangle P_aP_{a+k/2}\\,P_{a+k}$, and so $2|k$. Repeating this process on the legs of isosceles triangle ($\\overline{P_aP_{a+k/2}},\\,\\overline{P_{a+k}P_{a+k/2}}$), it follows that $k = 2^m$ for some positive integer $m$ (if we allow , then we can also let $m=0$). \nNow take the isosceles triangle $P_xP_yP_z,\\,0 \\le x < y < z < n$ in the triangulation that contains the center of $\\mathcal{P}$ in its interior; if a diagonal passes through the center, select either of the isosceles triangles with that diagonal as an edge. Without loss of generality, suppose $P_xP_y = P_yP_z$. From our previous result, it follows that there are $2^a$ edges of $P$ on the minor arcs of $P_xP_y,\\, P_yP_z$ and $2^b$ edges of $P$ on the minor arc of $P_zP_x$, for positive integers $a,\\,b$. Therefore, we can write\n\\[n = 2 \\cdot 2^a + 2^b = 2^{a+1} + 2^{b},\\] so $n$ must be the sum of two powers of $2$. \n\nWe now claim that this condition is sufficient. Suppose without loss of generality that $a+1 \\ge b$; then we rewrite this as \n\\[n = 2^{b}(2^{a-b+1}+1).\\]\nBy , it follows that we can cover all the desired $n$.\nFor $n = 3,4$, this is trivial. For $k>1$, we construct the diagonals of equal length $\\overline{P_0P_{2^{k-1}}}$ and $\\overline{P_{2^{k-1}+1}P_0}$. This partitions $\\mathcal{P}$ into $3$ regions: an isosceles $\\triangle P_0P_{2^{k-1}}P_{2^{k-1}+1}$, and two other regions. For these two regions, we can recursively construct the isosceles triangles defined above in the second paragraph. It follows that we have constructed $2(2^{k-1}-1) + (1) = 2^k-1 = n-2$ isosceles triangles with non-intersecting diagonals, as desired. \nWe construct the diagonals $\\overline{P_0P_2},\\ \\overline{P_2P_4},\\ \\ldots \\overline{P_{2n-2}P_0}$. This partitions $\\mathcal{P}$ into $n$ isosceles triangles of the form $\\triangle P_{2k}P_{2k+1}P_{2k+2}$, as well as a central regular polygon with $n$ sides. However, we know that there exists a triangulation for the $n$-sided polygon that yields $n-2$ isosceles triangles. Thus, we have created $(n) + (n-2) = 2n-2$ isosceles triangles with non-intersecting diagonals, as desired. \nIn summary, the answer is all $n$ that can be written in the form $2^{a+1} + 2^{b},\\, a,b \\ge 0$. Alternatively, this condition can be expressed as either $n=2^{k},\\, k \\ge 2$ (this is the case when $a+1 = b$) or $n$ is the sum of two distinct powers of $2$, where $1= 2^0$ is considered a power of $2$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1978_USAMO_Problems_1", "eval_func": null }, "input": { "text": "Given that $a,b,c,d,e$ are real numbers such that\n$a+b+c+d+e=8$,\n$a^2+b^2+c^2+d^2+e^2=16$.\nDetermine the maximum value of $e$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "By Cauchy Schwarz, we can see that $(1+1+1+1)(a^2+b^2+c^2+d^2)\\geq (a+b+c+d)^2$\nthus $4(16-e^2)\\geq (8-e)^2$ \nFinally, $e(5e-16) \\geq 0$ which means $\\frac{16}{5} \\geq e \\geq 0$\nso the maximum value of $e$ is $\\frac{16}{5}$.", "Seeing as we have an inequality with constraints, we can use Lagrange multipliers to solve this problem.\nWe get the following equations:\n$(1)\\hspace*{0.5cm} a+b+c+d+e=8\\\\ (2)\\hspace*{0.5cm} a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=16\\\\ (3)\\hspace*{0.5cm} 0=\\lambda+2a\\mu\\\\ (4)\\hspace*{0.5cm} 0=\\lambda+2b\\mu\\\\ (5)\\hspace*{0.5cm} 0=\\lambda+2c\\mu\\\\ (6)\\hspace*{0.5cm} 0=\\lambda+2d\\mu\\\\ (7)\\hspace*{0.5cm} 1=\\lambda+2e\\mu$\nIf $\\mu=0$, then $\\lambda=0$ according to $(6)$ and $\\lambda=1$ according to $(7)$, so $\\mu \\neq 0$. Setting the right sides of $(3)$ and $(4)$ equal yields $\\lambda+2a \\mu= \\lambda+2b \\mu \\implies 2a\\mu=2b \\mu \\implies a=b$. Similar steps yield that $a=b=c=d$. Thus, $(1)$ becomes $4d+e=8$ and $(2)$ becomes $4d^{2}+e^{2}=16$. Solving the system yields $e=0,\\frac{16}{5}$, so the maximum possible value of $e$ is $\\frac{16}{5}$.", "A re-writing of Solution 1 to avoid the use of Cauchy Schwarz. We have\n\\[(a+b+c+d)^2=(8-e)^2,\\] and \n\\[a^2+b^2+c^2+d^2=16-e^2.\\]\nThe second equation times 4, then minus the first equation, \n\\[(a-b)^2+(a-c)^2+(a-d)^2+(b-c)^2+(b-d)^2+(c-d)^2=4(16-e^2)-(8-e)^2.\\]\nThe rest follows." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1976_USAMO_Problems_3", "eval_func": null }, "input": { "text": "Determine all integral solutions of $a^2+b^2+c^2=a^2b^2$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Either $a^2=0$ or $a^2>0$. If $a^2=0$, then $b^2=c^2=0$. Symmetry applies for $b$ as well. If $a^2,b^2\\neq 0$, then $c^2\\neq 0$. Now we look at $a^2\\bmod{4}$:\n$a^2\\equiv 0\\bmod{4}$: Since a square is either 1 or 0 mod 4, then all the other squares are 0 mod 4. Let $a=2a_1$, $b=2b_1$, and $c=2c_1$. Thus $a_1^2+b_1^2+c_1^2=4a_1^2b_1^2$. Since the LHS is divisible by four, all the variables are divisible by 4, and we must do this over and over again, and from infinite descent, there are no non-zero solutions when $a^2\\equiv 0\\bmod{4}$.\n$a^2\\equiv 1\\bmod{4}$: Since $b^2\\neq 0\\bmod{4}$, $b^2\\equiv 1\\bmod{4}$, and $2+c^2\\equiv 1\\bmod{4}$. But for this to be true, $c^2\\equiv 3\\bmod{4}$, which is an impossibility. Thus there are no non-zero solutions when $a^2\\equiv 1\\bmod{4}$.\nThus the only solution is the solution above: $(a,b,c)=0$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2016_USAMO_Problems_3", "eval_func": null }, "input": { "text": "Let $\\triangle ABC$ be an acute triangle, and let $I_B, I_C,$ and $O$ denote its $B$-excenter, $C$-excenter, and circumcenter, respectively. Points $E$ and $Y$ are selected on $\\overline{AC}$ such that $\\angle ABY = \\angle CBY$ and $\\overline{BE}\\perp\\overline{AC}.$ Similarly, points $F$ and $Z$ are selected on $\\overline{AB}$ such that $\\angle ACZ = \\angle BCZ$ and $\\overline{CF}\\perp\\overline{AB}.$\nLines $I_B F$ and $I_C E$ meet at $P.$ Prove that $\\overline{PO}$ and $\\overline{YZ}$ are perpendicular.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "This problem can be proved in the following two steps.\n1. Let $I_A$ be the $A$-excenter, then $I_A,O,$ and $P$ are colinear. This can be proved by the Trigonometric Form of Ceva's Theorem for $\\triangle I_AI_BI_C.$\n2. Show that $I_AY^2-I_AZ^2=OY^2-OZ^2,$ which implies $\\overline{OI_A}\\perp\\overline{YZ}.$ This can be proved by multiple applications of the Pythagorean Thm.", "We find point $T$ on line $YZ,$ we prove that $TY \\perp OI_A$ and state that $P$ is the point $X(24)$ from ENCYCLOPEDIA OF TRIANGLE, therefore $P \\in OI_A.$\nLet $\\omega$ be circumcircle of $\\triangle ABC$ centered at $O.$ \nLet $Y_1,$ and $Z_1$ be crosspoints of $\\omega$ and $BY,$ and $CZ,$ respectively.\nLet $T$ be crosspoint of $YZ$ and $Y_1 Z_1.$\nIn accordance the Pascal theorem for pentagon $AZ_1BCY_1,$ $AT$ is tangent to $\\omega$ at $A.$\nLet $I_A, I_B, I_C$ be $A, B,$ and $C$-excenters of $\\triangle ABC.$\nDenote \\[a = BC, b = AC, c = AB, 2\\alpha = \\angle CAB, 2\\beta = \\angle ABC, 2\\gamma = \\angle ACB,\\]\n\\[\\psi = 90^\\circ \u2013 \\gamma + \\beta, X = AI_A \\cap \\omega, X_1 = BC \\cap AI_A,\\]\n\\[I = BI_B \\cap CI_C, U= YZ \\cap AI_A, W = Y_1Z_1 \\cap AI_A,\\]\n$V$ is the foot ot perpendicular from $O$ to $AI_A.$\n$I$ is ortocenter of $\\triangle I_A I_B I_C$ and incenter of $\\triangle ABC.$\n$\\omega$ is the Nine\u2013point circle of $\\triangle I_A I_B I_C.$\n$Y_1$ is the midpoint of $II_B, Z_1$ is the midpoint of $II_C$ in accordance with property of Nine\u2013point circle $\\implies$\n\\[Y_1Z_1 || I_B AI_C || VO, IW = AW \\implies TW \\perp AI.\\]\n\\[\\angle AXC = 180 ^\\circ \u2013 2\\gamma \u2013 \\alpha = 90 ^\\circ \u2013 \\gamma + \\beta = \\psi.\\]\n\\[\\angle TAI = \\angle VOA = 2\\beta + \\alpha = 90 ^\\circ \u2013 \\gamma + \\beta = \\psi.\\]\n\\[I_A X_1 = IX_1 = BX_1 = 2R \\sin \\alpha \\implies\\]\n\\[\\cot \\angle OI_A A = \\frac {VI_A}{VO} = \\frac {R \\sin \\psi + 2R \\sin \\alpha}{R \\cos \\psi} = \\tan \\psi + \\frac{2 \\sin\\alpha}{\\cos \\psi}.\\]\n\\[CX = \\frac {ab}{b+c} \\implies \\frac {AI}{IX}= \\frac {AC}{CX}= \\frac {b+c}{a} \\implies AI = AX \\frac {b+c}{a+b+c},\\]\n\\[AW = \\frac {AI}{2}, UW = AU \u2013 AW,\\] \nIn $\\triangle ABC$ segment $YZ$ cross segment $AX \\implies \\frac {AU}{UX} = \\frac {m + nk}{k+1},$ where $n = \\frac {a}{b}, m = \\frac{a}{c}, k=\\frac {b}{c},$\n\\[\\frac {AU}{UX} = \\frac{2a}{b+c} \\implies AU = AX \\cdot \\frac {b+c}{2a +b +c}.\\]\n\\[\\frac {AU \u2013 AW}{AW} = \\frac {b+c} {2a + b + c}.\\]\n\\[\\cot \\angle UTW = \\frac {TW}{UW} = \\frac {AW \\cdot \\tan \\psi}{AU \u2013 AW} = \\tan \\psi \\cdot \\frac {2a +b+c}{b+c} =\\]\n\\[=\\tan \\psi \\cdot \\frac {2a}{b+c} + \\tan \\psi = \\frac {2 \\sin \\alpha}{\\sin \\psi} \\tan \\psi + \\tan \\psi = \\tan \\psi + \\frac{2 \\sin\\alpha}{\\cos \\psi}\\]\n\\[\\implies \\angle UTW = \\angle OI_AA .\\]\n\\[TW \\perp AI_A \\implies TYZ \\perp OI_A.\\]\nLet $\\triangle II_B I_C$ be the base triangle with orthocenter $I_A,$ center of Nine-points circle $O \\implies OI_A$ be the Euler line of $\\triangle II_B I_C.$\n$\\triangle ABC$ is orthic triangle of $\\triangle II_B I_C,$\n$\\triangle DEF$ is orthic-of-orthic triangle.\n$P$ is perspector of base triangle and orthic-of-orthic triangle. \nTherefore $P$ is point $X(24)$ of ENCYCLOPEDIA OF TRIANGLE CENTERS which lies on Euler line of the base triangle.\n[]\n \\[\\frac {AU}{UX} = \\frac {m + nk}{k + 1}.\\]\n\n\\[\\frac {[AYZ]}{[ABC]} = \\frac {AZ \\cdot AY}{AB \\cdot AC} = \\frac {1}{(n + 1) \\cdot (m+1)},\\]\n\\[\\frac {[BXZ]}{[ABC]} = \\frac {BZ \\cdot BX}{AB \\cdot BC} = \\frac {n}{(n + 1) \\cdot (k+1)},\\]\n\\[\\frac {[CXY]}{[ABC]} = \\frac {CY \\cdot CX}{AC \\cdot BC} = \\frac {mk}{(m + 1) \\cdot (k+1)},\\]\n\\[\\frac {[XYZ]}{[ABC]} = 1 - \\frac {[AYZ]}{[ABC]} \u2013 \\frac {[BXZ]}{[ABC]} \u2013 \\frac {[CXY]}{[ABC]} = \\frac {m+nk}{(m + 1) \\cdot (k+1)\\cdot (n+1)},\\]\n\\[\\frac {AU}{UX} = \\frac {[AYZ]}{[XYZ]} = \\frac {m + nk}{k + 1}.\\]" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1976_USAMO_Problems_5", "eval_func": null }, "input": { "text": "If $P(x)$, $Q(x)$, $R(x)$, and $S(x)$ are all polynomials such that\n\\[P(x^5) + xQ(x^5) + x^2 R(x^5) = (x^4 + x^3 + x^2 + x +1) S(x),\\]\nprove that $x-1$ is a factor of $P(x)$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "In general we will show that if $m$ is an integer less than $n$ and $P_0, \\dotsc, P_{m-1}$ and $S$ are polynomials satisfying\n\\[P_0(x^n) + x P_1(x^n) + \\dotsb + x^{m-1} P_{m-1}(x^n) = \\sum_{k=0}^{n-1} x^k S(x),\\]\nthen $P_k(1) = 0$, for all integers $0 \\le k \\le m-1$. For the problem, we may set $n=5$, $m=3$, and then note that since $P(1)= 0$, $(x-1)$ is a factor of $P(x)$.\nIndeed, let $\\omega_1, \\dotsc, \\omega_{n-1}$ be the $n$th roots of unity other than 1. Then for all integers $1 \\le i \\le n-1$,\n\\[\\sum_{k=0}^{m-1} \\omega_i^k P_k(\\omega_i^n) = \\sum_{k=0}^{m-1} \\omega_i^k P_k(1) = \\sum_{k=0}^{n-1} \\omega_i^k S(\\omega_i) = \\frac{\\omega_i^{n-1} - 1}{\\omega_i - 1} S(\\omega_i) = 0 ,\\]\nfor all integers $1 \\le i \\le n$. This means that the $(m-1)$th degree polynomial\n\\[\\sum_{k=0}^{m-1} x^k P_k(1)\\]\nhas $n-1 > m-1$ distinct roots. Therefore all its coefficients must be zero, so $P_k = 0$ for all integers $0 \\le k \\le m-1$, as desired.", "Let $\\zeta, \\xi, \\rho$ be three distinct primitive fifth roots of unity.\nSetting $x = \\zeta, \\xi$, we have\n\\begin{align*} P(1) + \\zeta Q(1) + \\zeta^2 R(1) &= \\frac{\\zeta^5-1}{\\zeta-1} S(\\zeta) = 0, \\\\ P(1) + \\xi Q(1) + \\xi^2 R(1) &= \\frac{\\xi^5-1}{\\xi-1} S(\\xi) = 0 . \\end{align*}\nThese equations imply that\n\\[\\zeta Q(1) + \\zeta^2 R(1) = \\xi Q(1) + \\xi^2 R(1),\\]\nor\n\\[Q(1) = - ( \\zeta + \\xi)R(1).\\]\nBut by symmetry,\n\\[Q(1) = - (\\zeta + \\rho)R(1) .\\]\nSince $\\xi \\neq \\rho$, it follows that $Q(1) = R(1) = 0$. Then, as noted above,\n\\[P(1) = P(1) + \\zeta Q(1) + \\zeta^2 R(1) = 0,\\]\nso $(x-1)$ is a factor of $P(x)$, as desired.", "Let $z, z^2, z^3$ be three of the 5th roots of unity not equal to one that satisfy $1 + z + z^2 + z^3 + z^4 = 0$ as a result. Plugging them into the equation gives the linear system of equations in $(A(1), B(1), C(1))$:\n\\[A(1) + zB(1) + z^2C(1) = 0\\]\n\\[A(1) + z^2B(1) + z^4C(1) = 0\\]\n\\[A(1) + z^3B(1) + z^6C(1) = 0\\]\nDirect observation gives $(A(1), B(1), C(1)) = (0,0,0)$ as a solution, and there are no others because the system of equations is linear. Hence, $A(1) = 0$, and so $(x-1)$ is a factor of $A(x)$, as desired." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2001_USAMO_Problems_5", "eval_func": null }, "input": { "text": "Let $S$ be a set of integers (not necessarily positive) such that\n(a) there exist $a,b \\in S$ with $\\gcd(a,b) = \\gcd(a - 2,b - 2) = 1$;\n(b) if $x$ and $y$ are elements of $S$ (possibly equal), then $x^2 - y$ also belongs to $S$. \nProve that $S$ is the set of all integers.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "In the solution below we use the expression $S$ $x\\mapsto f(x)$ to mean that if $x$ belongs to $S$ then $f(x)$ also belongs to $S$. If $c,d\\in S$, then by (B), $S$ is stable under $x\\mapsto c^2 - x$ and $x\\mapsto d^2 - x$, hence stable under $x\\mapsto c^2 - (d^2 - x) = x + (c^2 - d^2)$. Similarly $S$ is stable under $x\\mapsto x + (d^2 - c^2)$. Hence $S$ is stable under $x\\mapsto x + n$ and $x\\mapsto x - n$ whenever $n$ is an integer linear combination of numbers of the form $c^2 - d^2$ with $c,d\\in S$. In particular, this holds for $n = m$, where $m = \\gcd\\{c^2 - d^2\u00a0: c,d\\in S\\}$.\nSince $S\\neq \\emptyset$ by (A), it suffices to prove that $m = 1$. For the sake of contradiction, assume that $m\\neq 1$. Let $p$ be a prime dividing $m$. Then $c^2 - d^2\\equiv 0\\pmod{p}$ for all $c,d\\in S$. In other words, for each $c,d\\in S$, either $d\\equiv c\\pmod{p}$ or $d\\equiv -c\\pmod{p}$. Given $c\\in S$, $c^2 - c\\in S$ by (B), so $c^2 - c\\equiv c\\pmod{p}$ or $c^2 - c\\equiv -c\\pmod{p}$. Hence\n\\[\\text{For each }c\\in S\\text{, either }c\\equiv 0\\pmod{p}\\text{ or }c\\equiv 2\\pmod{p}.\\qquad\\qquad (*)\\]\nBy (A), there exist some $a$ and $b$ in $S$ such that $\\gcd(a,b) = 1$, that is, at least one of $a$ or $b$ cannot be divisible by $p$. Denote such an element of $S$ by $\\alpha$: thus, $\\alpha\\not\\equiv 0\\pmod{p}$. Similarly, by (A), $\\gcd(a - 2, b - 2) = 1$, so $p$ cannot divide both $a - 2$ and $b - 2$. Thus, there is an element of $S$, call it $\\beta$, such that $\\beta\\not\\equiv 2\\pmod{p}$. By $(*)$, $\\alpha\\equiv 2\\pmod{p}$ and $\\beta\\equiv 0\\pmod{p}$. By (B), $\\beta^2 - \\alpha\\in S$. Taking $c = \\beta^2 - \\alpha$ in $(*)$ yields either $-2\\equiv 0\\pmod{p}$ or $-2\\equiv 2\\pmod{p}$, so $p = 2$. Now $(*)$ says that all elements of $S$ are even, contradicting (A). Hence our assumption is false and $S$ is the set of all integers." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1990_USAMO_Problems_4", "eval_func": null }, "input": { "text": "Find, with proof, the number of positive integers whose base-$n$ representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by $\\pm 1$ from some digit further to the left. (Your answer should be an explicit function of $n$ in simplest form.)", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let a $k$-good sequence be a sequence of distinct integers $\\{ a_i \\}_{i=1}^k$ such that for all integers $2\\le i \\le k$, $a_i$ differs from some preceding term by $\\pm 1$.\n Let $a$ be an integer. Then there are $2^{k-1}$ $k$-good sequences starting on $a$, and furthermore, the terms of each of these sequences constitute a permutation of $k$ consecutive integers.\n We induct on $k$. For $k=1$, the lemma is trivially true. Now, suppose the lemma holds for $k$. If $\\{ a_i \\}_{i=1}^{k+1}$ is a $(k+1)$-good sequence, then $\\{ a_i \\}_{i=1}^k$ is a $k$-good sequence which starts on $a$, so it is a permutation of $k$ consecutive integers, say $m, \\dotsc, M$. Then the only possibilities for $a_{k+1}$ are $m-1$ and $M+1$; either way, $\\{ a_i \\}_{i=1}^{k+1}$ constitutes a permutation of $k+1$ consecutive integers. Since there are $2^k$ possible sequences $\\{a_i\\}_{i=1}^k$, and 2 choices of $a_{k+1}$ for each of these sequences, it also follows that there are $2^k \\cdot 2 = 2^{k+1}$ $(k+1)$-good sequences which start on $a$. Thus the lemma holds by induction. $\\blacksquare$\nWe now consider the number of desired positive integers with $k$ digits. Evidently, $k$ must be less than or equal to $n$. We also note that the digits of such an integer must constitute a $k$-good sequence. Since the minimum of this sequence can be any of the digits $0, \\dotsc, n-k$, unless the minimum is 0 and is the first digit (in which case the only possible sequence is an increasing arithmetic sequence), and there are $2^{k-1}$ $k$-good sequences up to translation, it follows that there are $(n-k+1) 2^{k-1}-1$ desired positive integers with $k$ digits. Thus the total number of desired positive integers is\n\\begin{align*} \\sum_{k=1}^n \\bigl[ (n-k+1) 2^{k-1}-1 \\bigr] &= -n + \\sum_{k=1}^n \\sum_{j=k}^n 2^{k-1} = -n + \\sum_{j=1}^n \\sum_{k=1}^j 2^{k-1} \\\\ &= -n + \\sum_{j=1}^n (2^k-1) = - 2n -1 + \\sum_{j=0}^n 2^k, \\end{align*}\nwhich is equal to $2^{n+1} - 2(n+1)$, our answer. $\\blacksquare$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1999_USAMO_Problems_2", "eval_func": null }, "input": { "text": "Let $ABCD$ be a cyclic quadrilateral. Prove that \\[|AB - CD| + |AD - BC| \\geq 2|AC - BD|.\\]", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let arc $AB$ of the circumscribed circle (which we assume WLOG has radius 0.5) have value $2x$, $BC$ have $2y$, $CD$ have $2z$, and $DA$ have $2w$. \nThen our inequality reduces to, for $x+y+z+w = 180^\\circ$: \\[|\\sin x - \\sin z| + |\\sin y - \\sin w| \\ge 2|\\sin (x+y) - \\sin (y+z)|.\\]\nThis is equivalent to by sum-to-product and use of $\\cos x = \\sin (90^\\circ - x)$:\n\\[|\\sin \\frac{x-z}{2} \\sin \\frac{y+w}{2}| + |\\sin \\frac{y-w}{2} \\sin \\frac{x+z}{2}| \\ge 2|\\sin \\frac{x-z}{2} \\sin \\frac{y-w}{2}|.\\]\nClearly $90^\\circ \\ge \\frac{x+z}{2} > \\frac{x-z}{2} \\ge 0^\\circ$. As sine is increasing over $[0, \\pi/2]$, $|\\sin \\frac{x+z}{2}| > |\\sin \\frac{x-z}{2}|$.\nSimilarly, $|\\sin \\frac{y+w}{2}| > |\\sin \\frac{y-w}{2}|$. The result now follows after multiplying the first inequality by $|\\sin \\frac{x-z}{2}|$, the second by $|\\sin \\frac{y-w}{2}|$, and adding. (Equality holds if and only if $x=z$ and $y=w$, ie. $ABCD$ is a parallelogram.)" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1975_USAMO_Problems_3", "eval_func": null }, "input": { "text": "If $P(x)$ denotes a polynomial of degree $n$ such that \\[P(k)=\\frac{k}{k+1}\\] for $k=0,1,2,\\ldots,n$, determine $P(n+1)$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $Q(x) = (x+1)P(x) - x$, and clearly, $Q(x)$ has a degree of $n+1$. \nThen, for $k=0,1,2,\\ldots,n$, $Q(k) = (k+1)P(k) - k = (k+1)\\cdot \\dfrac{k}{k+1} - k = 0$. \nThus, $k=0,1,2,\\ldots,n$ are the roots of $Q(x)$. \nSince these are all $n+1$ of the roots of the $n+1^{\\text{th}}$ degree polynomial, by the , we can write $Q(x)$ as \\[Q(x) = c(x)(x-1)(x-2) \\cdots (x-n)\\] where $c$ is a constant. \nThus, \\[(x+1)P(x) - x = c(x)(x-1)(x-2) \\cdots (x-n).\\]\nWe plug in $x = -1$ to cancel the $(x+1)P(x)$ and find $c$: \n\\begin{align*} -(-1) &= c(-1)(-1-1)(-1-2) \\cdots (-1-n) \\\\ 1 &= c(-1)^{n+1}(1)(2) \\cdots (n+1) \\\\ c &= (-1)^{n+1}\\dfrac{1}{(n+1)!} \\\\ \\end{align*}\nFinally, plugging in $x = n+1$ to find $P(n+1)$ gives: \n\\begin{align*} Q(n+1)&=(n+2)P(n+1)-(n+1)\\\\ (-1)^{n+1}\\dfrac{1}{(n+1)!}\\cdot(n+1)! &=(n+2)P(n+1)-(n+1)\\\\ (-1)^{n+1}&=(n+2)P(n+1)-(n+1)\\\\ (-1)^{n+1}+(n+1)&=(n+2)P(n+1)\\\\ P(n+1) &= \\dfrac{(-1)^{n+1} + (n+1)}{n+2}\\\\ \\end{align*}\nIf $n$ is even, this simplifies to $P(n+1) = \\dfrac{n}{n+2}$. If $n$ is odd, this simplifies to $P(n+1) = 1$. $\\Box$", "It is fairly natural to use Lagrange's Interpolation Formula on this problem:\n\\begin{align*} P(n+1) &= \\sum_{k=0}^n \\frac{k}{k+1} \\prod_{j \\ne k} \\frac{n+1-j}{k-j} \\\\ &= \\sum_{k=0}^n \\frac{k}{k+1} \\cdot \\frac{\\frac{(n+1)!}{n+1-k}}{k(k-1)(k-2) \\dots 1\\cdot (-1)(-2) \\dots (k-n)} \\\\ &= \\sum_{k=0}^n \\frac{k}{k+1} (-1)^{n-k}\\cdot \\frac{(n+1)!}{k!(n+1-k)!} \\\\ &= \\sum_{k=0}^n (-1)^{n-k} \\binom{n+1}{k} - \\sum_{k=0}^n \\frac{(n+1)!(-1)^{n-k}}{(k+1)!(n+1-k)!} \\\\ &= -\\left(\\sum_{k=0}^{n+1} (-1)^{n+1-k} \\binom{n+1}{k} - 1\\right) + \\frac{1}{n+2} \\cdot \\sum_{k=0}^n (-1)^{n+1-k} \\binom{n+2}{k+1} \\\\ &= 1 + \\frac{1}{n+2} \\left(\\sum_{k=-1}^{n+1} (-1)^{n+2 - (k+1)} \\binom{n+2}{k+1} - (-1)^{n+2} - 1\\right) \\\\ &= \\boxed{1 - \\frac{(-1)^n + 1}{n+2}} \\end{align*}\nthrough usage of the Binomial Theorem. $\\square$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1995_USAMO_Problems_5", "eval_func": null }, "input": { "text": "Suppose that in a certain society, each pair of persons can be classified as either or . We shall say that each member of an amicable pair is a of the other, and each member of a hostile pair is a of the other. Suppose that the society has $\\, n \\,$ persons and $\\, q \\,$ amicable pairs, and that for every set of three persons, at least one pair is hostile. Prove that there is at least one member of the society whose foes include $\\, q(1 - 4q/n^2) \\,$ or fewer amicable pairs.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Consider the graph with two people joined if they are friends.\nFor each person $X$, let $A(X)$ be the set of its friends and $B(X)$ the set of its foes. Note that any edge goes either: from $X$ to $A(X)$ (type 1), from $A(X)$ to $B(X)$ (type 2) or from a point of $B(X)$ to another (type 3), but there's no edge joining two points of $A(X)$ (since they would form a triangle with $X$). Let the number of type 1, type 2, type 3 edges of $X$ be $x_1, x_2, x_3$ respectively, so that $x_1$ is the degree of $X$ and we want to show that for some $X$, we have $x_3 \\leq q(1-4q/n^2)$.\nSince each edge is of one of those types, we have $x_1+x_2+x_3=q$. Thus\n\\[x_3 \\leq q(1-4q/n^2) \\Longleftrightarrow \\\\ x_3 \\leq q-4q^2/n^2 \\Longleftrightarrow \\\\ q-x_1-x_2 \\leq q-4q^2/n^2 \\Longleftrightarrow \\\\ x_1+x_2 \\geq 4q^2/n^2.\\]\nThat is, what we want is equivalent to proving that for some vertex $X$, the set of edges touching either $X$ or a vertex joined to $X$ is at least $4q^2/n^2$. Obviously now we'll sum $x_1+x_2$ over all vertices $X$. In the resulting sum, an edge joining $X, Y$ is counted once for each edge that $X, Y$ have, that is, it is counted $D(X)+D(Y)$ times, where $D(X)=x_1$ is the degree of $X$. Thus each vertex $X$ contributes to the overall sum with $D(X)$ for each edge it has, and since it has $D(X)$ edges, it contributes with $D(X)^2$. Thus the considered sum is equal to\n\\[\\sum_{X \\in G} D(X)^2 \\geq \\frac{(\\sum_{X \\in G} D(X))^2}{n}=\\frac{(2q)^2}{n}=4q^2/n.\\]\n(That's Cauchy.) Since we are summing over $n$ vertices, one of the summands is at least $4q^2/n^2$ by pigeonhole, which is what we wanted to prove." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2014_USAMO_Problems_3", "eval_func": null }, "input": { "text": "Prove that there exists an infinite set of points \\[\\ldots,\\,\\,\\,\\,P_{-3},\\,\\,\\,\\,P_{-2},\\,\\,\\,\\,P_{-1},\\,\\,\\,\\,P_0,\\,\\,\\,\\,P_1,\\,\\,\\,\\,P_2,\\,\\,\\,\\,P_3,\\,\\,\\,\\,\\ldots\\] in the plane with the following property: For any three distinct integers $a,b,$ and $c$, points $P_a$, $P_b$, and $P_c$ are collinear if and only if $a+b+c=2014$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Consider an elliptic curve with a generator $g$, such that $g$ is not a root of $0$. By repeatedly adding $g$ to itself under the standard group operation, with can build $g, 2g, 3g, \\ldots$ as well as $-g, -2g, -3g, \\ldots$. If we let \\[P_k = (3k-2014)g\\] then we can observe that collinearity between $P_a$, $P_b$, and $P_c$ occurs only if $P_a + P_b + P_c = 0$ (by definition of the group operation), which is equivalent to $(3a-2014)g + (3b-2014)g + (3c-2014)g = (3a+3b+3c-3*2014)g = 0$, or $3a + 3b + 3c = 3*2014$, or $a + b + c = 2014$. We know that all these points $P_k$ exist because $3k-2014$ is never 0 for integer $k$, so that none of these points need to be point at infinity (the identity element of the group).", "Consider letting $P_x$ be the point $(x, f(x))$, where $f(x) = x^3 - 2014x^2$. Then if three points $P_a, P_b, P_c$ are on the same line $y = mx + p$, they must be the solutions to the equation $x^3 - 2014x^2 = mx + p$ (i.e. the intersection of $f$ and the line). By Vieta's Formulas, the sum of the roots to this equation is 2014. Because each value of x corresponds to a different one of a, b, and c by definition of $P_x$, $a + b + c = 2014$. Conversely, if $a + b + c = 2014$, they must be the solutions to $(x-a)(x-b)(x-c) = x^3 - 2014x^2 - mx - p = 0$ for some real $m$ and $p$. Clearly, then, $P_a, P_b, P_c$ must all lie on the line $y = mx + p$. Hence, our setting $P_x = f(x)$ produces a valid infinite set of points." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2006_USAMO_Problems_2", "eval_func": null }, "input": { "text": "For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k+1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $N/2$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let one optimal set of integers be $\\{a_1,\\dots,a_{2k+1}\\}$ with $a_1 > a_2 > \\cdots > a_{2k+1} > 0$.\nThe two conditions can now be rewritten as $a_1+\\cdots + a_k \\leq N/2$ and $a_1+\\cdots +a_{2k+1} > N$.\nSubtracting, we get that $a_{k+1}+\\cdots + a_{2k+1} > N/2$, and hence $a_{k+1}+\\cdots + a_{2k+1} > a_1+\\cdots + a_k$.\nIn words, the sum of the $k+1$ smallest numbers must exceed the sum of the $k$ largest ones.\nLet $a_{k+1}=C$. As all the numbers are distinct integers, we must have $\\forall i \\in\\{1,\\dots,k\\}:~ a_{k+1-i} \\geq C+i$, and also $\\forall i \\in\\{1,\\dots,k\\}:~ a_{k+1+i} \\leq C-i$.\nThus we get that $a_1+\\cdots + a_k \\geq kC + \\dfrac{k(k+1)}2$, and $a_{k+1}+\\cdots + a_{2k+1} \\leq (k+1)C - \\dfrac{k(k+1)}2$.\nAs we want the second sum to be larger, clearly we must have $(k+1)C - \\dfrac{k(k+1)}2 > kC + \\dfrac{k(k+1)}2$.\nThis simplifies to $C > k(k+1)$.\nHence we get that:\n\\begin{align*} N & \\geq 2(a_1+\\cdots + a_k) \\\\ & \\geq 2\\left( kC + \\dfrac{k(k+1)}2 \\right) \\\\ & = 2kC + k(k+1) \\\\ & \\geq 2k(k^2+k+1) + k(k+1) \\\\ & = 2k^3 + 3k^2 + 3k \\end{align*}\nOn the other hand, for the set $\\{ k^2+k+1+i ~|~ i\\in\\{-k,\\dots,k\\} \\, \\}$ the sum of the largest $k$ elements is exactly $k^3 + k^2 + k + \\dfrac{k(k+1)}2$, and the sum of the entire set is $(k^2+k+1)(2k+1) = 2k^3 + 3k^2 + 3k + 1$, which is more than twice the sum of the largest set.\nHence the smallest possible $N$ is $\\boxed{ N = 2k^3 + 3k^2 + 3k }$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2010_USAMO_Problems_6", "eval_func": null }, "input": { "text": "A blackboard contains 68 pairs of nonzero integers. Suppose that for each positive integer $k$ at most one of the pairs $(k, k)$ and $(-k, -k)$ is written on the blackboard. A student erases some of the 136 integers, subject to the condition that no two erased integers may add to 0. The student then scores one point for each of the 68 pairs in which at least one integer is erased. Determine, with proof, the largest number $N$ of points that the student can guarantee to score regardless of which 68 pairs have been written on the board.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Answer: 43\n\nAttainability: Consider 8 distinct positive numbers. Let there be 5 pairs for each of the numbers including 2 clones of that number. Let there also be 28 pairs that include the negatives of those numbers such that each negative associates with another negative once and exactly once (in graph theoretic terms, a K8). Let x be the number of positives chosen out of these 8 (assume the other chosen numbers are negatives) for erasing; then, the number of points the student scores is $28 + 5x - {x \\choose 2}$, which is maximized at x=5 and x=6, and the maximum value is $43$. Choosing the first 5 numbers as positive and the other three as negative attains this. Hence, 43 is a possible maximum possible score.\n\nBounding: We use expected values. WLOG all the pairs of numbers with both numbers identical have only positive values. Consider flipping a weighted coin whether to choose the positive number or its negation for each positive number; it chooses positive with probability p. The pairs with both numbers same are chosen with probability p, the pairs (k, -k) are chosen with probability 1, and the pairs (x, y) for distinct x and y that sum to a nonzero number are chosen with probability $1-p^2$. We are trying to minimize the expected value, so we can assume that no pairs (k, -k) exist. Let A be the number of (k, k) pairs, and 68-A be the number of (x, y) pairs. The expected number of points scored is $Ap + (68-A)(1-p^2)$. We want to prove this is larger than 42 at all times for some choice of p. If $A < 36$, $1/2$ works for p to give this bound. If $A > 36$, $5/8$ works for p for p to give the desired bound. If $A = 36$, we can use $3/5$ for p to get the desired bound. Hence, in any case, the expected value for the number of points scored across all of these weighted processes is larger than 42, so there exists some case that gives a score of 43. Hence, bounding is complete. We are done with both parts. Q.E.D." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2004_USAMO_Problems_1", "eval_func": null }, "input": { "text": "Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that\n\\[\\frac {1}{3}|AB^3 - AD^3| \\le |BC^3 - CD^3| \\le 3|AB^3 - AD^3|.\\]\nWhen does equality hold?", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "It suffices to show that \\[|BC^3-CD^3| \\leq 3|AB^3-AD^3|,\\] because the other side of the inequality follows by symmetry. By Pitot's Theorem, we have $AB-AD=BC-CD$. WLOG, let $AB \\geq AD$. Then we have that $BC \\geq CD$, so we must show $BC^3-CD^3 \\leq 3(AB^3-AD^3)$. If $AB=AD$, we are done. Therefore, we will assume for the rest of the proof that $AB>AD$, $BC>CD$.\nThen we would like to show that \\[BC^2+BC\\cdot CD+CD^2 \\leq 3(AB^2+AB\\cdot AD+AD^2).\\]\nBy the Law of Cosines, we have that \\[BD^2=BC^2+k_1(BC\\cdot CD)+CD^2=AB^2+k_2(AB\\cdot AD)+AD^2,\\] where $-1 \\leq k_1, k_2 \\leq 1$ (this is because $60^{\\circ} \\leq \\angle BAD,\\angle BCD \\leq 120^{\\circ}$). Therefore, \\[3(BC^2-BC\\cdot CD+CD^2) \\leq 3(BC^2+k_1(BC\\cdot CD)+CD^2)=3(AB^2+k_2(AB\\cdot AD)+AD^2)\\leq 3(AB^2+AB\\cdot AD+AD^2).\\] Then we wish to show that \\[3(BC^2-BC\\cdot CD+CD^2)-(BC^2+BC\\cdot CD+CD^2) \\geq 0 \\rightarrow 2(BC-CD)^2 \\geq 0,\\] and we are done. $\\blacksquare$", "By a well-known property of tangential quadrilaterals, the sum of the two pairs of opposite sides are equal; hence $a + c = b + d \\Rightarrow a - b = d - c \\Rightarrow |a - b| = |d - c|$ Now we factor the desired expression into $\\frac {|d - c|(c^2 + d^2 + cd)}{3} \\le|a - b|(a^2 + b^2 + ab)\\le 3|d - c|(c^2 + d^2 + cd)$. Temporarily discarding the case where $a = b$ and $c = d$, we can divide through by the $|a - b| = |d - c|$ to get the simplified expression $(c^2 + d^2 + cd)/3\\le a^2 + b^2 + ab\\le 3(c^2 + d^2 + cd)$.\nNow, draw diagonal $BD$. By the law of cosines, $c^2 + d^2 + 2cd\\cos A = BD$. Since each of the interior and exterior angles of the quadrilateral is at least 60 degrees, we have that $A\\in [60^{\\circ},120^{\\circ}]$. Cosine is monotonically decreasing on this interval, so by setting $A$ at the extreme values, we see that $c^2 + d^2 - cd\\le BD^2 \\le c^2 + d^2 + cd$. Applying the law of cosines analogously to $a$ and $b$, we see that $a^2 + b^2 - ab\\le BD^2 \\le a^2 + b^2 + ab$; we hence have $c^2 + d^2 - cd\\le BD^2 \\le a^2 + b^2 + ab$ and $a^2 + b^2 - ab\\le BD^2 \\le c^2 + d^2 + cd$. \nWe wrap up first by considering the second inequality. Because $c^2 + d^2 - cd\\le BD^2 \\le a^2 + b^2 + ab$, $\\text{RHS}\\ge 3(a^2 + b^2 - ab)$. This latter expression is of course greater than or equal to $a^2 + b^2 + ab$ because the inequality can be rearranged to $2(a - b)^2\\ge 0$, which is always true. Multiply the first inequality by $3$ and we see that it is simply the second inequality with the variables swapped; hence by symmetry it is true as well.\nEquality occurs when $a = b$ and $c = d$, or when $ABCD$ is a kite." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1973_USAMO_Problems_2", "eval_func": null }, "input": { "text": "Let $\\{X_n\\}$ and $\\{Y_n\\}$ denote two sequences of integers defined as follows:\nThus, the first few terms of the sequences are:\nProve that, except for the \"1\", there is no term which occurs in both sequences.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We can look at each sequence $\\bmod{8}$:\nThe third and fourth terms are $3$ and $5$ $\\bmod{8}$. Plugging into the formula, we see that the next term is $11\\equiv 3\\bmod{8}$, and plugging $5$ and $3$, we get that the next term is $13\\equiv 5\\bmod{8}$. Thus the sequence $X$ repeats, and the pattern is $3,5,3,5,\\dots$.\nThe first and second terms are $1$ and $7$ $\\bmod{8}$. Plugging into the formula, we see that the next term is $17\\equiv 1\\bmod{8}$, and plugging $7$ and $1$, we get that the next term is $23\\equiv 7\\bmod{8}$. Thus the sequence $Y$ repeats, and the pattern is $1,7,1,7,\\dots$.\n\nCombining both results, we see that $X_i$ and $Y_j$ are not congruent $\\bmod{8}$ when $i\\geq 3$ and $j\\geq 2$. Thus after the \"1\", the terms of each sequence are not equal." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2001_USAMO_Problems_6", "eval_func": null }, "input": { "text": "Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We label each upper case point with the corresponding lower case letter as its assigned number. The key step is the following lemma.\n: If $ABCD$ is an isosceles trapezoid, then $a + c = b + d$.\n: Assume without loss of generality that $BC\\parallel AD$, and that rays $AB$ and $DC$ meet at $P$. Let $I$ be the incenter of triangle $PAC$, and let line $l$ bisect $\\angle APD$. Then $I$ is on $l$, so reflecting everything across line $l$ shows that $I$ is also the incenter of triangle $PDB$. Therefore,\n\\[\\frac{p + a + c}{3} = i = \\frac{p + b + d}{3}.\\]\nHence $a + c = b + d$, as desired.$\\blacksquare$\nFor any two distinct points $A_1$ and $A_2$ in the plane, we construct a regular pentagon $A_1A_2A_3A_4A_5$. Applying the lemma to isosceles trapezoids $A_1A_3A_4A_5$ and $A_2A_3A_4A_5$ yields\n\\[a_1 + a_4 = a_3 + a_5\\quad\\text{and}\\quad a_2 + a_4 = a_3 + a_5.\\]\nHence $a_1 = a_2$. Since $A_1$ and $A_2$ were arbitrary, all points in the plane are assigned the same number." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1985_USAMO_Problems_2", "eval_func": null }, "input": { "text": "Determine each real root of\n$x^4-(2\\cdot10^{10}+1)x^2-x+10^{20}+10^{10}-1=0$\ncorrect to four decimal places.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The equation can be re-written as \n\\begin{align}\\label{eqn1} (x+10^5)^2(x-10^5)^2 -(x+10^5)(x-10^5) -x-1=0. \\end{align}\nWe first prove that the equation has no negative roots.\nLet $x\\le 0.$ The equation above can be further re-arranged as\n\\begin{align*}[(x+10^5)(x-10^5)+1][(x+10^5)(x-10^5)-2]=x-1.\\end{align*}\nThe right hand side of the equation is negative. Therefore \\[[(x+10^5)(x-10^5)+1][(x+10^5)(x-10^5)-2)]<0,\\] and we have\n$-1<(x+10^5)(x-10^5) <2.$ Then the left hand side of the equation is bounded by\n\\[|[(x+10^5)(x-10^5)+1][(x+10^5)(x-10^5)-2]|\\le 3\\times 3.\\]\nHowever, since $|(x+10^5)(x-10^5)|\\le 2$ and $x<0,$ it follows that $|x+10^5| <\\frac{2}{|x-10^5|}<2\\times 10^{-5}$ for negative $x.$ Then $x<2\\times 10^{-5}-10^5.$ The right hand side of the equation is then a large negative number. It cannot be equal to the left hand side which is bounded by 9.\nNow let $x>0.$ When $x=10^5,$ the left hand side of equation (1) is negative. Therefore the equation has real roots on both side of $10^5$, as its leading coefficient is positive. We will prove that $x=10^5$ is a good approximation of the roots (within $10^{-2}$). In fact, we can solve the \"quadratic\" equation (1) for $(x+10^5)(x-10^5)$:\n\\[(x+10^5)(x-10^5)=\\frac{1\\pm\\sqrt{1+4(x+1)}}{2}.\\]\nThen \\[x-10^5=\\frac{1\\pm\\sqrt{1+4(x+1)}}{2(x+10^5)}.\\]\nEasy to see that\n$|x-10^5| <1$ for positve $x.$ Therefore, $10^5-110^5,$ we choose \n\\[\\tilde{x}_2=10^5+\\frac{1+\\sqrt{1+4(10^5+1)}}{2(10^5+10^5)}.\\]\nWe can similarly prove it has the desired approximation." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1990_USAMO_Problems_3", "eval_func": null }, "input": { "text": "Suppose that necklace $\\, A \\,$ has 14 beads and necklace $\\, B \\,$ has \n19. Prove that for any odd integer $n \\geq 1$, there is a way to number \neach of the 33 beads with an integer from the sequence\n\\[\\{ n, n+1, n+2, \\dots, n+32 \\}\\]\nso that each integer is used once, and adjacent beads correspond to \nrelatively prime integers. (Here a \"necklace\" is viewed as a circle in \nwhich each bead is adjacent to two other beads.)", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "For every positive odd integer $n$, there exists a nonnegative integer $k \\le 17$ such that $n+k$ is relatively prime to $n+k+15$, $n+k+1$ is relatively prime to $n+k+14$.\n Consider the positive integers $n, n+1, n+2$. Note that at most one of these is divisible by 3, and at most one is divisible by 5. Therefore one of these, say $n+a$, is divisible by neither, and is therefore relatively prime to 15. Furthermore, $n+a+1$ and $n+(a+15)+1$ have different residues mod 13, so one of them, say $n+b+1$, is relatively prime to 13. Since $n+b \\equiv n+a \\pmod{15}$, $n+b$ is relatively prime to 15. But\n\\[\\gcd(n+b,n+b+15) = \\gcd\\bigl[ n+b,(n+b+15) - (n+b) \\bigr] = \\gcd(n+b,15) = 1,\\]\nso $n+b$ is relatively prime to $n+b+15$. Also,\n\\[\\gcd(n+b+1, n+b+14) = \\gcd \\bigl[ n+b+1, (n+b+14)-(n+b+1) \\bigr] = \\gcd( n+b+1, 13) ,\\]\nso $n+b+1$ is relatively prime to $n+b+14$. Finally, $b \\le a+15 \\le 17$. It follows that setting $k=b$ satisfies the lemma. $\\blacksquare$\nLet $k$ be an integer as described in the lemma. We place the integers $n, \\dotsc, n+k, n+k+15, \\dotsc, n+32$ on the necklace with 19 beads, and the integers $n+k+1, \\dotsc, n+k+14$ on the necklace with 14 beads, in those orders. Since $n$ is odd, $n$ is relatively prime to $n+32 = n+2^5$. By definition, $n+k$ and $n+k+15$ are relatively prime, as are $n+k+1$ and $n+k+14$; finally, $a$ and $a+1$ are relatively prime for all integers $a$. It follows that each bead in this arrangement is relatively prime to its neighbors. $\\blacksquare$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1975_USAMO_Problems_5", "eval_func": null }, "input": { "text": "A deck of $n$ playing cards, which contains three aces, is shuffled at random (it is assumed that all possible card distributions are equally likely). The cards are then turned up one by one from the top until the second ace appears. Prove that the expected (average) number of cards to be turned up is $(n+1)/2$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We begin by induction. Our base case is naturally when $n = 3$, as there can be no less than $3$ cards in the deck. The only way to turn up the second ace is to turn up the first, and then turn up the second, which requires $2$ moves. This indeed is equal to $(3 + 1)/2$. Next, we assume that the expected number of cards required to pull the second ace from a deck of n cards is in fact $(n + 1)/2$. Now, we see that to simulate the case with $n + 1$ cards, we simply must add a card to our deck. By our assumption, we can expect to turn up $(n + 1)/2$ cards to be turned up, including our second ace. Thus ahead of our second ace, we expect $(n - 1)/2$ cards , and after it $(n - 1)/2$ as well. When we add our card, it is ahead of the second ace half the time, and behind if half the time. Therefore we expect to add one card to the number of cards drawn half the time, and add nothing for half the time. Hence we expect to add half a card, and the expected value is now $((n + 1)/2) + 1/2)$, or $(n + 2)/2$, the desired outcome for the $n + 1$ case.", "Although we can induct as in the situation above, we can use a more direct approach.\nConsider a deck. The second ace is in the $k$th place. If we draw the deck backwards (which has equal probability as drawing it forwards), then we will draw it in the $n+1-k$th place, since the $n$th place will become the first, the $n-1$th the second, etc.\nAlso, note that when second ace is still the second ace when we draw the deck backwards. (we need to note this, else this argument holds for the first ace and third ace as well)\nNow, this implies that the expected location of the second ace is in the $(k+n+1-k)/2=(n+1)/2$ place.", "Like Solution 2, this solution is also a direct solution.\nThe three aces divide the deck into four piles with a non-negative number of cards. Let $a$, $b$, $c$, and $d$ be the number of cards in the first, second, third, and fourth piles, respectively, given a random deck $D$. Also, let $E(a), E(b), E(c), E(d)$ denote the expected value of $a$, $b$, $c$, and $d$ over all decks $D$. Then $a + b + c + d = n - 3$ (the deck minus the three aces).\nFor every deck $D$, there must exist other decks that permute the values of $a$, $b$, $c$, $d$. (i.e. if there is a deck with $a = 2, b = 3, c = 1, d = 4,$ there exists a deck with $a = 1, b = 2, c = 3, d = 4$.) Therefore, the value of $a$ over all decks $D$ is a permutation of the value of $b$ over all such decks, and as a result $E(a) = E(b)$. A similar result holds for $b$, $c$, and $d$, and so $E(a) = E(b) = E(c) = E(d)$. Because \\[E(a) + E(b) + E(c) + E(d) = E(a + b + c + d) = E(n-3) = n-3,\\] $E(a) = E(b) = \\frac{n-3}{4}$. We want to compute the number of cards up to and including the second ace, which is just \\[E(a) + 1 + E(b) + 1 = \\frac{n-3}{2} + 2 = \\frac{n+1}{2},\\] as desired." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2022_AIME_II_Problems_12", "eval_func": null }, "input": { "text": "Let $a, b, x,$ and $y$ be real numbers with $a>4$ and $b>1$ such that\\[\\frac{x^2}{a^2}+\\frac{y^2}{a^2-16}=\\frac{(x-20)^2}{b^2-1}+\\frac{(y-11)^2}{b^2}=1.\\]Find the least possible value of $a+b.$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Denote $P = \\left( x , y \\right)$.\nBecause $\\frac{x^2}{a^2}+\\frac{y^2}{a^2-16} = 1$, $P$ is on an ellipse whose center is $\\left( 0 , 0 \\right)$ and foci are $\\left( - 4 , 0 \\right)$ and $\\left( 4 , 0 \\right)$.\nHence, the sum of distance from $P$ to $\\left( - 4 , 0 \\right)$ and $\\left( 4 , 0 \\right)$ is equal to twice the major axis of this ellipse, $2a$.\nBecause $\\frac{(x-20)^2}{b^2-1}+\\frac{(y-11)^2}{b^2} = 1$, $P$ is on an ellipse whose center is $\\left( 20 , 11 \\right)$ and foci are $\\left( 20 , 10 \\right)$ and $\\left( 20 , 12 \\right)$.\nHence, the sum of distance from $P$ to $\\left( 20 , 10 \\right)$ and $\\left( 20 , 12 \\right)$ is equal to twice the major axis of this ellipse, $2b$.\nTherefore, $2a + 2b$ is the sum of the distance from $P$ to four foci of these two ellipses.\nTo make this minimized, $P$ is the intersection point of the line that passes through $\\left( - 4 , 0 \\right)$ and $\\left( 20 , 10 \\right)$, and the line that passes through $\\left( 4 , 0 \\right)$ and $\\left( 20 , 12 \\right)$.\nThe distance between $\\left( - 4 , 0 \\right)$ and $\\left( 20 , 10 \\right)$ is $\\sqrt{\\left( 20 + 4 \\right)^2 + \\left( 10 - 0 \\right)^2} = 26$.\nThe distance between $\\left( 4 , 0 \\right)$ and $\\left( 20 , 12 \\right)$ is $\\sqrt{\\left( 20 - 4 \\right)^2 + \\left( 12 - 0 \\right)^2} = 20$.\nHence, $2 a + 2 b = 26 + 20 = 46$.\nTherefore, $a + b = \\boxed{\\textbf{(023) }}.$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2012_AIME_I_Problems_1", "eval_func": null }, "input": { "text": "Find the number of positive integers with three not necessarily distinct digits, $abc$, with $a \\neq 0$ and $c \\neq 0$ such that both $abc$ and $cba$ are multiples of $4$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "A positive integer is divisible by $4$ if and only if its last two digits are divisible by $4.$ For any value of $b$, there are two possible values for $a$ and $c$, since we find that if $b$ is even, $a$ and $c$ must be either $4$ or $8$, and if $b$ is odd, $a$ and $c$ must be either $2$ or $6$. There are thus $2 \\cdot 2 = 4$ ways to choose $a$ and $c$ for each $b,$ and $10$ ways to choose $b$ since $b$ can be any digit. The final answer is then $4 \\cdot 10 = \\boxed{040}$.", "A number is divisible by four if its last two digits are divisible by 4. Thus, we require that $10b + a$ and $10b + c$ are both divisible by $4$. If $b$ is odd, then $a$ and $c$ must both be $2 \\pmod 4$ meaning that $a$ and $c$ are $2$ or $6$. If $b$ is even, then $a$ and $c$ must be $0 \\pmod 4$ meaning that $a$ and $c$ are $4$ or $8$. For each choice of $b$ there are $2$ choices for $a$ and $2$ for $c$ for a total of $10 \\cdot 2 \\cdot 2 = \\boxed{040}$ numbers.", "For this number to fit the requirements $bc$ and $ba$ must be divisible by 4. So $bc = 00, 04, 08, 12, 16, ... , 92, 96$ and so must $ba$ for each two digits of $bc$. There are two possibilities for $ba$ if $b$ is odd and three possibilities if $b$ is even. So there are $2^{2} \\cdot 5+3^{2} \\cdot 4 = 65$ possibilities but this overcounts when $a$ or $c = 0$. So when $bc = 00, 20, 40, 60, 80$ and the corresponding $ba$ should be removed, so $65 - 5 \\cdot 3 = 50$. But we are still overcounting when $b$ is even because then $a$ can be 0. So the answer is $50 - 10 \\cdot 1 = \\boxed{040}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2008_AIME_I_Problems_5", "eval_func": null }, "input": { "text": "A right circular cone has base radius $r$ and height $h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to its original position on the table after making $17$ complete rotations. The value of $h/r$ can be written in the form $m\\sqrt {n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The path is a circle with radius equal to the slant height of the cone, which is $\\sqrt {r^{2} + h^{2}}$. Thus, the length of the path is $2\\pi\\sqrt {r^{2} + h^{2}}$.\nAlso, the length of the path is 17 times the circumference of the base, which is $34r\\pi$. Setting these equal gives $\\sqrt {r^{2} + h^{2}} = 17r$, or $h^{2} = 288r^{2}$. Thus, $\\dfrac{h^{2}}{r^{2}} = 288$, and $\\dfrac{h}{r} = 12\\sqrt {2}$, giving an answer of $12 + 2 = \\boxed{014}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2003_AIME_I_Problems_6", "eval_func": null }, "input": { "text": "The sum of the areas of all triangles whose vertices are also vertices of a $1$ by $1$ by $1$ cube is $m + \\sqrt{n} + \\sqrt{p},$ where $m, n,$ and $p$ are integers. Find $m + n + p.$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Since there are $8$ vertices of a cube, there are ${8 \\choose 3} = 56$ total triangles to consider. They fall into three categories: there are those which are entirely contained within a single face of the cube (whose sides are two edges and one face diagonal), those which lie in a plane perpendicular to one face of the cube (whose sides are one edge, one face diagonal and one space diagonal of the cube) and those which lie in a plane oblique to the edges of the cube, whose sides are three face diagonals of the cube. \nEach face of the cube contains ${4\\choose 3} = 4$ triangles of the first type, and there are $6$ faces, so there are $24$ triangles of the first type. Each of these is a right triangle with legs of length $1$, so each triangle of the first type has area $\\frac 12$.\nEach edge of the cube is a side of exactly $2$ of the triangles of the second type, and there are $12$ edges, so there are $24$ triangles of the second type. Each of these is a right triangle with legs of length $1$ and $\\sqrt 2$, so each triangle of the second type has area $\\frac{\\sqrt{2}}{2}$.\nEach vertex of the cube is associated with exactly one triangle of the third type (whose vertices are its three neighbors), and there are $8$ vertices of the cube, so there are $8$ triangles of the third type. Each of the these is an equilateral triangle with sides of length $\\sqrt 2$, so each triangle of the third type has area $\\frac{\\sqrt 3}2$.\nThus the total area of all these triangles is $24 \\cdot \\frac12 + 24\\cdot\\frac{\\sqrt2}2 + 8\\cdot\\frac{\\sqrt3}2 = 12 + 12\\sqrt2 + 4\\sqrt3 = 12 + \\sqrt{288} + \\sqrt{48}$ and the answer is $12 + 288 + 48 = \\boxed{348}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2013_AIME_II_Problems_11", "eval_func": null }, "input": { "text": "Let $A = \\{1, 2, 3, 4, 5, 6, 7\\}$, and let $N$ be the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Find the remainder when $N$ is divided by $1000$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Any such function can be constructed by distributing the elements of $A$ on three tiers.\nThe bottom tier contains the constant value, $c=f(f(x))$ for any $x$. (Obviously $f(c)=c$.)\nThe middle tier contains $k$ elements $x\\ne c$ such that $f(x)=c$, where $1\\le k\\le 6$.\nThe top tier contains $6-k$ elements such that $f(x)$ equals an element on the middle tier.\nThere are $7$ choices for $c$. Then for a given $k$, there are $\\tbinom6k$ ways to choose the elements on the middle tier, and then $k^{6-k}$ ways to draw arrows down from elements on the top tier to elements on the middle tier.\nThus $N=7\\cdot\\sum_{k=1}^6\\tbinom6k\\cdot k^{6-k}=7399$, giving the answer $\\boxed{399}$.", "Define the three layers as domain $x$, codomain $f(x)$, and codomain $f(f(x))$. Each one of them is contained in the set $A$. We know that $f(f(x))$ is a constant function, or in other words, can only take on one value. So, we can start off by choosing that value $c$ in $7$ ways. So now, we choose the values that can be $f(x)$ for all those values should satisfy $f(f(x))=c$. Let's $S$ be that set of values. First things first, we must have $c$ to be part of $S$, for the $S$ is part of the domain of $x$. Since the values in $i\\in S$ all satisfy $f(i) = c$, we have $c$ to be a value that $f(x)$ can be. Now, for the elements other than $5$:\nIf we have $k$ elements other than $5$ that can be part of $S$, we will have $\\binom{6}{k}$ ways to choose those values. There will also be $k$ ways for each of the elements in $A$ other than $c$ and those in set $S$ (for when function $f$ is applied on those values, we already know it would be $c$). There are $6-k$ elements in $A$ other than $c$ and those in set $S$. Thus, there should be $6^{6-k}$ ways to match the domain $x$ to the values of $f(x)$. Summing up all possible values of $k$ ($[1,6]$), we have\n\\[\\sum_{k=1}^6 k^{6-k} = 6\\cdot 1 + 15\\cdot 16 + 20\\cdot 27 + 15\\cdot 16 + 6\\cdot 5 + 1) = 1057\\]\nMultiplying that by the original $7$ for the choice of $c$, we have $7 \\cdot 1057 = 7\\boxed{399}.$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2002_AIME_II_Problems_7", "eval_func": null }, "input": { "text": "It is known that, for all positive integers $k$,\nFind the smallest positive integer $k$ such that $1^2+2^2+3^2+\\ldots+k^2$ is a multiple of $200$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "$\\frac{k(k+1)(2k+1)}{6}$ is a multiple of $200$ if $k(k+1)(2k+1)$ is a multiple of $1200 = 2^4 \\cdot 3 \\cdot 5^2$. \nSo $16,3,25|k(k+1)(2k+1)$. \nSince $2k+1$ is always odd, and only one of $k$ and $k+1$ is even, either $k, k+1 \\equiv 0 \\pmod{16}$. \nThus, $k \\equiv 0, 15 \\pmod{16}$. \nIf $k \\equiv 0 \\pmod{3}$, then $3|k$. If $k \\equiv 1 \\pmod{3}$, then $3|2k+1$. If $k \\equiv 2 \\pmod{3}$, then $3|k+1$. \nThus, there are no restrictions on $k$ in $\\pmod{3}$. \nIt is easy to see that only one of $k$, $k+1$, and $2k+1$ is divisible by $5$. So either $k, k+1, 2k+1 \\equiv 0 \\pmod{25}$. \nThus, $k \\equiv 0, 24, 12 \\pmod{25}$. \nFrom the Chinese Remainder Theorem, $k \\equiv 0, 112, 224, 175, 287, 399 \\pmod{400}$. Thus, the smallest positive integer $k$ is $\\boxed{112}$.", "To elaborate, we write out all 6 possibilities of pairings. For example, we have\n$k \\equiv 24 \\pmod{25}$\n$k \\equiv 15 \\pmod{16}$\nis one pairing, and\n$k \\equiv 24 \\pmod{25}$\n$k \\equiv 0 \\pmod{16}$\nis another. We then solve this by writing the first as $16k+15 \\equiv 24 \\pmod{25}$ and then move the 15 to get $16k \\equiv 9 \\pmod{25}$.\nWe then list out all the mods of the multiples of $16$, and realize that each of these $6$ pairings can be generalized to become one of these multiples of $16$.\nThe chain is as follows:\n$16 \\pmod{25}$\n$7, 23, 14, 5, 21, 12, 3, 19, 10, 1, 17, 8, 24, 15, 6, 22, 13, 4, 20, 11, 27, 18, 9, 0,$ and then it loops.\nWe see that for the first equation we have $9 \\pmod {25}$ at the 24th position, so we then do $24(16)+15$ to get the first answer of 399.\nAgain, this is possible to repeat for all $6$ cases. CRT guarantees that we will have a solution before $25 \\times 16$ or $400$ and indeed we did\u00a0:P" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1987_AIME_Problems_11", "eval_func": null }, "input": { "text": "Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let us write down one such sum, with $m$ terms and first term $n + 1$:\n$3^{11} = (n + 1) + (n + 2) + \\ldots + (n + m) = \\frac{1}{2} m(2n + m + 1)$.\nThus $m(2n + m + 1) = 2 \\cdot 3^{11}$ so $m$ is a divisor of $2\\cdot 3^{11}$. However, because $n \\geq 0$ we have $m^2 < m(m + 1) \\leq 2\\cdot 3^{11}$ so $m < \\sqrt{2\\cdot 3^{11}} < 3^6$. Thus, we are looking for large factors of $2\\cdot 3^{11}$ which are less than $3^6$. The largest such factor is clearly $2\\cdot 3^5 = 486$; for this value of $m$ we do indeed have the valid expression $3^{11} = 122 + 123 + \\ldots + 607$, for which $k=\\boxed{486}$.", "First note that if $k$ is odd, and $n$ is the middle term, the sum equals $kn$. If $k$ is even, then we have the sum equal to $kn+k/2$, which will be even. Since $3^{11}$ is odd, we see that $k$ is odd. \nThus, we have $nk=3^{11} \\implies n=3^{11}/k$. Also, note $n-(k+1)/2=0 \\implies n=(k+1)/2.$ Subsituting $n=3^{11}/k$, we have $k^2+k=2*3^{11}$. Proceed as in solution 1.", "Proceed as in Solution 1 until it is noted that $m$ is a divisor of $2\\cdot 3^{11}$. The divisors of $2\\cdot 3^{11}$ are $3^{1} , 2\\cdot 3^{1} , 3^{2} , 2\\cdot 3^{2} , \\ldots , 2\\cdot 3^{10} , 3^{11}$. Note that the factors of $m(2n + m + 1)$ are of opposite parity (if $m$ is odd, then $(2n + m + 1)$ is even and vice versa). Thus, one of the two factors will be a power of three, and the other will be twice a power of three. $(2n + m + 1)$ will represent the greater factor while $m$ will represent the lesser factor. Given this information, we need to find the factor pair that maximizes the lesser of the two factors, as this will maximize the value of $m$. The factor pair which maximizes the lesser factor is $2\\cdot 3^{5}$ and $3^{6}$. It follows that $m$ = $2\\cdot 3^{5}$ = $\\boxed{486}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2003_AIME_I_Problems_12", "eval_func": null }, "input": { "text": "In convex quadrilateral $ABCD, \\angle A \\cong \\angle C, AB = CD = 180,$ and $AD \\neq BC.$ The perimeter of $ABCD$ is $640$. Find $\\lfloor 1000 \\cos A \\rfloor.$ (The notation $\\lfloor x \\rfloor$ means the greatest integer that is less than or equal to $x.$)", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "By the Law of Cosines on $\\triangle ABD$ at angle $A$ and on $\\triangle BCD$ at angle $C$ (note $\\angle C = \\angle A$),\n\\[180^2 + AD^2 - 360 \\cdot AD \\cos A = 180^2 + BC^2 - 360 \\cdot BC \\cos A\\]\n\\[(AD^2 - BC^2) = 360(AD - BC) \\cos A\\]\n\\[(AD - BC)(AD + BC) = 360(AD - BC) \\cos A\\]\n\\[(AD + BC) = 360 \\cos A\\]\nWe know that $AD + BC = 640 - 360 = 280$. $\\cos A = \\dfrac{280}{360} = \\dfrac{7}{9} = 0.777 \\ldots$\n$\\lfloor 1000 \\cos A \\rfloor = \\boxed{777}$.", "Notice that $AB = CD$, and $BD = DB$, and $\\angle{DAB} \\cong \\angle{BCD}$, so we have side-side-angle matching on triangles $ABD$ and $CDB$. Since the problem does not allow $\\triangle{ABD} \\cong \\triangle{CDB}$, we know that $\\angle{ADB}$ is not a right angle, and there is a unique other triangle with the matching side-side-angle.\nExtend $AD$ to $C'$ so that $\\triangle{ABC'}$ is isosceles with $AB = C'B$. Then notice that $\\triangle{DC'B}$ has matching side-side-angle, and yet $\\triangle{ADB} \\not\\cong \\triangle{C'DB}$ because $\\angle{ADB}$ is not right. Therefore $\\triangle{C'DB}$ is the unique triangle mentioned above, so $\\triangle{CDB}$ is congruent, in some order of vertices, to $\\triangle{C'DB}$. Since $\\triangle{CDB} \\cong \\triangle{C'DB}$ would imply $\\triangle{CDB} = \\triangle{C'DB}$, making quadrilateral $ABCD$ degenerate, we must have $\\triangle{CDB} \\cong \\triangle{C'BD}$.\nSince the perimeter of $ABCD$ is $640$, $AD + BC = 640 - 180 - 180 = 280$. Hence $280 = AD + BC = AD + DC'$. Drop the altitude of $\\triangle{ABC'}$ from $B$ and call the foot $P$. Then right triangle trigonometry on $\\triangle{APB}$ shows that $\\cos{A} = AP/AB = 140/180 = 7/9$, so $\\lfloor 1000 \\cos A \\rfloor = \\boxed{777}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1999_AIME_Problems_8", "eval_func": null }, "input": { "text": "Let $\\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\\ge a, y\\ge b, z\\ge c.$ Let $\\mathcal{S}$ consist of those triples in $\\mathcal{T}$ that support $\\left(\\frac 12,\\frac 13,\\frac 16\\right).$ The area of $\\mathcal{S}$ divided by the area of $\\mathcal{T}$ is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n.$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "This problem just requires a good diagram and strong 3D visualization.\n\nThe region in $(x,y,z)$ where $x \\ge \\frac{1}{2}, y \\ge \\frac{1}{3}$ is that of a little triangle on the bottom of the above diagram, of $y \\ge \\frac{1}{3}, z \\ge \\frac{1}{6}$ is the triangle at the right, and $x \\ge \\frac 12, z \\ge \\frac 16$ the triangle on the left, where the triangles are coplanar with the large equilateral triangle formed by $x+y+z=1,\\ x,y,z \\ge 0$. We can check that each of the three regions mentioned fall under exactly two of the inequalities and not the third. \n\nThe side length of the large equilateral triangle is $\\sqrt{2}$, which we can find using 45-45-90 $\\triangle$ with the axes. Using the formula $A = \\frac{s^2\\sqrt{3}}{4}$ for equilateral triangles, the area of the large triangle is $\\frac{(\\sqrt{2})^2\\sqrt{3}}{4} = \\frac{\\sqrt{3}}{2}$. Since the lines of the smaller triangles are parallel to those of the large triangle, by corresponding angles we see that all of the triangles are similar, so they are all equilateral triangles. We can solve for their side lengths easily by subtraction, and we get $\\frac{\\sqrt{2}}{6}, \\frac{\\sqrt{2}}{3}, \\frac{\\sqrt{2}}{2}$. Calculating their areas, we get $\\frac{\\sqrt{3}}{8}, \\frac{\\sqrt{3}}{18}, \\frac{\\sqrt{3}}{72}$. The ratio $\\frac{\\mathcal{S}}{\\mathcal{T}} = \\frac{\\frac{9\\sqrt{3} + 4\\sqrt{3} + \\sqrt{3}}{72}}{\\frac{\\sqrt{3}}{2}} = \\frac{14}{36} = \\frac{7}{18}$, and the answer is $m + n = \\boxed{025}$.\nTo simplify the problem, we could used the fact that the area ratios are equal to the side ratios squared, and we get $\\left(\\frac{1}{2}\\right)^2 + \\left(\\frac{1}{3}\\right)^2 + \\left(\\frac{1}{6}\\right)^2 = \\frac{14}{36} = \\frac{7}{18}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2011_AIME_I_Problems_12", "eval_func": null }, "input": { "text": "Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that $p$ does not exceed 1 percent.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $n$ be the number of women present, and let _ be some positive number of women between groups of men. Since the problem states that every man stands next to another man, there cannot be isolated men. Thus, there are five cases to consider, where $(k)$ refers to a consecutive group of $k$ men:\nFor the first case, we can place the three groups of men in between women. We can think of the groups of men as dividers splitting up the $n$ women. Since there are $n+1$ possible places to insert the dividers, and we need to choose any three of these locations, we have $\\dbinom{n+1}{3}$ ways.\nThe second, third, and fourth cases are like the first, only that we need to insert two dividers among the $n+1$ possible locations. Each gives us $\\dbinom{n+1}{2}$ ways, for a total of $3\\dbinom{n+1}{2}$ ways.\nThe last case gives us $\\dbinom{n+1}{1}=n+1$ ways.\nTherefore, the total number of possible ways where there are no isolated men is\n\\[\\dbinom{n+1}{3}+3\\dbinom{n+1}{2}+(n+1).\\]\nThe total number of ways where there is a group of at least four men together is the sum of the third, fourth, and fifth case, or\n\\[2\\dbinom{n+1}{2}+(n+1).\\]\nThus, we want to find the minimum possible value of $n$ where $n$ is a positive integer such that\n\\[\\dfrac{2\\dbinom{n+1}{2}+(n+1)}{\\dbinom{n+1}{3}+3\\dbinom{n+1}{2}+(n+1)}\\le\\dfrac{1}{100}.\\]\nAfter simplification, we arrive at \\[\\dfrac{6(n+1)}{n^2+8n+6}\\le\\dfrac{1}{100}.\\]\nSimplifying again, we see that we seek the smallest positive integer value of $n$ such that $n(n-592)\\ge594$. Clearly $n>592$, or the left side will not even be positive; we quickly see that $n=593$ is too small but $n=\\boxed{594}$ satisfies the inequality." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2015_AIME_II_Problems_1", "eval_func": null }, "input": { "text": "Let $N$ be the least positive integer that is both $22$ percent less than one integer and $16$ percent greater than another integer. Find the remainder when $N$ is divided by $1000$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "If $N$ is $22$ percent less than one integer $k$, then $N=\\frac{78}{100}k=\\frac{39}{50}k$. In addition, $N$ is $16$ percent greater than another integer $m$, so $N=\\frac{116}{100}m=\\frac{29}{25}m$. Therefore, $k$ is divisible by $50$ and $m$ is divisible by $25$. Setting these two equal, we have $\\frac{39}{50}k=\\frac{29}{25}m$. Multiplying by $50$ on both sides, we get $39k=58m$.\nThe smallest integers $k$ and $m$ that satisfy this are $k=1450$ and $m=975$, so $N=1131$. The answer is $\\boxed{131}$.", "Continuing from Solution 1, we have $N=\\frac{39}{50}k$ and $N=\\frac{29}{25}m$. It follows that $k=\\frac{50}{39}N$ and $m=\\frac{25}{29}N$. Both $m$ and $k$ have to be integers, so, in order for that to be true, $N$ has to cancel the denominators of both $\\frac{50}{39}$ and $\\frac{25}{29}$. In other words, $N$ is a multiple of both $29$ and $39$. That makes $N=\\operatorname{lcm}(29,39)=29\\cdot39=1131$. The answer is $\\boxed{131}$.", "We can express $N$ as $0.78a$ and $1.22b$, where $a$ and $b$ are some positive integers. $N=0.78a=1.22b\\implies100N=78a=122b.$ Let us try to find the smallest possible value of $100N$, first ignoring the integral constraint. \nObviously, we are just trying to find the LCM of $78$ and $116.$ They have no common factors but $2$, so we multiply $78$ and $116$ and divide by $2$ to get $4524.$ This is obviously not divisible by $100$, so this is not possible, as it would imply that $N=\\dfrac{4524}{100},$ which is not an integer. This can be simplified to $\\dfrac{1131}{25}$. \nWe know that any possible value of $N$ will be an integral multiple of this value; the smallest such $N$ is achieved by multiplying this value by $25.$ We arrive at $1131$, which is $\\boxed{131}\\mod1000.$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2024_AIME_II_Problems_5", "eval_func": null }, "input": { "text": "Let ABCDEF be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are extensions of segments AB, CD, and EF has side lengths 200, 240, and 300. Find the side length of the hexagon.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "(Sorry i have zero idea how to make drawings)\nDraw a good diagram!\nLet $AB \\cap DC$, $CD \\cap FE$, and $BA \\cap EF$ be P, Q, and R, respectively. Let $QR=200, RP=300, PQ=240$. Notice that all smaller triangles formed are all similar to the larger $(200,240,300)$ triangle. Let the side length of the hexagon be x. Triangle $\\triangle BCP \\sim \\triangle RQP$, so $\\frac{BC}{BP} =\\frac{x}{BP} =\\frac{200}{300} \\implies BP=\\frac{3x}{2}$. Triangle $\\triangle AFR \\sim \\triangle PQR$, so $\\frac{AF}{AR}=\\frac{x}{AR} = \\frac{240}{300} \\implies AR=\\frac{5x}{4}$. We know $RA+AB+BP=300$, so $\\frac{5}{4}x + x + \\frac{3}{2}x = 300$. Solving, we get $x=\\boxed{080}$. ", "Draw an accurate diagram using the allowed compass and ruler: Draw a to-scale diagram of the $(200,240,300)$ triangle (e.g. 10cm-12cm-15cm). Because of the nature of these lengths and the integer answer needed, it can be assumed that the side length of the hexagon will be divisible by 10. Therefore, a trial-and-error method can be set up, wherein line segments of length $n\\cdot 10$, scaled to the same scale as the triangle, can be drawn to represent the sides of the hexagon. For instance, side $BC$ would be drawn parallel to the side of the triangle of length 300, and would have endpoints on the other sides of the triangle. Using this method, it would be evident that line segments of length 80 units, scaled proportionally (4cm using the scale above), would create a perfect equilateral hexagon when drawn parallel to the sides of the triangle. $x=\\boxed{080}$. " ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2001_AIME_II_Problems_14", "eval_func": null }, "input": { "text": "There are $2n$ complex numbers that satisfy both $z^{28} - z^{8} - 1 = 0$ and $\\mid z \\mid = 1$. These numbers have the form $z_{m} = \\cos\\theta_{m} + i\\sin\\theta_{m}$, where $0\\leq\\theta_{1} < \\theta_{2} < \\ldots < \\theta_{2n} < 360$ and angles are measured in degrees. Find the value of $\\theta_{2} + \\theta_{4} + \\ldots + \\theta_{2n}$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "$z$ can be written in the form $\\text{cis\\,}\\theta$. Rearranging, we find that $\\text{cis\\,}{28}\\theta = \\text{cis\\,}{8}\\theta+1$\nSince the real part of $\\text{cis\\,}{28}\\theta$ is one more than the real part of $\\text{cis\\,} {8}\\theta$ and their imaginary parts are equal, it is clear that either $\\text{cis\\,}{28}\\theta = \\frac{1}{2}+\\frac {\\sqrt{3}}{2}i$ and $\\text{cis\\,} {8}\\theta = -\\frac{1}{2}+\\frac {\\sqrt{3}}{2}i$, or $\\text{cis\\,}{28}\\theta = \\frac{1}{2} - \\frac{\\sqrt{3}}{2}i$ and $\\text{cis\\,} {8}\\theta = -\\frac{1}{2}- \\frac{\\sqrt{3}}{2}i$\nSetting up and solving equations, $Z^{28}= \\text{cis\\,}{60^\\circ}$ and $Z^8= \\text{cis\\,}{120^\\circ}$, we see that the solutions common to both equations have arguments $15^\\circ , 105^\\circ, 195^\\circ,$ and $\\ 285^\\circ$. We can figure this out by adding 360 repeatedly to the number 60 to try and see if it will satisfy what we need. We realize that it does not work in the integer values.\nAgain setting up equations ($Z^{28}= \\text{cis\\,}{300^\\circ}$ and $Z^{8} = \\text{cis\\,}{240^\\circ}$) we see that the common solutions have arguments of $75^\\circ, 165^\\circ, 255^\\circ,$ and $345^\\circ$\nListing all of these values, we find that $\\theta_{2} + \\theta_{4} + \\ldots + \\theta_{2n}$ is equal to $(75 + 165 + 255 + 345) ^\\circ$ which is equal to $\\boxed {840}$ degrees. We only want the sum of a certain number of theta, not all of it." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1989_AIME_Problems_12", "eval_func": null }, "input": { "text": "Let $ABCD$ be a tetrahedron with $AB=41$, $AC=7$, $AD=18$, $BC=36$, $BD=27$, and $CD=13$, as shown in the figure. Let $d$ be the distance between the midpoints of edges $AB$ and $CD$. Find $d^{2}$.\n[asy] defaultpen(fontsize(10)+0.8); size(175); pair A,B,C,D,M,P,Q; C=origin; B=(8,0); D=IP(CR(C,6.5),CR(B,8)); A=(4,-3); P=midpoint(A--B); Q=midpoint(C--D); draw(B--C--D--B--A--C^^A--D); draw(D--P--C^^P--Q, gray+dashed+0.5); pen p=fontsize(12)+linewidth(3); dot(\"$A$\",A,down,p); dot(\"$B$\",B,right,p); dot(\"$C$\",C,left,p); dot(\"$D$\",D,up,p); dot(\"$M$\",P,dir(-45),p); dot(\"$N$\",Q,0.2*(Q-P),p); label(\"$27$\",B--D,2*dir(30),fontsize(10)); label(\"$7$\",A--C,2*dir(210),fontsize(10)); label(\"$18$\",A--D,1.5*dir(30),fontsize(10)); label(\"$36$\",(3,0),up,fontsize(10)); [/asy]", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Call the midpoint of $\\overline{AB}$ $M$ and the midpoint of $\\overline{CD}$ $N$. $d$ is the median of triangle $\\triangle CDM$. The formula for the length of a median is $m=\\sqrt{\\frac{2a^2+2b^2-c^2}{4}}$, where $a$, $b$, and $c$ are the side lengths of triangle, and $c$ is the side that is bisected by median $m$. The formula is a direct result of the Law of Cosines applied twice with the angles formed by the median (Stewart's Theorem). We can also get this formula from the parallelogram law, that the sum of the squares of the diagonals is equal to the squares of the sides of a parallelogram.\nWe first find $CM$, which is the median of $\\triangle CAB$.\n\\[CM=\\sqrt{\\frac{98+2592-1681}{4}}=\\frac{\\sqrt{1009}}{2}\\]\nNow we must find $DM$, which is the median of $\\triangle DAB$.\n\\[DM=\\frac{\\sqrt{425}}{2}\\]\nNow that we know the sides of $\\triangle CDM$, we proceed to find the length of $d$.\n\\[d=\\frac{\\sqrt{548}}{2} \\Longrightarrow d^2=\\frac{548}{4}=\\boxed{137}\\]" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1997_AIME_Problems_4", "eval_func": null }, "input": { "text": "circles of radii $5, 5, 8,$ and $\\frac mn$ are mutually externally tangent, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "If (in the diagram above) we draw the line going through the centers of the circles with radii $8$ and $\\frac mn = r$, that line is the perpendicular bisector of the segment connecting the centers of the two circles with radii $5$. Then we form two right triangles, of lengths $5, x, 5+r$ and $5, 8+r+x, 13$, wher $x$ is the distance between the center of the circle in question and the segment connecting the centers of the two circles of radii $5$. By the Pythagorean Theorem, we now have two equations with two unknowns:\n\\begin{eqnarray*} 5^2 + x^2 &=& (5+r)^2 \\\\ x &=& \\sqrt{10r + r^2} \\\\ && \\\\ (8 + r + \\sqrt{10r+r^2})^2 + 5^2 &=& 13^2\\\\ 8 + r + \\sqrt{10r+r^2} &=& 12\\\\ \\sqrt{10r+r^2}&=& 4-r\\\\ 10r+r^2 &=& 16 - 8r + r^2\\\\ r &=& \\frac{8}{9} \\end{eqnarray*}\nSo $m+n = \\boxed{17}$.\n\nNOTE: It can be seen that there is no apparent need to use the variable x as a 5,12,13 right triangle has been formed.", "We may also use Descartes' theorem, $k_4=k_1+k_2+k_3\\pm 2\\sqrt{k_1k_2+k_2k_3+k_3k_1}$ where each of $k_i$ is the curvature of a circle with radius $r_i$, and the curvature is defined as $k_i=\\frac{1}{r_i}$. The larger solution for $k_4$ will give the curvature of the circle externally tangent to the other circles, while the smaller solution will give the curvature for the circle internally tangent to each of the other circles. Using Descartes' theorem, we get $k_4=\\frac15+\\frac15+\\frac18+2\\sqrt{\\frac{1}{40}+\\frac{1}{40}+\\frac{1}{25}}=\\frac{21}{40}+2\\sqrt{\\frac{45}{500}}=\\frac{45}{40}$. Thus, $r_4=\\frac{1}{k_4}=\\frac{40}{45}=\\frac89$, and the answer is $\\boxed{017}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1998_AIME_Problems_13", "eval_func": null }, "input": { "text": "If $\\{a_1,a_2,a_3,\\ldots,a_n\\}$ is a set of real numbers, indexed so that $a_1 < a_2 < a_3 < \\cdots < a_n,$ its complex power sum is defined to be $a_1i + a_2i^2+ a_3i^3 + \\cdots + a_ni^n,$ where $i^2 = - 1.$ Let $S_n$ be the sum of the complex power sums of all nonempty subsets of $\\{1,2,\\ldots,n\\}.$ Given that $S_8 = - 176 - 64i$ and $S_9 = p + qi,$ where $p$ and $q$ are integers, find $|p| + |q|.$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We note that the number of subsets (for now, including the empty subset, which we will just define to have a power sum of zero) with $9$ in it is equal to the number of subsets without a $9$. To easily see this, take all possible subsets of $\\{1,2,\\ldots,8\\}$. Since the sets are ordered, a $9$ must go at the end; hence we can just append a $9$ to any of those subsets to get a new one. \nNow that we have drawn that bijection, we can calculate the complex power sum recursively. Since appending a $9$ to a subset doesn't change anything about that subset's complex power sum besides adding an additional term, we have that $S_9 = 2S_8 + T_9$, where $T_9$ refers to the sum of all of the $9i^x$. \nIt a subset of size 1 has a 9, then its power sum must be $9i$, and there is only $1$ of these such subsets. There are ${8\\choose1}$ with $9\\cdot i^2$, ${8\\choose2}$ with $9\\cdot i^3$, and so forth. So $T_9 =\\sum_{k=0}^{8} 9{8\\choose{k}}i^{k+1}$. This is exactly the binomial expansion of $9i \\cdot (1+i)^8$. We can use De Moivre's Theorem to calculate the power: $(\\sqrt{2})^8\\cos{8\\cdot45} = 16$. Hence $T_9 = 16\\cdot9i = 144i$, and $S_9 = 2S_8 + 144i = 2(-176 -64i) + 144i = -352 + 16i$. Thus, $|p| + |q| = |-352| + |16| = 368$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2011_AIME_II_Problems_9", "eval_func": null }, "input": { "text": "Let $x_1, x_2, ... , x_6$ be non-negative real numbers such that $x_1 +x_2 +x_3 +x_4 +x_5 +x_6 =1$, and $x_1 x_3 x_5 +x_2 x_4 x_6 \\ge {\\frac{1}{540}}$. Let $p$ and $q$ be positive relatively prime integers such that $\\frac{p}{q}$ is the maximum possible value of\n$x_1 x_2 x_3 + x_2 x_3 x_4 +x_3 x_4 x_5 +x_4 x_5 x_6 +x_5 x_6 x_1 +x_6 x_1 x_2$. Find $p+q$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Note that neither the constraint nor the expression we need to maximize involves products $x_i x_j$ with $i \\equiv j \\pmod 3$. Factoring out say $x_1$ and $x_4$ we see that the constraint is $x_1(x_3x_5) + x_4(x_2x_6) \\ge {\\frac1{540}}$, while the expression we want to maximize is $x_1(x_2x_3 + x_5x_6 + x_6x_2) + x_4(x_2x_3 + x_5x_6 + x_3x_5)$. Adding the left side of the constraint to the expression, we get: $(x_1 + x_4)(x_2x_3 + x_5x_6 + x_6x_2 + x_3x_5) = (x_1 + x_4)(x_2 + x_5)(x_3 + x_6)$. This new expression is the product of three non-negative terms whose sum is equal to 1. By AM-GM this product is at most $\\frac1{27}$. Since we have added at least $\\frac{1}{540}$ the desired maximum is at most $\\frac1{27} - \\frac1{540} =\\frac{19}{540}$. It is easy to see that this upper bound can in fact be achieved by ensuring that the constraint expression is equal to $\\frac1{540}$ with $x_1 + x_4 = x_2 + x_5 = x_3 + x_6 =\\frac13$\u2014for example, by choosing $x_1$ and $x_2$ small enough\u2014so our answer is $540 + 19 = \\fbox{559}.$\nAn example is:\n\\begin{align*} x_3 &= x_6 = \\frac16 \\\\ x_1 &= x_2 = \\frac{5 - \\sqrt{20}}{30} \\\\ x_5 &= x_4 = \\frac{5 + \\sqrt{20}}{30} \\end{align*}\nAnother example is \n\\begin{align*} x_1 = x_3 = \\frac{1}{3} \\\\ x_2 = \\frac{19}{60}, \\ x_5 = \\frac{1}{60} \\\\ x_4 &= x_6 = 0 \\end{align*}", "There's a symmetry between $x_1, x_3, x_5$ and $x_2,x_4,x_6$. Therefore, a good guess is that $a = x_1 = x_3 = x_5$ and $b = x_2 = x_4 = x_6$, at which point we know that $a+b = 1/3$, $a^3+b^3 \\geq 1/540$, and we are trying to maximize $3a^2b+3ab^2$. Then,\n\\[3a^2b+3ab^2 = (a+b)^3-a^3-b^3 \\leq \\frac{1}{27} - \\frac{1}{540} = \\boxed{\\frac{19}{540}}\\] which is the answer.\nThis solution is extremely lucky; if you attempt to solve for $a$ and $b$ you receive complex answers (which contradict the problem statement), but the final answer is correct." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2000_AIME_I_Problems_12", "eval_func": null }, "input": { "text": "Given a function $f$ for which\n\\[f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)\\]\nholds for all real $x,$ what is the largest number of different values that can appear in the list $f(0),f(1),f(2),\\ldots,f(999)?$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "\\begin{align*}f(2158 - x) = f(x) &= f(3214 - (2158 - x)) &= f(1056 + x)\\\\ f(398 - x) = f(x) &= f(2158 - (398 - x)) &= f(1760 + x)\\end{align*}\nSince $\\mathrm{gcd}(1056, 1760) = 352$ we can conclude that (by the Euclidean algorithm)\n\\[f(x) = f(352 + x)\\]\nSo we need only to consider one period $f(0), f(1), ... f(351)$, which can have at most $352$ distinct values which determine the value of $f(x)$ at all other integers. \nBut we also know that $f(x) = f(46 - x) = f(398 - x)$, so the values $x = 24, 25, ... 46$ and $x = 200, 201, ... 351$ are repeated. This gives a total of\n\\[352 - (46 - 24 + 1) - (351 - 200 + 1) = \\boxed{ 177 }\\]\ndistinct values.\nTo show that it is possible to have $f(23), f(24), \\ldots, f(199)$ distinct, we try to find a function which fulfills the given conditions. A bit of trial and error would lead to cosine the function: $f(x) = \\cos \\left(\\frac{360}{352}(x-23)\\right)$ (in degrees)." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2004_AIME_II_Problems_3", "eval_func": null }, "input": { "text": "A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly $231$ of the 1-cm cubes cannot be seen. Find the smallest possible value of $N.$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The $231$ cubes which are not visible must lie below exactly one layer of cubes. Thus, they form a rectangular solid which is one unit shorter in each dimension. If the original block has dimensions $l \\times m \\times n$, we must have $(l - 1)\\times(m-1) \\times(n - 1) = 231$. The prime factorization of $231 = 3\\cdot7\\cdot11$, so we have a variety of possibilities; for instance, $l - 1 = 1$ and $m - 1 = 11$ and $n - 1 = 3 \\cdot 7$, among others. However, it should be fairly clear that the way to minimize $l\\cdot m\\cdot n$ is to make $l$ and $m$ and $n$ as close together as possible, which occurs when the smaller block is $3 \\times 7 \\times 11$. Then the extra layer makes the entire block $4\\times8\\times12$, and $N= \\boxed{384}$.\nAn alternate way to visualize the problem is to count the blocks that can be seen and subtract the blocks that cannot be seen. In the given block with dimensions $l\\times m \\times n$, the three faces have $lm$, $mn$, and $ln$ blocks each. However, $l$ blocks along the first edge, $m$ blocks along the second edge, and $n$ blocks along the third edge were counted twice, so they must be subtracted. After subtracting these three edges, 1 block has not been counted - it was added three times on each face, but subtracted three times on each side. Thus, the total number of visible cubes is $lm+mn+ln-l-m-n+1$, and the total number of invisible cubes is $lmn-lm-mn-ln+l+m+n-1$, which can be factored into $(l-1)(m-1)(n-1)$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2001_AIME_II_Problems_2", "eval_func": null }, "input": { "text": "Each of the $2001$ students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between $80$ percent and $85$ percent of the school population, and the number who study French is between $30$ percent and $40$ percent. Let $m$ be the smallest number of students who could study both languages, and let $M$ be the largest number of students who could study both languages. Find $M-m$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $S$ be the percent of people who study Spanish, $F$ be the number of people who study French, and let $S \\cap F$ be the number of students who study both. Then $\\left\\lceil 80\\% \\cdot 2001 \\right\\rceil = 1601 \\le S \\le \\left\\lfloor 85\\% \\cdot 2001 \\right\\rfloor = 1700$, and $\\left\\lceil 30\\% \\cdot 2001 \\right\\rceil = 601 \\le F \\le \\left\\lfloor 40\\% \\cdot 2001 \\right\\rfloor = 800$. By the Principle of Inclusion-Exclusion,\n\\[S+F- S \\cap F = S \\cup F = 2001\\]\nFor $m = S \\cap F$ to be smallest, $S$ and $F$ must be minimized.\n\\[1601 + 601 - m = 2001 \\Longrightarrow m = 201\\]\nFor $M = S \\cap F$ to be largest, $S$ and $F$ must be maximized.\n\\[1700 + 800 - M = 2001 \\Longrightarrow M = 499\\]\nTherefore, the answer is $M - m = 499 - 201 = \\boxed{298}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2017_AIME_I_Problems_14", "eval_func": null }, "input": { "text": "Let $a > 1$ and $x > 1$ satisfy $\\log_a(\\log_a(\\log_a 2) + \\log_a 24 - 128) = 128$ and $\\log_a(\\log_a x) = 256$. Find the remainder when $x$ is divided by $1000$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The first condition implies\n\\[a^{128} = \\log_a\\log_a 2 + \\log_a 24 - 128\\]\n\\[128+a^{128} = \\log_a\\log_a 2^{24}\\]\n\\[a^{a^{128}a^{a^{128}}} = 2^{24}\\]\n\\[\\left(a^{a^{128}}\\right)^{\\left(a^{a^{128}}\\right)} = 2^{24} = 8^8\\]\nSo $a^{a^{128}} = 8$. \nPutting each side to the power of $128$:\n\\[\\left(a^{128}\\right)^{\\left(a^{128}\\right)} = 8^{128} = 64^{64},\\]\nso $a^{128} = 64 \\implies a = 2^{\\frac{3}{64}}$. Specifically,\n\\[\\log_a(x) = \\frac{\\log_2(x)}{\\log_2(a)} = \\frac{64}{3}\\log_2(x)\\]\nso we have that\n\\[256 = \\log_a(\\log_a(x)) = \\frac{64}{3}\\log_2\\left(\\frac{64}{3}\\log_2(x)\\right)\\]\n\\[12 = \\log_2\\left(\\frac{64}{3}\\log_2(x)\\right)\\]\n\\[2^{12} = \\frac{64}{3}\\log_2(x)\\]\n\\[192 = \\log_2(x)\\]\n\\[x = 2^{192}\\]\nWe only wish to find $x\\bmod 1000$. To do this, we note that $x\\equiv 0\\bmod 8$ and now, by the Chinese Remainder Theorem, wish only to find $x\\bmod 125$. By Euler's Totient Theorem:\n\\[2^{\\phi(125)} = 2^{100} \\equiv 1\\bmod 125\\]\nso\n\\[2^{192} \\equiv \\frac{1}{2^8} \\equiv \\frac{1}{256} \\equiv \\frac{1}{6} \\bmod 125\\]\nso we only need to find the inverse of $6 \\bmod 125$. It is easy to realize that $6\\cdot 21 = 126 \\equiv 1\\bmod 125$, so \n\\[x\\equiv 21\\bmod 125, x\\equiv 0\\bmod 8.\\]\nUsing Chinese Remainder Theorem, we get that $x\\equiv \\boxed{896}\\bmod 1000$, finishing the solution.", "$\\log_a(\\log_a(\\log_a 2) + \\log_a 24 - 128) = 128$\n$\\implies \\log_a(\\log_a 2))+\\log_a(24)=a^{128}+128$\n$\\implies \\log_a(\\log_a 2^{24})=a^{128}+128$\n$\\implies 2^{24}=a^{a^{(a^{128}+128)}}$\nObviously letting $a=2^y$ will simplify a lot and to make the $a^{128}$ term simpler, let $a=2^{\\frac{y}{128}}$. Then, \n$2^{24}=2^{\\frac{y}{128} \\cdot 2^{\\frac{y}{128} \\cdot (2^y+128)}}=2^{\\frac{y}{128} \\cdot 2^{y \\cdot (2^{y-7}+1)}}$\n$\\implies 24=\\frac{y}{128} \\cdot 2^{y \\cdot (2^{y-7}+1)}$\n$\\implies 3 \\cdot 2^{10}=y \\cdot 2^{y \\cdot (2^{y-7}+1)}$\nObviously, $y$ is $3$ times a power of $2$. (It just makes sense.) Testing, we see $y=6$ satisfy the equation so $a=2^{\\frac{3}{64}}$. Therefore, $x=2^{192} \\equiv \\boxed{896} \\pmod{1000}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1989_AIME_Problems_7", "eval_func": null }, "input": { "text": "If the integer $k$ is added to each of the numbers $36$, $300$, and $596$, one obtains the squares of three consecutive terms of an arithmetic series. Find $k$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Call the terms of the arithmetic progression $a,\\ a + d,\\ a + 2d$, making their squares $a^2,\\ a^2 + 2ad + d^2,\\ a^2 + 4ad + 4d^2$.\nWe know that $a^2 = 36 + k$ and $(a + d)^2 = 300 + k$, and subtracting these two we get $264 = 2ad + d^2$ (1). Similarly, using $(a + d)^2 = 300 + k$ and $(a + 2d)^2 = 596 + k$, subtraction yields $296 = 2ad + 3d^2$ (2). \nSubtracting the first equation from the second, we get $2d^2 = 32$, so $d = 4$. Substituting backwards yields that $a = 31$ and $k = \\boxed{925}$.", "Since terms in an arithmetic progression have constant differences, \n\\[\\sqrt{300+k}-\\sqrt{36+k}=\\sqrt{596+k}-\\sqrt{300+k}\\]\n\\[\\implies 2\\sqrt{300+k} = \\sqrt{596+k}+\\sqrt{36+k}\\]\n\\[\\implies 4k+1200=596+k+36+k+2\\sqrt{(596+k)(36+k)}\\]\n\\[\\implies 2k+568=2\\sqrt{(596+k)(36+k)}\\]\n\\[\\implies k+284=\\sqrt{(596+k)(36+k)}\\]\n\\[\\implies k^2+568k+80656=k^2+632k+21456\\]\n\\[\\implies 568k+80656=632k+21456\\]\n\\[\\implies 64k = 59200\\]\n\\[\\implies k = \\boxed{925}\\]" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1999_AIME_Problems_5", "eval_func": null }, "input": { "text": "For any positive integer $x_{}$, let $S(x)$ be the sum of the digits of $x_{}$, and let $T(x)$ be $|S(x+2)-S(x)|.$ For example, $T(199)=|S(201)-S(199)|=|3-19|=16.$ How many values of $T(x)$ do not exceed 1999?", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "For most values of $x$, $T(x)$ will equal $2$. For those that don't, the difference must be bumping the number up a ten, a hundred, etc. If we take $T(a999)$ as an example,\n\\[|(a + 1) + 0 + 0 + 1 - (a + 9 + 9 + 9)| = |2 - 9(3)|\\]\nAnd in general, the values of $T(x)$ will then be in the form of $|2 - 9n| = 9n - 2$. From $7$ to $1999$, there are $\\left\\lceil \\frac{1999 - 7}{9}\\right\\rceil = 222$ solutions; including $2$ and there are a total of $\\boxed{223}$ solutions." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1988_AIME_Problems_2", "eval_func": null }, "input": { "text": "For any positive integer $k$, let $f_1(k)$ denote the square of the sum of the digits of $k$. For $n \\ge 2$, let $f_n(k) = f_1(f_{n - 1}(k))$. Find $f_{1988}(11)$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We see that $f_{1}(11)=4$\n$f_2(11) = f_1(4)=16$\n$f_3(11) = f_1(16)=49$\n$f_4(11) = f_1(49)=169$\n$f_5(11) = f_1(169)=256$\n$f_6(11) = f_1(256)=169$\nNote that this revolves between the two numbers. Since $1988$ is even, we thus have $f_{1988}(11) = f_{4}(11) = \\boxed{169}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2022_AIME_I_Problems_4", "eval_func": null }, "input": { "text": "Let $w = \\dfrac{\\sqrt{3} + i}{2}$ and $z = \\dfrac{-1 + i\\sqrt{3}}{2},$ where $i = \\sqrt{-1}.$ Find the number of ordered pairs $(r,s)$ of positive integers not exceeding $100$ that satisfy the equation $i \\cdot w^r = z^s.$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We rewrite $w$ and $z$ in polar form:\n\\begin{align*} w &= e^{i\\cdot\\frac{\\pi}{6}}, \\\\ z &= e^{i\\cdot\\frac{2\\pi}{3}}. \\end{align*}\nThe equation $i \\cdot w^r = z^s$ becomes\n\\begin{align*} e^{i\\cdot\\frac{\\pi}{2}} \\cdot \\left(e^{i\\cdot\\frac{\\pi}{6}}\\right)^r &= \\left(e^{i\\cdot\\frac{2\\pi}{3}}\\right)^s \\\\ e^{i\\left(\\frac{\\pi}{2}+\\frac{\\pi}{6}r\\right)} &= e^{i\\left(\\frac{2\\pi}{3}s\\right)} \\\\ \\frac{\\pi}{2}+\\frac{\\pi}{6}r &= \\frac{2\\pi}{3}s+2\\pi k \\\\ 3+r &= 4s+12k \\\\ 3+r &= 4(s+3k). \\end{align*}\nfor some integer $k.$\nSince $4\\leq 3+r\\leq 103$ and $4\\mid 3+r,$ we conclude that\n\\begin{align*} 3+r &\\in \\{4,8,12,\\ldots,100\\}, \\\\ s+3k &\\in \\{1,2,3,\\ldots,25\\}. \\end{align*}\nNote that the values for $s+3k$ and the values for $r$ have one-to-one correspondence.\nWe apply casework to the values for $s+3k:$\nTogether, the answer is $264+306+264=\\boxed{834}.$", "First we recognize that $w = \\operatorname{cis}(30^{\\circ})$ and $z = \\operatorname{cis}(120^{\\circ})$ because the cosine and sine sums of those angles give the values of $w$ and $z$, respectively. By De Moivre's theorem, $\\operatorname{cis}(\\theta)^n = \\operatorname{cis}(n\\theta)$. When you multiply by $i$, we can think of that as rotating the complex number $90^{\\circ}$ counterclockwise in the complex plane. Therefore, by the equation we know that $30r + 90$ and $120s$ land on the same angle.\nThis means that\n\\[30r + 90 \\equiv 120s \\pmod{360},\\]\nwhich we can simplify to \n\\[r+3 \\equiv 4s \\pmod{12}.\\]\nNotice that this means that $r$ cycles by $12$ for every value of $s$. This is because once $r$ hits $12$, we get an angle of $360^{\\circ}$ and the angle laps onto itself again. By a similar reasoning, $s$ laps itself every $3$ times, which is much easier to count. By listing the possible values out, we get the pairs $(r,s)$:\n\\[\\begin{array}{cccccccc} (1,1) & (5,2) & (9,3) & (13,1) & (17,2) & (21,3) & \\ldots & (97,1) \\\\ (1,4) & (5,5) & (9,6) & (13,4) & (17,5) & (21,6) & \\ldots & (97,4) \\\\ (1,7) & (5,8) & (9,9) & (13,7) & (17,8) & (21,9) & \\ldots & (97,7) \\\\ [-1ex] \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ (1,100) & (5,98) & (9,99) & (13,100) & (17,98) & (21,99) & \\ldots & (97,100) \\end{array}\\]\nWe have $25$ columns in total: $34$ values for the first column, $33$ for the second, $33$ for the third, and then $34$ for the fourth, $33$ for the fifth, $33$ for the sixth, etc. Therefore, this cycle repeats every $3$ columns and our total sum is $(34+33+33) \\cdot 8 + 34 = 100 \\cdot 8 + 34 = \\boxed{834}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1996_AIME_Problems_15", "eval_func": null }, "input": { "text": "In parallelogram $ABCD$, let $O$ be the intersection of diagonals $\\overline{AC}$ and $\\overline{BD}$. Angles $CAB$ and $DBC$ are each twice as large as angle $DBA$, and angle $ACB$ is $r$ times as large as angle $AOB$. Find $\\lfloor 1000r \\rfloor$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $\\theta = \\angle DBA$. Then $\\angle CAB = \\angle DBC = 2\\theta$, $\\angle AOB = 180 - 3\\theta$, and $\\angle ACB = 180 - 5\\theta$. Since $ABCD$ is a parallelogram, it follows that $OA = OC$. By the Law of Sines on $\\triangle ABO,\\, \\triangle BCO$,\nDividing the two equalities yields \n\\[\\frac{\\sin 2\\theta}{\\sin \\theta} = \\frac{\\sin (180 - 5\\theta)}{\\sin 2\\theta} \\Longrightarrow \\sin^2 2\\theta = \\sin 5\\theta \\sin \\theta.\\]\nPythagorean and product-to-sum identities yield\n\\[1 - \\cos^2 2 \\theta = \\frac{\\cos 4\\theta - \\cos 6 \\theta}{2},\\]\nand the double and triple angle ($\\cos 3x = 4\\cos^3 x - 3\\cos x$) formulas further simplify this to\n\\[4\\cos^3 2\\theta - 4\\cos^2 2\\theta - 3\\cos 2\\theta + 3 = (4\\cos^2 2\\theta - 3)(\\cos 2\\theta - 1) = 0\\]\nThe only value of $\\theta$ that fits in this context comes from $4 \\cos^2 2\\theta - 3 = 0 \\Longrightarrow \\cos 2\\theta = \\frac{\\sqrt{3}}{2} \\Longrightarrow \\theta = 15^{\\circ}$. The answer is $\\lfloor 1000r \\rfloor = \\left\\lfloor 1000 \\cdot \\frac{180 - 5\\theta}{180 - 3\\theta} \\right\\rfloor = \\left \\lfloor \\frac{7000}{9} \\right \\rfloor = \\boxed{777}$.", "Define $\\theta$ as above. Since $\\angle CAB = \\angle CBO$, it follows that $\\triangle COB \\sim \\triangle CBA$, and so $\\frac{CO}{BC} = \\frac{BC}{AC} \\Longrightarrow BC^2 = AC \\cdot CO = 2CO^2 \\Longrightarrow BC = CO\\sqrt{2}$. The Law of Sines on $\\triangle BOC$ yields that \n\\[\\frac{BC}{CO} = \\frac{\\sin (180-3\\theta)}{\\sin 2\\theta} = \\frac{\\sin 3\\theta}{\\sin 2\\theta} = \\sqrt{2}\\]\nExpanding using the sine double and triple angle formulas, we have \n\\[2\\sqrt {2} \\sin \\theta \\cos \\theta = \\sin\\theta( - 4\\sin ^2 \\theta + 3) \\Longrightarrow \\sin \\theta\\left(4\\cos^2\\theta - 2\\sqrt {2} \\cos \\theta - 1\\right) = 0.\\]\nBy the quadratic formula, we have $\\cos \\theta = \\frac {2\\sqrt {2} \\pm \\sqrt {8 + 4 \\cdot 1 \\cdot 4}}{8} = \\frac {\\sqrt {6} \\pm \\sqrt {2}}{4}$, so $\\theta = 15^{\\circ}$ (as the other roots are too large to make sense in context). The answer follows as above.", "We will focus on $\\triangle ABC$. Let $\\angle ABO = x$, so $\\angle BAO = \\angle OBC = 2x$. Draw the perpendicular from $C$ intersecting $AB$ at $H$. Without loss of generality, let $AO = CO = 1$. Then $HO = 1$, since $O$ is the circumcenter of $\\triangle AHC$. Then $\\angle OHA = 2x$. \nBy the Exterior Angle Theorem, $\\angle COB = 3x$ and $\\angle COH = 4x$. That implies that $\\angle HOB = x$. That makes $HO = HB = 1$. Then since by AA ($\\angle HBC = \\angle HOC = 3x$ and reflexive on $\\angle OCB$), $\\triangle OCB \\sim \\triangle BCA$.\nThen by the Pythagorean Theorem, $1^2 + HC^2 = \\left(\\sqrt {2}\\right)^2\\implies HC = 1$. That makes $\\triangle HOC$ equilateral. Then $\\angle HOC = 4x = 60 \\implies x = 15$. The answer follows as above." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2000_AIME_II_Problems_5", "eval_func": null }, "input": { "text": "Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "There are $\\binom{8}{5}$ ways to choose the rings, and there are $5!$ distinct arrangements to order the rings [we order them so that the first ring is the bottom-most on the first finger that actually has a ring, and so forth]. The number of ways to distribute the rings among the fingers is equivalent the number of ways we can drop five balls into 4 urns, or similarly dropping five balls into four compartments split by three dividers. The number of ways to arrange those dividers and balls is just $\\binom {8}{3}$.\nMultiplying gives the answer: $\\binom{8}{5}\\binom{8}{3}5! = 376320$, and the three leftmost digits are $\\boxed{376}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1993_AIME_Problems_9", "eval_func": null }, "input": { "text": "Two thousand points are given on a circle. Label one of the points $1$. From this point, count $2$ points in the clockwise direction and label this point $2$. From the point labeled $2$, count $3$ points in the clockwise direction and label this point $3$. (See figure.) Continue this process until the labels $1,2,3\\dots,1993\\,$ are all used. Some of the points on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as $1993$?", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The label $1993$ will occur on the $\\frac12(1993)(1994) \\pmod{2000}$th point around the circle. (Starting from 1) A number $n$ will only occupy the same point on the circle if $\\frac12(n)(n + 1)\\equiv \\frac12(1993)(1994) \\pmod{2000}$. \nSimplifying this expression, we see that $(1993)(1994) - (n)(n + 1) = (1993 - n)(1994 + n)\\equiv 0\\pmod{2000}$. Therefore, one of $1993 - n$ or $1994 + n$ is odd, and each of them must be a multiple of $125$ or $16$. \nFor $1993 - n$ to be a multiple of $125$ and $1994 + n$ to be a multiple of $16$, $n \\equiv 118 \\pmod {125}$ and $n\\equiv 6 \\pmod {16}$. The smallest $n$ for this case is $118$. \nIn order for $1993 - n$ to be a multiple of $16$ and $1994 + n$ to be a multiple of $125$, $n\\equiv 9\\pmod{16}$ and $n\\equiv 6\\pmod{125}$. The smallest $n$ for this case is larger than $118$, so $\\boxed{118}$ is our answer.\n One can just substitute $1993\\equiv-7\\pmod{2000}$ and $1994\\equiv-6\\pmod{2000}$ to simplify calculations.", "Two labels $a$ and $b$ occur on the same point if $\\ a(a+1)/2\\equiv \\ b(b+1)/2\\pmod{2000}$. If we assume the final answer be $n$, then we have $\\frac12(n)(n + 1)\\equiv \\frac12(1993)(1994) \\pmod{2000}$.\nMultiply $2$ on both side we have $(1993)(1994) - (n)(n + 1) = (1993 - n)(1994 + n)\\equiv 0\\pmod{4000}$. As they have different parities, the even one must be divisible by $32$. As $(1993 - n)+(1994 + n)\\equiv 2\\pmod{5}$, one of them is divisible by $5$, which indicates it's divisible by $125$.\nWhich leads to four different cases: $1993-n\\equiv 0\\pmod{4000}$\u00a0; $1994+n\\equiv 0\\pmod{4000}$\u00a0; $1993-n\\equiv 0\\pmod{32}$ and $1994+n\\equiv 0\\pmod{125}$\u00a0; $1993-n\\equiv 0\\pmod{125}$ and $1994+n\\equiv 0\\pmod{32}$. Which leads to $n\\equiv 1993,2006,3881$ and $118\\pmod{4000}$ respectively, and only $n=118$ satisfied.Therefore answer is $\\boxed{118}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2008_AIME_II_Problems_1", "eval_func": null }, "input": { "text": "Let $N = 100^2 + 99^2 - 98^2 - 97^2 + 96^2 + \\cdots + 4^2 + 3^2 - 2^2 - 1^2$, where the additions and subtractions alternate in pairs. Find the remainder when $N$ is divided by $1000$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Rewriting this sequence with more terms, we have\nFactoring this expression yields\nNext, we get\nThen,\nDividing $10100$ by $1000$ yields a remainder of $\\boxed{100}$.", "Since we want the remainder when $N$ is divided by $1000$, we may ignore the $100^2$ term. Then, applying the difference of squares factorization to consecutive terms,", "By observation, we realize that the sequence \\[(a+3)^2 + (a+2)^2 - (a+1)^2 - a^2\\] alternates every 4 terms. Simplifying, we get \\[(a+3)^2 + (a+2)^2 - (a+1)^2 - a^2 = 8a + 12\\], turning $N$ into a arithmetic sequence with 25 terms, them being $1, 5, 9, \\dots ,97$, as the series $8a + 12$ alternates every 4 terms.\nApplying the sum of arithmetic sequence formula, we get\nSo the answer would be \\[\\frac{10100}{1000} = \\boxed{100}\\].\n" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2009_AIME_II_Problems_2", "eval_func": null }, "input": { "text": "Suppose that $a$, $b$, and $c$ are positive real numbers such that $a^{\\log_3 7} = 27$, $b^{\\log_7 11} = 49$, and $c^{\\log_{11}25} = \\sqrt{11}$. Find\n\\[a^{(\\log_3 7)^2} + b^{(\\log_7 11)^2} + c^{(\\log_{11} 25)^2}.\\]", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "First, we have:\n\\[x^{(\\log_y z)^2} = x^{\\left( (\\log_y z)^2 \\right) } = x^{(\\log_y z) \\cdot (\\log_y z) } = \\left( x^{\\log_y z} \\right)^{\\log_y z}\\]\nNow, let $x=y^w$, then we have:\n\\[x^{\\log_y z} = \\left( y^w \\right)^{\\log_y z} = y^{w\\log_y z} = y^{\\log_y (z^w)} = z^w\\]\nThis is all we need to evaluate the given formula. Note that in our case we have $27=3^3$, $49=7^2$, and $\\sqrt{11}=11^{1/2}$. We can now compute:\n\\[a^{(\\log_3 7)^2} = \\left( a^{\\log_3 7} \\right)^{\\log_3 7} = 27^{\\log_3 7} = (3^3)^{\\log_3 7} = 7^3 = 343\\]\nSimilarly, we get\n\\[b^{(\\log_7 11)^2} = (7^2)^{\\log_7 11} = 11^2 = 121\\]\nand\n\\[c^{(\\log_{11} 25)^2} = (11^{1/2})^{\\log_{11} 25} = 25^{1/2} = 5\\]\nand therefore the answer is $343+121+5 = \\boxed{469}$.", "We know from the first three equations that $\\log_a27 = \\log_37$, $\\log_b49 = \\log_711$, and $\\log_c\\sqrt{11} = \\log_{11}25$. Substituting, we find\n\\[a^{(\\log_a27)(\\log_37)} + b^{(\\log_b49)(\\log_711)} + c^{(\\log_c\\sqrt {11})(\\log_{11}25)}.\\]\nWe know that $x^{\\log_xy} =y$, so we find\n\\[27^{\\log_37} + 49^{\\log_711} + \\sqrt {11}^{\\log_{11}25}\\]\n\\[(3^{\\log_37})^3 + (7^{\\log_711})^2 + ({11^{\\log_{11}25}})^{1/2}.\\]\nThe $3$ and the $\\log_37$ cancel to make $7$, and we can do this for the other two terms. Thus, our answer is\n\\[7^3 + 11^2 + 25^{1/2}\\]\n\\[= 343 + 121 + 5\\]\n\\[= \\boxed {469}.\\]" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2002_AIME_I_Problems_2", "eval_func": null }, "input": { "text": "The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\\dfrac{1}{2}(\\sqrt{p}-q)$ where $p$ and $q$ are positive integers. Find $p+q$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let the radius of the circles be $r$. The longer dimension of the rectangle can be written as $14r$, and by the Pythagorean Theorem, we find that the shorter dimension is $2r\\left(\\sqrt{3}+1\\right)$.\nTherefore, $\\frac{14r}{2r\\left(\\sqrt{3}+1\\right)}= \\frac{7}{\\sqrt{3} + 1} \\cdot \\left[\\frac{\\sqrt{3}-1}{\\sqrt{3}-1}\\right] = \\frac{1}{2}\\left(7\\sqrt{3} - 7\\right) = \\frac{1}{2}\\left(\\sqrt{p}-q\\right)$. Thus we have $p=147$ and $q=7$, so $p+q=\\boxed{154}$.", "Since we only care about the between the longer side and shorter side, we can set the longer side to $14$. So, this means that each of the radii is $1$. Now, we connect the radii of three circles such that they form an equilateral triangle with side length 4. Obviously, the height this triangle is $2\\sqrt{3}$, and the shorter side if the triangle is therefore $2\\sqrt{3}+2$ and we use simplification similar to as showed above, and we reach the result $\\frac{1}{2} \\cdot (\\sqrt{147}-7)$ and the final answer is $147+7 = \\boxed{154}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2023_AIME_II_Problems_10", "eval_func": null }, "input": { "text": "Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2 \\times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3.$ One way to do this is shown below. Find the number of positive integer divisors of $N.$\n\\[\\begin{array}{|c|c|c|c|c|c|} \\hline \\,1\\, & \\,3\\, & \\,5\\, & \\,7\\, & \\,9\\, & 11 \\\\ \\hline \\,2\\, & \\,4\\, & \\,6\\, & \\,8\\, & 10 & 12 \\\\ \\hline \\end{array}\\]", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We replace the numbers which are 0 mod(3) to 0, 1 mod(3) to 1, and 2 mod(3) to 2.\nThen, the problem is equivalent to arranging 4 0's,4 1's, and 4 2's into the grid (and then multiplying by $4!^3$ to account for replacing the remainder numbers with actual numbers) such that no 2 of the same numbers are adjacent. \nThen, the numbers which are vertically connected must be different. 2 1s must be connected with 2 2s, and 2 1s must be connected with 2 2s (vertically), as if there are less 1s connected with either, then there will be 2s or 3s which must be connected within the same number.\nFor instance if we did did:\n- 12 x 3\n- 13 x 1\nThen we would be left with 23 x1, and 2 remaining 3s which aren't supposed to be connected, so it is impossible. Similar logic works for why all 1s can't be connected with all 2s. \n\nThus, we are left with the problem of re-arranging 2x 12 pairs, 2x 13 pairs, 2x 23 pairs. Notice that the pairs can be re-arranged horizontally in any configuration, but when 2 pairs are placed adjacent there is only 1 configuration for the rightmost pair to be set after the leftmost one has been placed.\nWe have $\\frac{6!}{2!2!2!}=90$ ways to horizontally re-arrange the pairs, with 2 ways to set the initial leftmost column. Thus, there are 180 ways to arrange the pairs. Accounting for the initial simplification of the problem with 1-12 -> 0-3, we obtain the answer is: \n\\[2488320=2^{11}\\cdot3^5\\cdot5^1\\]\nThe number of divisors is: $12\\cdot6\\cdot2=\\boxed{144}.$", "Let's carry out an archaeological study, that is, we will find the bones (the base) and \"build up the meat.\"\n1. Let \"bones\" of number $X$ be $X \\pmod 3.$ Then the \u201cskeleton\u201d of the original table is\n\\[1 0 2 1 0 2\\]\n\\[2 1 0 2 1 0\\]\nBy condition, the table cannot have a column of two identical numbers (the difference of such numbers is a multiple of $3).$\nThere are $4$ zeros, $4$ ones and $4$ twos in the table (the order of the numbers in the columns is not important).\nTherefore, there cannot be more than two identical columns (otherwise there will be four identical numbers in the remaining three, that is, the numbers in one column are the same).\nAny such table has exactly $2$ columns $(0,1), 2$ columns $(0,2)$ and $2$ columns $(1,2).$\nThe number of possible tables of six elements of three types is \n\\[m = \\frac {6!}{2! \\cdot 2! \\cdot 2!} = 2 \\cdot 3^2 \\cdot 5.\\]\nThe number of possible tables of six elements, if the order of the digits is important, is \n\\[M = 2^6 \\cdot m = 2^7 \\cdot 3^2 \\cdot 5.\\]\n2. We are looking for the total number of options.\nThe column $(0,1)$ can contain the following $4^2 = 16$ pairs of numbers (the order is not important, it is already taken into account)\n\\[(1,3), (1,6), (1,9), (1, 12), (4.3), (4.6), (4.9) (4.12), (7.3), (7.6), (7.9), (7.12), (11,3), (11,6), (11,9), (11,12).\\]\nThe second column $(0,1)$ can contain $3^2 = 9$ pairs of numbers (excluding the two that are in the first column).\nSimilarly with the other two columns, i.e. the total number of choices \\[M \\cdot 16 \\cdot 9 \\cdot 3 = 2^{11} \\cdot 3^5 \\cdot 5.\\]\nNumber of divisors \\[(11+1) \\cdot (5+1) \\cdot (1 + 1) = \\boxed{144}.\\]" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2012_AIME_II_Problems_7", "eval_func": null }, "input": { "text": "Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Let $N$ be the 1000th number in $S$. Find the remainder when $N$ is divided by $1000$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Okay, an exercise in counting (lots of binomials to calculate!). In base 2, the first number is $11111111$, which is the only way to choose 8 1's out of 8 spaces, or $\\binom{8}{8}$. What about 9 spaces? Well, all told, there are $\\binom{9}{8}=9$, which includes the first 1. Similarly, for 10 spaces, there are $\\binom{10}{8}=45,$ which includes the first 9. For 11 spaces, there are $\\binom{11}{8}=165$, which includes the first 45. You're getting the handle. For 12 spaces, there are $\\binom{12}{8}=495$, which includes the first 165; for 13 spaces, there are $\\binom{13}{8}=13 \\cdot 99 > 1000$, so we now know that $N$ has exactly 13 spaces, so the $2^{12}$ digit is 1. \nNow we just proceed with the other 12 spaces with 7 1's, and we're looking for the $1000-495=505th$ number. Well, $\\binom{11}{7}=330$, so we know that the $2^{11}$ digit also is 1, and we're left with finding the $505-330=175th$ number with 11 spaces and 6 1's. Now $\\binom{10}{6}=210,$ which is too big, but $\\binom{9}{6}=84.$ Thus, the $2^9$ digit is 1, and we're now looking for the $175-84=91st$ number with 9 spaces and 5 1's. Continuing the same process, $\\binom{8}{5}=56$, so the $2^8$ digit is 1, and we're left to look for the $91-56=35th$ number with 8 spaces and 4 1's. But here $\\binom{7}{4}=35$, so N must be the last or largest 7-digit number with 4 1's. Thus the last 8 digits of $N$ must be $01111000$, and to summarize, $N=1101101111000$ in base $2$. Therefore, $N = 8+16+32+64+256+512+2048+4096 \\equiv 32 \\pmod{1000}$, and the answer is $\\boxed{032}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2012_AIME_II_Problems_11", "eval_func": null }, "input": { "text": "Let $f_1(x) = \\frac23 - \\frac3{3x+1}$, and for $n \\ge 2$, define $f_n(x) = f_1(f_{n-1}(x))$. The value of $x$ that satisfies $f_{1001}(x) = x-3$ can be expressed in the form $\\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "After evaluating the first few values of $f_k (x)$, we obtain $f_4(x) = f_1(x) = \\frac{2}{3} - \\frac{3}{3x+1} = \\frac{6x-7}{9x+3}$. Since $1001 \\equiv 2 \\mod 3$, $f_{1001}(x) = f_2(x) = \\frac{3x+7}{6-9x}$. We set this equal to $x-3$, i.e.\n\n$\\frac{3x+7}{6-9x} = x-3 \\Rightarrow x = \\frac{5}{3}$. The answer is thus $5+3 = \\boxed{008}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2010_AIME_II_Problems_4", "eval_func": null }, "input": { "text": "Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks $400$ feet or less to the new gate be a fraction $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "There are $12 \\cdot 11 = 132$ possible situations ($12$ choices for the initially assigned gate, and $11$ choices for which gate Dave's flight was changed to). We are to count the situations in which the two gates are at most $400$ feet apart.\nIf we number the gates $1$ through $12$, then gates $1$ and $12$ have four other gates within $400$ feet, gates $2$ and $11$ have five, gates $3$ and $10$ have six, gates $4$ and $9$ have have seven, and gates $5$, $6$, $7$, $8$ have eight. Therefore, the number of valid gate assignments is \\[2\\cdot(4+5+6+7)+4\\cdot8 = 2 \\cdot 22 + 4 \\cdot 8 = 76\\] so the probability is $\\frac{76}{132} = \\frac{19}{33}$. The answer is $19 + 33 = \\boxed{052}$.", "As before, derive that there are $132$ possibilities for Dave's original and replacement gates.\nNow suppose that Dave has to walk $100k$ feet to get to his new gate. This means that Dave's old and new gates must be $k$ gates apart. (For example, a $100$ foot walk would consist of the two gates being adjacent to each other.) There are $12-k$ ways to pick two gates which are $k$ gates apart, and $2$ possibilities for gate assignments, for a total of $2(12-k)$ possible assignments for each $k$.\nAs a result, the total number of valid gate arrangements is \\[2\\cdot 11 + 2\\cdot 10 + 2\\cdot 9 + 2\\cdot 8 = 76\\] and so the requested probability is $\\tfrac{19}{33}$ for a final answer of $\\boxed{052}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1984_AIME_Problems_9", "eval_func": null }, "input": { "text": "In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is $15\\mbox{cm}^2$ and the area of face $ABD$ is $12 \\mbox { cm}^2$. These two faces meet each other at a $30^\\circ$ angle. Find the volume of the tetrahedron in $\\mbox{cm}^3$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Position face $ABC$ on the bottom. Since $[\\triangle ABD] = 12 = \\frac{1}{2} \\cdot AB \\cdot h_{ABD}$, we find that $h_{ABD} = 8$. Because the problem does not specify, we may assume both $ABC$ and $ABD$ to be isosceles triangles. Thus, the height of $ABD$ forms a $30-60-90$ with the height of the tetrahedron. So, $h = \\frac{1}{2} (8) = 4$. The volume of the tetrahedron is thus $\\frac{1}{3}Bh = \\frac{1}{3} \\cdot15 \\cdot 4 = \\boxed{020}$.", "It is clear that $DX=8$ and $CX=10$ where $X$ is the foot of the perpendicular from $D$ and $C$ to side $AB$. Thus $[DXC]=\\frac{ab\\sin{c}}{2}=20=5 \\cdot h \\rightarrow h = 4$ where h is the height of the tetrahedron from $D$. Hence, the volume of the tetrahedron is $\\frac{bh}{3}=15\\cdot \\frac{4}{3}=\\boxed{020}$\n\n(Note this actually isn't rigorous because they never proved that the height from $D$ to $XC$ is the altitude of the tetrahedron.", "Make faces $ABC$ and $ABD$ right triangles. This makes everything a lot easier. Then do everything in solution 1.", "We can use 3D coordinates.\nLet $A = (0, 0, 0)$ and $B = (3, 0, 0).$ WLOG, let $D = \\left(\\frac{3}{2}, 8, 0\\right)$, because the area of $\\Delta{ABD} = 12$ and the tetrahedron area won't change if we put it somewhere else with $y=8.$\nTo find $C$, we can again let the $x$-coordinate be $\\frac{3}{2}$ for simplicity. Note that $C$ is $10$ units away from $AB$ because the area of $\\Delta{ABC}$ is $15$. Since the angle between $ABD$ and $ABC$ is $30^\\circ$, we can form a 30-60-90 triangle between $A$, $B$, and an altitude dropped from $C$ onto face $ABD$. Since $10$ is the hypotenuse, we get $5\\sqrt{3}$ and $5$ as legs. Then $y=5\\sqrt{3}$ and $z=5$, so $C = \\left(\\frac{3}{2}, 5\\sqrt{3}, 5\\right).$\n(I highly advise you to draw both the tetrahedron and 30-60-90 triangle to get a better perspective.)\nNow, we can move onto vectors. To find the volume of the tetrahedron, we use the formula $\\frac{1}{3}Bh.$ Letting $\\Delta{ABC}$ be the base we have $B = 15$ (from the problem statement). We need to find the distance between $D$ and $ABC$, and to do this, we should find the projection of $D$ onto face $ABC$.\nNote that we can simplify this to projecting $D$ onto $\\mathbf{\\overrightarrow{C}}.$ This is because we know the projection will have the same $x$-coordinate as $D$ and $C$, as both are $\\frac{3}{2}.$ Now we find $\\text{proj}_{\\mathbf{\\overrightarrow{D}}} \\mathbf{\\overrightarrow{C}}$, or plugging in our coordinates, $\\text{proj}_{\\langle\\frac{3}{2}, 5\\sqrt{3}, 5\\rangle} \\left\\langle\\frac{3}{2}, 8, 0\\right\\rangle$. \nLet the $x$-coordinates for both be $0$ for simplicity, because we can always add a $\\frac{3}{2}$ at the end. Using the projection formula, we get \\[\\langle 0, 6, 2\\sqrt{3}\\rangle.\\]\nFinally, we calculate the distance between $\\left(\\frac{3}{2}, 6, 2\\sqrt{3}\\right)$ and $D$ to be $4$. So the height is $4$, and plugging into our tetrahedron formula we get \\[\\frac{1}{3}\\cdot 15\\cdot 4 = \\boxed{20}.\\]" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2007_AIME_I_Problems_3", "eval_func": null }, "input": { "text": "The complex number $z$ is equal to $9+bi$, where $b$ is a positive real number and $i^{2}=-1$. Given that the imaginary parts of $z^{2}$ and $z^{3}$ are the same, what is $b$ equal to?", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Squaring, we find that $(9 + bi)^2 = 81 + 18bi - b^2$. Cubing and ignoring the real parts of the result, we find that $(81 + 18bi - b^2)(9 + bi) = \\ldots + (9\\cdot 18 + 81)bi - b^3i$.\nSetting these two equal, we get that $18bi = 243bi - b^3i$, so $b(b^2 - 225) = 0$ and $b = -15, 0, 15$. Since $b > 0$, the solution is $\\boxed{015}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2006_AIME_II_Problems_11", "eval_func": null }, "input": { "text": "A sequence is defined as follows $a_1=a_2=a_3=1,$ and, for all positive integers $n, a_{n+3}=a_{n+2}+a_{n+1}+a_n.$ Given that $a_{28}=6090307, a_{29}=11201821,$ and $a_{30}=20603361,$ find the remainder when $\\sum^{28}_{k=1} a_k$ is divided by 1000.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Define the sum as $s$. Since $a_n\\ = a_{n + 3} - a_{n + 2} - a_{n + 1}$, the sum will be:\nThus $s = \\frac{a_{28} + a_{30}}{2}$, and $a_{28},\\,a_{30}$ are both given; the last four digits of their sum is $3668$, and half of that is $1834$. Therefore, the answer is $\\boxed{834}$.", "Since the problem only asks for the first 28 terms and we only need to calculate mod 1000, we simply bash the first 28 terms:\n\n$a_{1}\\equiv 1 \\pmod {1000} \\\\ a_{2}\\equiv 1 \\pmod {1000} \\\\ a_{3}\\equiv 1 \\pmod {1000} \\\\ a_{4}\\equiv 3 \\pmod {1000} \\\\ a_{5}\\equiv 5 \\pmod {1000} \\\\ \\cdots \\\\ a_{25} \\equiv 793 \\pmod {1000} \\\\ a_{26} \\equiv 281 \\pmod {1000} \\\\ a_{27} \\equiv 233 \\pmod {1000} \\\\ a_{28} \\equiv 307 \\pmod {1000}$\nAdding all the residues shows the sum is congruent to $\\boxed{834}$ mod $1000$.", "All terms in the sequence are sums of previous terms, so the sum of all terms up to a certain point must be some linear combination of the first three terms. Also, we are given $a_{28}, a_{29},$ and $a_{30}$, so we can guess that there is some way to use them in a formula. Namely, we guess that there exists some $p, q, r$ such that $\\sum_{k=1}^{n}{a_k} = pa_n+qa_{n+1}+ra_{n+2}$. From here, we list out the first few terms of the sequence and the cumulative sums, and with a little bit of substitution and algebra we see that $(p, q, r) = (\\frac{1}{2}, 0, \\frac{1}{2})$, at least for the first few terms. From this, we have that $\\sum_{k=1}^{28}{a_k} = \\frac{a_{28}+a_{30}}{2} \\equiv{\\boxed{834}}(\\mod 1000)$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1996_AIME_Problems_5", "eval_func": null }, "input": { "text": "Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a$, $b$, and $c$, and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$, $b+c$, and $c+a$. Find $t$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "By Vieta's formulas on the polynomial $P(x) = x^3+3x^2+4x-11 = (x-a)(x-b)(x-c) = 0$, we have $a + b + c = s = -3$, $ab + bc + ca = 4$, and $abc = 11$. Then\nThis is just the definition for $-P(-3) = \\boxed{023}$.\nAlternatively, we can expand the expression to get\n\\begin{align*} t &= -(-3-a)(-3-b)(-3-c)\\\\ &= (a+3)(b+3)(c+3)\\\\ &= abc + 3[ab + bc + ca] + 9[a + b + c] + 27\\\\ t &= 11 + 3(4) + 9(-3) + 27 = 23\\end{align*}", "Each term in the expansion of $(a+b)(b+c)(c+a)$ has a total degree of 3. Another way to get terms with degree 3 is to multiply out $(a+b+c)(ab+bc+ca)$. Expanding both of these expressions and comparing them shows that:\n$(a+b)(b+c)(c+a) = (ab+bc+ca)(a+b+c)-abc$\n$t = -(a+b)(b+c)(c+a) = abc-(ab+bc+ca)(a+b+c) = 11-(4)(-3) = 23$\n\nA way to realize $(a+b)(b+c)(c+a) = (ab+bc+ca)(a+b+c)-abc$:\n$(a + b)(b+c)(c+a) = a^2b + a^2c + b^2a + b^2c + c^2a + c^2b + abc + abc$\n(Add an extra $abc$)\n$a^2b + a^2c + abc + b^2a + b^2c + abc + c^2a + c^2b + abc - abc =$\n$a(ab + bc + ac) + b(ab + bc + ac) + c(ab + bc + ac) - abc =$\n$(a + b + c)(ab + bc + ac) - abc = (-3)(4) - 11 = -23$.\nThe value of $t$ is the negation of this, which is $-(-23) = \\boxed{023}$", "We have that $x^3+3x^2+4x-11=0$ for roots $a,b,c.$\nIn the second cubic function $x^3+rx^2+sx+t=0,$ the roots are $a+b,b+c,c+a.$\nBy Vieta's formulae, we see that $t= -(a+b)(b+c)(a+c).$\nAs we know that the sum of the roots of the first polynomial, $a+b+c$ is $-3$ by applying Vieta's again.\nUsing this fact, we can rewrite $t$ as $-(-3-a)(-3-b)(-3-c) = (a+3)(b+3)(c+3).$\nSeeing this, we can find the value of the product of the roots by applying this to the first equation.\nThis can be done by setting $x'=a+3,$ so, from this, we see that we should plug in $x'-3$ for $x$ in $x^3+3x^2+4x-11=0$.\nAfter simplifying, we get that the polynomial is $x'^3 - 6x'^2 + 13x' - 23 = 0.$\nGiven that the product of the roots of this equation is equivalent to our desired value for $t$, we can apply Vieta's formulae for a third time to find that $t = -(\\frac{-23}{1}) = \\boxed{023}.$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2008_AIME_II_Problems_6", "eval_func": null }, "input": { "text": "The sequence $\\{a_n\\}$ is defined by\n\\[a_0 = 1,a_1 = 1, \\text{ and } a_n = a_{n - 1} + \\frac {a_{n - 1}^2}{a_{n - 2}}\\text{ for }n\\ge2.\\]\nThe sequence $\\{b_n\\}$ is defined by\n\\[b_0 = 1,b_1 = 3, \\text{ and } b_n = b_{n - 1} + \\frac {b_{n - 1}^2}{b_{n - 2}}\\text{ for }n\\ge2.\\]\nFind $\\frac {b_{32}}{a_{32}}$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Rearranging the definitions, we have\n\\[\\frac{a_n}{a_{n-1}} = \\frac{a_{n-1}}{a_{n-2}} + 1,\\quad \\frac{b_n}{b_{n-1}} = \\frac{b_{n-1}}{b_{n-2}} + 1\\]\nfrom which it follows that $\\frac{a_n}{a_{n-1}} = 1+ \\frac{a_{n-1}}{a_{n-2}} = \\cdots = (n-1) + \\frac{a_{1}}{a_0} = n$ and $\\frac{b_n}{b_{n-1}} = (n-1) + \\frac{b_{1}}{b_0} = n+2$. These recursions, $a_{n} = na_{n-1}$ and $b_{n} = (n+2)b_{n-1}$, respectively, correspond to the explicit functions $a_n = n!$ and $b_n = \\frac{(n+2)!}{2}$ (after applying our initial conditions). It follows that $\\frac{b_{32}}{a_{32}} = \\frac{\\frac{34!}{2}}{32!} = \\frac{34 \\cdot 33}{2} = \\boxed{561}$.\nFrom this, we can determine that the sequence $\\frac {b_n}{a_n}$ corresponds to the triangular numbers." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1990_AIME_Problems_3", "eval_func": null }, "input": { "text": "Let $P_1^{}$ be a regular polygon and $P_2^{}$ be a regular polygon $(r\\geq s\\geq 3)$ such that each interior angle of $P_1^{}$ is $\\frac{59}{58}$ as large as each interior angle of $P_2^{}$. What's the largest possible value of $s_{}^{}$?", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The formula for the interior angle of a regular sided polygon is $\\frac{(n-2)180}{n}$. \nThus, $\\frac{\\frac{(r-2)180}{r}}{\\frac{(s-2)180}{s}} = \\frac{59}{58}$. Cross multiplying and simplifying, we get $\\frac{58(r-2)}{r} = \\frac{59(s-2)}{s}$. Cross multiply and combine like terms again to yield $58rs - 58 \\cdot 2s = 59rs - 59 \\cdot 2r \\Longrightarrow 118r - 116s = rs$. Solving for $r$, we get $r = \\frac{116s}{118 - s}$.\n$r \\ge 0$ and $s \\ge 0$, making the numerator of the fraction positive. To make the denominator positive, $s < 118$; the largest possible value of $s$ is $117$.\nThis is achievable because the denominator is $1$, making $r$ a positive number $116 \\cdot 117$ and $s = \\boxed{117}$.", "Like above, use the formula for the interior angles of a regular sided polygon.\n\n$\\frac{(r-2)180}{r} = \\frac{59}{58} * \\frac{(s-2)180}{s}$\n\n$59 * 180 * (s-2) * r = 58 * 180 * (r-2) * s$\n\n$59 * (rs - 2r) = 58 * (rs - 2s)$\n\n$rs - 118r = -116s$\n\n$rs = 118r-116s$\n\nThis equation tells us $s$ divides $118r$. If $s$ specifically divides 118 then the highest it can be is 118. However, this gives an equation with no solution. The second largest possibility in this case is $s=59$, which does give a solution: $s=59, r=116$. Although, the problem asks for $s$, not $r$. The only conceivable reasoning behind this is that $r$ is greater than 1000. This prompts us to look into the second case, where $s$ divides $r$. Make $r = s * k$. Rewrite the equation using this new information.\n\n$s * s * k = 118 * s * k - 116 * s$\n\n$s * k = 118 * k - 116$\n\nNow we now k divides 116. The larger k is, the larger s will be, so we set k to be the maximum: 116.\n\n$s * 116 = 118 * 116 - 116$\n\n$s = 118 - 1$\n\n$s = \\boxed{117}$\n", "As in above, we have $rs = 118r - 116s.$ This means that $rs + 116s - 118r = 0.$ Using SFFT we obtain $s(r+116) - 118(r+116) = -118 \\cdot 116 \\implies (s-118)(r+116) = -118 \\cdot 116.$ Since $r+116$ is always positive, we know thta $s-118$ must be negative. Therefore the maximum value of $s$ must be $\\boxed{117}$ which indeed yields an integral value of $r.$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2005_AIME_I_Problems_1", "eval_func": null }, "input": { "text": "Six congruent circles form a ring with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle $C$ with radius 30. Let $K$ be the area of the region inside circle $C$ and outside of the six circles in the ring. Find $\\lfloor K \\rfloor$ (the floor function).", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Define the radii of the six congruent circles as $r$. If we draw all of the radii to the points of external tangency, we get a regular hexagon. If we connect the vertices of the hexagon to the center of the circle $C$, we form several equilateral triangles. The length of each side of the triangle is $2r$. Notice that the radius of circle $C$ is equal to the length of the side of the triangle plus $r$. Thus, the radius of $C$ has a length of $3r = 30$, and so $r = 10$. $K = 30^2\\pi - 6(10^2\\pi) = 300\\pi$, so $\\lfloor 300\\pi \\rfloor = \\boxed{942}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1994_AIME_Problems_10", "eval_func": null }, "input": { "text": "In triangle $ABC,\\,$ angle $C$ is a right angle and the altitude from $C\\,$ meets $\\overline{AB}\\,$ at $D.\\,$ The lengths of the sides of $\\triangle ABC\\,$ are integers, $BD=29^3,\\,$ and $\\cos B=m/n\\,$, where $m\\,$ and $n\\,$ are relatively prime positive integers. Find $m+n.\\,$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Since $\\triangle ABC \\sim \\triangle CBD$, we have $\\frac{BC}{AB} = \\frac{29^3}{BC} \\Longrightarrow BC^2 = 29^3 AB$. It follows that $29^2 | BC$ and $29 | AB$, so $BC$ and $AB$ are in the form $29^2 x$ and $29 x^2$, respectively, where x is an integer. \nBy the Pythagorean Theorem, we find that $AC^2 + BC^2 = AB^2 \\Longrightarrow (29^2x)^2 + AC^2 = (29 x^2)^2$, so $29x | AC$. Letting $y = AC / 29x$, we obtain after dividing through by $(29x)^2$, $29^2 = x^2 - y^2 = (x-y)(x+y)$. As $x,y \\in \\mathbb{Z}$, the pairs of factors of $29^2$ are $(1,29^2)(29,29)$; clearly $y = \\frac{AC}{29x} \\neq 0$, so $x-y = 1, x+y= 29^2$. Then, $x = \\frac{1+29^2}{2} = 421$.\nThus, $\\cos B = \\frac{BC}{AB} = \\frac{29^2 x}{29x^2} = \\frac{29}{421}$, and $m+n = \\boxed{450}$.", "We will solve for $\\cos B$ using $\\triangle CBD$, which gives us $\\cos B = \\frac{29^3}{BC}$. By the Pythagorean Theorem on $\\triangle CBD$, we have $BC^2 - DC^2 = (BC + DC)(BC - DC) = 29^6$. Trying out factors of $29^6$, we can either guess and check or just guess to find that $BC + DC = 29^4$ and $BC - DC = 29^2$ (The other pairs give answers over 999). Adding these, we have $2BC = 29^4 + 29^2$ and $\\frac{29^3}{BC} = \\frac{2*29^3}{29^2 (29^2 +1)} = \\frac{58}{842} = \\frac{29}{421}$, and our answer is $\\boxed{450}$.", "Using similar right triangles, we identify that $CD = \\sqrt{AD \\cdot BD}$. Let $AD$ be $29 \\cdot k^2$, to avoid too many radicals, getting $CD = k \\cdot 29^2$. Next we know that $AC = \\sqrt{AB \\cdot AD}$ and that $BC = \\sqrt{AB \\cdot BD}$. Applying the logic with the established values of k, we get $AC = 29k \\cdot \\sqrt{29^2 + k^2}$ and $BC = 29^2 \\cdot \\sqrt{29^2 + k^2}$. Next we look to the integer requirement. Since $k$ is both outside and inside square roots, we know it must be an integer to keep all sides as integers. Let $y$ be $\\sqrt{29^2 + k^2}$, thus $29^2 = y^2 - k^2$, and $29^2 = (y + k)(y - k)$. Since $29$ is prime, and $k$ cannot be zero, we find $k = 420$ and $y = 421$ as the smallest integers satisfying this quadratic Diophantine equation. Then, since $cos B$ = $\\frac{29}{\\sqrt{29^2 + k^2}}$. Plugging in we get $cos B = \\frac{29}{421}$, thus our answer is $\\boxed{450}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2015_AIME_I_Problems_1", "eval_func": null }, "input": { "text": "The expressions $A$ = $1 \\times 2 + 3 \\times 4 + 5 \\times 6 + \\cdots + 37 \\times 38 + 39$ and $B$ = $1 + 2 \\times 3 + 4 \\times 5 + \\cdots + 36 \\times 37 + 38 \\times 39$ are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers $A$ and $B$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We have \\[|A-B|=|1+3(4-2)+5(6-4)+ \\cdots + 37(38-36)-39(1-38)|\\]\\[\\implies |2(1+3+5+7+ \\cdots +37)-1-39(37)|\\]\\[\\implies |361(2)-1-39(37)|=|722-1-1443|=|-722|\\implies \\boxed{722}\\]", "We see that\n$A=(1\\times 2)+(3\\times 4)+(5\\times 6)+\\cdots +(35\\times 36)+(37\\times 38)+39$\nand\n$B=1+(2\\times 3)+(4\\times 5)+(6\\times 7)+\\cdots +(36\\times 37)+(38\\times 39)$.\nTherefore,\n$B-A=-38+(2\\times 2)+(2\\times 4)+(2\\times 6)+\\cdots +(2\\times 36)+(2\\times 38)$\n$=-38+4\\times (1+2+3+\\cdots+19)$\n$=-38+4\\times\\frac{20\\cdot 19}{2}=-38+760=\\boxed{722}.$", "For those that aren't shrewd enough to recognize the above, we may use Newton's Little Formula to semi-bash the equations.\nWe write down the pairs of numbers after multiplication and solve each layer:\n\n\\[2, 12, 30, 56, 90...(39)\\]\n\\[6, 18, 26, 34...\\]\n\\[8, 8, 8...\\]\nand\n\\[(1) 6, 20, 42, 72...\\]\n\\[14, 22, 30...\\]\n\\[8, 8, 8...\\]\n\nThen we use Newton's Little Formula for the sum of $n$ terms in a sequence.\nNotice that there are $19$ terms in each sequence, plus the tails of $39$ and $1$ on the first and second equations, respectively.\n\nSo,\n\n\\[2\\binom{19}{1}+10\\binom{19}{2}+8\\binom{19}{3}+1\\]\n\n\\[6\\binom{19}{1}+14\\binom{19}{2}+8\\binom{19}{3}+39\\]\n\nSubtracting $A$ from $B$ gives:\n\n\\[4\\binom{19}{1}+4\\binom{19}{2}-38\\]\n\nWhich unsurprisingly gives us $\\boxed{722}.$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2008_AIME_I_Problems_10", "eval_func": null }, "input": { "text": "Let $ABCD$ be an isosceles trapezoid with $\\overline{AD}||\\overline{BC}$ whose angle at the longer base $\\overline{AD}$ is $\\dfrac{\\pi}{3}$. The diagonals have length $10\\sqrt {21}$, and point $E$ is at distances $10\\sqrt {7}$ and $30\\sqrt {7}$ from vertices $A$ and $D$, respectively. Let $F$ be the foot of the altitude from $C$ to $\\overline{AD}$. The distance $EF$ can be expressed in the form $m\\sqrt {n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "$AD = 20\\sqrt{7}$.\n By the triangle inequality, we can immediately see that $AD \\geq 20\\sqrt{7}$. However, notice that $10\\sqrt{21} = 20\\sqrt{7}\\cdot\\sin\\frac{\\pi}{3}$, so by the law of sines, when $AD = 20\\sqrt{7}$, $\\angle ACD$ is right and the circle centered at $A$ with radius $10\\sqrt{21}$, which we will call $\\omega$, is tangent to $\\overline{CD}$. Thus, if $AD$ were increased, $\\overline{CD}$ would have to be moved even farther outwards from $A$ to maintain the angle of $\\frac{\\pi}{3}$ and $\\omega$ could not touch it, a contradiction.\n Again, use the triangle inequality to obtain $AD \\geq 20\\sqrt{7}$. Let $x = AD$ and $y = CD$. By the law of cosines on $\\triangle ADC$, $2100 = x^2+y^2-xy \\iff y^2-xy+(x^2-2100) = 0$. Viewing this as a quadratic in $y$, the discriminant $\\Delta$ must satisfy $\\Delta = x^2-4(x^2-2100) = 8400-3x^2 \\geq 0 \\iff x \\leq 20\\sqrt{7}$. Combining these two inequalities yields the desired conclusion.\nThis observation tells us that $E$, $A$, and $D$ are collinear, in that order.\nThen, $\\triangle ADC$ and $\\triangle ACF$ are $30-60-90$ triangles. Hence $AF = 15\\sqrt {7}$, and\nFinally, the answer is $25+7=\\boxed{032}$.", "Extend $\\overline {AB}$ through $B$, to meet $\\overline {DC}$ (extended through $C$) at $G$. $ADG$ is an equilateral triangle because of the angle conditions on the base.\nIf $\\overline {GC} = x$ then $\\overline {CD} = 40\\sqrt{7}-x$, because $\\overline{AD}$ and therefore $\\overline{GD}$ $= 40\\sqrt{7}$.\nBy simple angle chasing, $CFD$ is a 30-60-90 triangle and thus $\\overline{FD} = \\frac{40\\sqrt{7}-x}{2}$, \nand $\\overline{CF} = \\frac{40\\sqrt{21} - \\sqrt{3}x}{2}$\nSimilarly $CAF$ is a 30-60-90 triangle and thus $\\overline{CF} = \\frac{10\\sqrt{21}}{2} = 5\\sqrt{21}$.\nEquating and solving for $x$, $x = 30\\sqrt{7}$ and thus $\\overline{FD} = \\frac{40\\sqrt{7}-x}{2} = 5\\sqrt{7}$.\n$\\overline{ED}-\\overline{FD} = \\overline{EF}$\n$30\\sqrt{7} - 5\\sqrt{7} = 25\\sqrt{7}$ and $25 + 7 = \\boxed{032}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2006_AIME_I_Problems_4", "eval_func": null }, "input": { "text": "Let $N$ be the number of consecutive $0$'s at the right end of the decimal representation of the product $1!2!3!4!\\cdots99!100!.$ Find the remainder when $N$ is divided by $1000$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "A number in decimal notation ends in a zero for each power of ten which divides it. Thus, we need to count both the number of 5s and the number of 2s dividing into our given expression. Since there are clearly more 2s than 5s, it is sufficient to count the number of 5s.\nOne way to do this is as follows: $96$ of the numbers $1!,\\ 2!,\\ 3!,\\ 100!$ have a factor of $5$. $91$ have a factor of $10$. $86$ have a factor of $15$. And so on. This gives us an initial count of $96 + 91 + 86 + \\ldots + 1$. Summing this arithmetic series of $20$ terms, we get $970$. However, we have neglected some powers of $5$ - every $n!$ term for $n\\geq25$ has an additional power of $5$ dividing it, for $76$ extra; every n! for $n\\geq 50$ has one more in addition to that, for a total of $51$ extra; and similarly there are $26$ extra from those larger than $75$ and $1$ extra from $100$. Thus, our final total is $970 + 76 + 51 + 26 + 1 = 1124$, and the answer is $\\boxed{124}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2003_AIME_II_Problems_7", "eval_func": null }, "input": { "text": "Find the area of rhombus $ABCD$ given that the circumradii of triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectively.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The diagonals of the rhombus perpendicularly bisect each other. Call half of diagonal BD $a$ and half of diagonal AC $b$. The length of the four sides of the rhombus is $\\sqrt{a^2+b^2}$. \nThe area of any triangle can be expressed as $\\frac{a\\cdot b\\cdot c}{4R}$, where $a$, $b$, and $c$ are the sides and $R$ is the circumradius. Thus, the area of $\\triangle ABD$ is $ab=2a(a^2+b^2)/(4\\cdot12.5)$. Also, the area of $\\triangle ABC$ is $ab=2b(a^2+b^2)/(4\\cdot25)$. Setting these two expressions equal to each other and simplifying gives $b=2a$. Substitution yields $a=10$ and $b=20$, so the area of the rhombus is $20\\cdot40/2=\\boxed{400}$.", "Let $\\theta=\\angle BDA$. Let $AB=BC=CD=x$. By the extended law of sines,\n\\[\\frac{x}{\\sin\\theta}=25\\]\nSince $AC\\perp BD$, $\\angle CAD=90-\\theta$, so\n\\[\\frac{x}{\\sin(90-\\theta)=\\cos\\theta}=50\\]\nHence $x=25\\sin\\theta=50\\cos\\theta$. Solving $\\tan\\theta=2$, $\\sin\\theta=\\frac{2}{\\sqrt{5}}, \\cos\\theta=\\frac{1}{\\sqrt{5}}$. Thus\n\\[x=25\\frac{2}{\\sqrt{5}}\\implies x^2=500\\]\nThe height of the rhombus is $x\\sin(2\\theta)=2x\\sin\\theta\\cos\\theta$, so we want\n\\[2x^2\\sin\\theta\\cos\\theta=\\boxed{400}\\]" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1987_AIME_Problems_7", "eval_func": null }, "input": { "text": "Let $[r,s]$ denote the least common multiple of positive integers $r$ and $s$. Find the number of ordered triples $(a,b,c)$ of positive integers for which $[a,b] = 1000$, $[b,c] = 2000$, and $[c,a] = 2000$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "It's clear that we must have $a = 2^j5^k$, $b = 2^m 5^n$ and $c = 2^p5^q$ for some nonnegative integers $j, k, m, n, p, q$. Dealing first with the powers of 2: from the given conditions, $\\max(j, m) = 3$, $\\max(m, p) = \\max(p, j) = 4$. Thus we must have $p = 4$ and at least one of $m, j$ equal to 3. This gives 7 possible triples $(j, m, p)$: $(0, 3, 4), (1, 3, 4), (2, 3, 4), (3, 3, 4), (3, 2, 4), (3, 1, 4)$ and $(3, 0, 4)$.\nNow, for the powers of 5: we have $\\max(k, n) = \\max(n, q) = \\max(q, k) = 3$. Thus, at least two of $k, n, q$ must be equal to 3, and the other can take any value between 0 and 3. This gives us a total of 10 possible triples: $(3, 3, 3)$ and three possibilities of each of the forms $(3, 3, n)$, $(3, n, 3)$ and $(n, 3, 3)$.\nSince the exponents of 2 and 5 must satisfy these conditions independently, we have a total of $7 \\cdot 10 = 70$ possible valid triples.", "$1000 = 2^35^3$ and $2000 = 2^45^3$. By looking at the prime factorization of $2000$, $c$ must have a factor of $2^4$. If $c$ has a factor of $5^3$, then there are two cases: either (1) $a$ or $b = 5^32^3$, or (2) one of $a$ and $b$ has a factor of $5^3$ and the other a factor of $2^3$. For case 1, the other number will be in the form of $2^x5^y$, so there are $4 \\cdot 4 = 16$ possible such numbers; since this can be either $a$ or $b$ there are a total of $2(16)-1=31$ possibilities. For case 2, $a$ and $b$ are in the form of $2^35^x$ and $2^y5^3$, with $x < 3$ and $y < 3$ (if they were equal to 3, it would overlap with case 1). Thus, there are $2(3 \\cdot 3) = 18$ cases.\nIf $c$ does not have a factor of $5^3$, then at least one of $a$ and $b$ must be $2^35^3$, and both must have a factor of $5^3$. Then, there are $4$ solutions possible just considering $a = 2^35^3$, and a total of $4 \\cdot 2 - 1 = 7$ possibilities. Multiplying by three, as $0 \\le c \\le 2$, there are $7 \\cdot 3 = 21$. Together, that makes $31 + 18 + 21 = 70$ solutions for $(a, b, c)$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1987_AIME_Problems_8", "eval_func": null }, "input": { "text": "What is the largest positive integer $n$ for which there is a unique integer $k$ such that $\\frac{8}{15} < \\frac{n}{n + k} < \\frac{7}{13}$?", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Multiplying out all of the denominators, we get:\n\\begin{align*}104(n+k) &< 195n< 105(n+k)\\\\ 0 &< 91n - 104k < n + k\\end{align*}\nSince $91n - 104k < n + k$, $k > \\frac{6}{7}n$. Also, $0 < 91n - 104k$, so $k < \\frac{7n}{8}$. Thus, $48n < 56k < 49n$. $k$ is unique if it is within a maximum range of $112$, so $n = 112$.", "Flip all of the fractions for\n\\[\\begin{array}{ccccc}\\frac{15}{8} &>& \\frac{k + n}{n} &>& \\frac{13}{7}\\\\ 105n &>& 56 (k + n)& >& 104n\\\\ 49n &>& 56k& >& 48n\\end{array}\\]\nContinue as in Solution 1.", "Flip the fractions and subtract one from all sides to yield \\[\\frac{7}{8}>\\frac{k}{n}>\\frac{6}{7}.\\] Multiply both sides by $56n$ to get \\[49n>56k>48n.\\] This is equivalent to find the largest value of $n$ such that there is only one multiple of 56 within the open interval between $48n$ and $49n$. If $n=112,$ then $98>k>96$ and $k=97$ is the unique value. For $n\\geq 113,$ there is at least $(49\\cdot 113-48\\cdot 113)-1=112$ possible numbers for $k$ and there is one $k$ every 56 numbers. Hence, there must be at least two values of $k$ that work. So, the largest value of $n$ is $\\boxed{112}$.", "Notice that in order for $k$ to be unique, $\\frac{n}{n + k+ 1} \\le \\frac{8}{15}$ and $\\frac{n}{n+ k-1} \\ge \\frac{7}{13}$ must be true. Solving these inequalities for $k$ yields $\\frac{7}{6}(k-1) \\le n \\le \\frac{8}{7}(k+1)$.\nThus, we want to find $k$ such that $\\frac{7}{6}(k-1)\\le \\frac{8}{7}(k+1)$. Solving this inequality yields $k \\le 97$, and plugging this into $\\frac{n}{n+k} < \\frac{7}{13}$ in the original equation yields $n \\le 112$ so the answer is $\\boxed{112}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2001_AIME_II_Problems_12", "eval_func": null }, "input": { "text": "Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra $P_{i}$ is defined recursively as follows: $P_{0}$ is a regular tetrahedron whose volume is 1. To obtain $P_{i + 1}$, replace the midpoint triangle of every face of $P_{i}$ by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of $P_{3}$ is $\\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "On the first construction, $P_1$, four new tetrahedra will be constructed with side lengths $\\frac 12$ of the original one. Since the ratio of the volume of similar polygons is the cube of the ratio of their corresponding lengths, it follows that each of these new tetrahedra will have volume $\\left(\\frac 12\\right)^3 = \\frac 18$. The total volume added here is then $\\Delta P_1 = 4 \\cdot \\frac 18 = \\frac 12$.\nWe now note that for each midpoint triangle we construct in step $P_{i}$, there are now $6$ places to construct new midpoint triangles for step $P_{i+1}$. The outward tetrahedron for the midpoint triangle provides $3$ of the faces, while the three equilateral triangles surrounding the midpoint triangle provide the other $3$. This is because if you read this question carefully, it asks to add new tetrahedra to each face of $P_{i}$ which also includes the ones that were left over when we did the previous addition of tetrahedra. However, the volume of the tetrahedra being constructed decrease by a factor of $\\frac 18$. Thus we have the recursion $\\Delta P_{i+1} = \\frac{6}{8} \\Delta P_i$, and so $\\Delta P_i = \\frac 12 \\cdot \\left(\\frac{3}{4}\\right)^{i-1} P_1$.\nThe volume of $P_3 = P_0 + \\Delta P_1 + \\Delta P_2 + \\Delta P_3 = 1 + \\frac 12 + \\frac 38 + \\frac 9{32} = \\frac{69}{32}$, and $m+n=\\boxed{101}$. Note that the summation was in fact a geometric series." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1956_AHSME_Problems_23", "eval_func": null }, "input": { "text": "About the equation $ax^2 - 2x\\sqrt {2} + c = 0$, with $a$ and $c$ real constants, \nwe are told that the discriminant is zero. The roots are necessarily: \n$\\textbf{(A)}\\ \\text{equal and integral}\\qquad \\textbf{(B)}\\ \\text{equal and rational}\\qquad \\textbf{(C)}\\ \\text{equal and real} \\\\ \\textbf{(D)}\\ \\text{equal and irrational} \\qquad \\textbf{(E)}\\ \\text{equal and imaginary}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Plugging into the quadratic formula, we get\n\\[x = \\frac{2\\sqrt{2} \\pm \\sqrt{8-4ac}}{2a}.\\]\nThe discriminant is equal to 0, so this simplifies to $x = \\frac{2\\sqrt{2}}{2a}=\\frac{\\sqrt{2}}{a}.$ Because we are given that $a$ is real, $x$ is always real, and the answer is $\\boxed{\\textbf{(C)}}.$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1958_AHSME_Problems_31", "eval_func": null }, "input": { "text": "The altitude drawn to the base of an isosceles triangle is $8$, and the perimeter $32$. The area of the triangle is:\n$\\textbf{(A)}\\ 56\\qquad \\textbf{(B)}\\ 48\\qquad \\textbf{(C)}\\ 40\\qquad \\textbf{(D)}\\ 32\\qquad \\textbf{(E)}\\ 24$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Consider the half of the triangle that is left of the altitude. Using the information given in the problem, we can determine that this is a right triangle with one leg of length 8 whose other two sides sum to 16. Through either setting the other leg as $x$ and the hypotenuse as $16-x$ and using the Pythagorean Theorem, or by recognizing this $6-8-10$ triangle, we find that the other leg has length 6. So the triangle's total area is 48, and our answer is $\\fbox{E}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1958_AHSME_Problems_28", "eval_func": null }, "input": { "text": "A $16$-quart radiator is filled with water. Four quarts are removed and replaced with pure antifreeze liquid. Then four quarts of the mixture are removed and replaced with pure antifreeze. This is done a third and a fourth time. The fractional part of the final mixture that is water is:\n$\\textbf{(A)}\\ \\frac{1}{4}\\qquad \\textbf{(B)}\\ \\frac{81}{256}\\qquad \\textbf{(C)}\\ \\frac{27}{64}\\qquad \\textbf{(D)}\\ \\frac{37}{64}\\qquad \\textbf{(E)}\\ \\frac{175}{256}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Every time the process is done, $\\frac{3}{4}$ of the mixture is replaced with antifreeze. That means that $\\frac{3}{4}$ of the water is replaced by antifreeze, and the amount of water in the mixture after the fourth time is $\\left(\\frac{3}{4}\\right)^4 = \\boxed{\\textbf{(B) }\\frac{81}{256}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1990_AHSME_Problems_19", "eval_func": null }, "input": { "text": "For how many integers $N$ between $1$ and $1990$ is the improper fraction $\\frac{N^2+7}{N+4}$ $\\underline{not}$ in lowest terms?\n$\\text{(A) } 0\\quad \\text{(B) } 86\\quad \\text{(C) } 90\\quad \\text{(D) } 104\\quad \\text{(E) } 105$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "What we want to know is for how many $n$ is \\[\\gcd(n^2+7, n+4) > 1.\\] We start by setting \\[n+4 \\equiv 0 \\mod m\\] for some arbitrary $m$. This shows that $m$ evenly divides $n+4$. Next we want to see under which conditions $m$ also divides $n^2 + 7$. We know from the previous statement that \\[n \\equiv -4 \\mod m\\] and thus \\[n^2 \\equiv (-4)^2 \\equiv 16 \\mod m.\\] Next we simply add $7$ to get \\[n^2 + 7 \\equiv 23 \\mod m.\\] However, we also want \\[n^2 + 7 \\equiv 0 \\mod m\\] which leads to \\[n^2 + 7\\equiv 23 \\equiv 0 \\mod m\\] from the previous statement. From that statement, we get that $m$ divides $23$ evenly. Since $23$ is prime and we're looking for a GCD greater than 1, $m$ must be $23$. Going back to our original statement, we can set \\[n+4=23x\\] for some positive integer x, and \\[n=23x-4.\\] Finally, we must find the largest $x$ such that \\[23x-4<1990.\\] This is a simple linear inequality for which the answer is $x=86$, or $\\fbox{B}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1994_AHSME_Problems_24", "eval_func": null }, "input": { "text": "A sample consisting of five observations has an arithmetic mean of $10$ and a median of $12$. The smallest value that the range (largest observation minus smallest) can assume for such a sample is\n$\\textbf{(A)}\\ 2 \\qquad\\textbf{(B)}\\ 3 \\qquad\\textbf{(C)}\\ 5 \\qquad\\textbf{(D)}\\ 7 \\qquad\\textbf{(E)}\\ 10$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The minimum range occurs in the set $\\{7,7,12,12,12\\}$, so the answer is $\\boxed{\\textbf{(C)}\\ 5}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1999_AHSME_Problems_15", "eval_func": null }, "input": { "text": "Let $x$ be a real number such that $\\sec x - \\tan x = 2$. Then $\\sec x + \\tan x =$\n$\\textbf{(A)}\\ 0.1 \\qquad \\textbf{(B)}\\ 0.2 \\qquad \\textbf{(C)}\\ 0.3 \\qquad \\textbf{(D)}\\ 0.4 \\qquad \\textbf{(E)}\\ 0.5$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "$(\\sec x - \\tan x)(\\sec x + \\tan x) = \\sec^{2} x - \\tan^{2} x = 1$, so $\\sec x + \\tan x = \\boxed{\\textbf{(E)}\\ 0.5}$.", "Note that $\\sec x - \\tan x = (1-\\sin x)/\\cos x$ and $\\sec x + \\tan x = (1+\\sin x)/\\cos x$. Let $(1+\\sin x)/\\cos x = y$. Multiplying, we get $(1-\\sin^{2}x)/\\cos^{2}x = 1$.Then, $2y = 1$. $\\sec x + \\tan x = \\boxed{\\textbf{(E)}\\ 0.5}$. " ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1968_AHSME_Problems_26", "eval_func": null }, "input": { "text": "Let $S=2+4+6+\\cdots +2N$, where $N$ is the smallest positive integer such that $S>1,000,000$. Then the sum of the digits of $N$ is:\n$\\text{(A) } 27\\quad \\text{(B) } 12\\quad \\text{(C) } 6\\quad \\text{(D) } 2\\quad \\text{(E) } 1$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Note that $S = 2(1+2+3+...+N) = N(N +1)$. It follows that $N = 1000$, so the sum of the digits of $N$ is $\\fbox{E}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1984_AHSME_Problems_22", "eval_func": null }, "input": { "text": "Let $a$ and $c$ be fixed . For each $t$ let $(x_t, y_t)$ be the of the $y=ax^2+bx+c$. If the set of the vertices $(x_t, y_t)$ for all real numbers of $t$ is graphed on the , the is\n$\\mathrm{(A) \\ } \\text{a straight line} \\qquad \\mathrm{(B) \\ } \\text{a parabola} \\qquad \\mathrm{(C) \\ } \\text{part, but not all, of a parabola} \\qquad \\mathrm{(D) \\ } \\text{one branch of a hyperbola} \\qquad$ $\\mathrm{(E) \\ } \\text{None of these}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The x-coordinate of the vertex of a parabola is $-\\frac{b}{2a}$, so $x_t=-\\frac{b}{2a}$. Plugging this into $y=ax^2+bx+c$ yields $y=-\\frac{b^2}{4a^2}+c$, so $y_t=-\\frac{b^2}{4a^2}+c$. Notice that $y_t=-\\frac{b^2}{4a^2}+c=-a(-\\frac{b}{2a})^2+c=-ax_t^2+c$, so all of the vertices are on a parabola. However, we have only showed that all of the points in the locus of vertices are on a parabola, we have not shown whether or not all points on the parabola are on the locus. Assume we are given an $x_t$ on the parabola. $-\\frac{b}{2a}=x_t$, $b=-2ax_t$, so a unique $b$, and therefore a unique vertex, is determined for each point on the parabola. We therefore conclude that every point in the locus is on the parabola and every point on the parabola is in the locus, and the graph of the locus is the same as the graph of the parabola, $\\boxed{\\text{B}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1951_AHSME_Problems_21", "eval_func": null }, "input": { "text": "Given: $x > 0, y > 0, x > y$ and $z\\ne 0$. The inequality which is not always correct is:\n$\\textbf{(A)}\\ x + z > y + z \\qquad\\textbf{(B)}\\ x - z > y - z \\qquad\\textbf{(C)}\\ xz > yz$\n$\\textbf{(D)}\\ \\frac {x}{z^2} > \\frac {y}{z^2} \\qquad\\textbf{(E)}\\ xz^2 > yz^2$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "$\\textbf{(A)}\\ x + z > y + z\\implies x>y$, just subtract $z$ from both sides\n$\\textbf{(B)}\\ x - z > y - z\\implies x>y$, just add $z$ to both sides\n$\\textbf{(C)}\\ xz > yz\\implies x>y\\text{ if }x>0$, so that means that our desired answer is $\\boxed{\\textbf{(C)}\\ xz > yz}$. \nAs a check:\n$\\textbf{(D)}\\ \\frac {x}{z^2} > \\frac {y}{z^2}\\implies x>y$, we can divide $z^2$ safely and without worry because $z^2>0$.\n$\\textbf{(E)}\\ xz^2 > yz^2\\implies x>y$, similar reasoning as above but instead, multiply $z^2$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1954_AHSME_Problems_38", "eval_func": null }, "input": { "text": "If $\\log 2 = .3010$ and $\\log 3 = .4771$, the value of $x$ when $3^{x+3} = 135$ is approximately\n$\\textbf{(A) \\ } 5 \\qquad \\textbf{(B) \\ } 1.47 \\qquad \\textbf{(C) \\ } 1.67 \\qquad \\textbf{(D) \\ } 1.78 \\qquad \\textbf{(E) \\ } 1.63$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Taking the logarithm in base $3$ of both sides, we get $x+3 = \\log_3 135$. Using the property $\\log ab = \\log a + \\log b$, we get $x+3 = \\log_3 5 + \\log_3 3^3$, or $x = \\log_3 5$. Converting into base $10$ gives $x = \\frac{\\log 5}{\\log 3} = \\frac{1 - \\log 2}{\\log 3}$. Now, plugging in the values yeilds $\\boxed{\\textbf{(B) \\ } 1.47 }$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1958_AHSME_Problems_4", "eval_func": null }, "input": { "text": "In the expression $\\frac{x + 1}{x - 1}$ each $x$ is replaced by $\\frac{x + 1}{x - 1}$. The resulting expression, evaluated for $x = \\frac{1}{2}$, equals:\n$\\textbf{(A)}\\ 3\\qquad \\textbf{(B)}\\ -3\\qquad \\textbf{(C)}\\ 1\\qquad \\textbf{(D)}\\ -1\\qquad \\textbf{(E)}\\ \\text{none of these}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "When $x=\\frac{1}{2}$, $\\frac{x+1}{x-1}=-3$, substituting $-3$ for $x$ in the original equation we get:\n$\\frac{-3+1}{-3-1}=\\frac{-2}{-4}=\\frac{1}{2}\\implies \\boxed{\\mathbf{(E)}\\text{ None of these}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1952_AHSME_Problems_36", "eval_func": null }, "input": { "text": "To be continuous at $x = - 1$, the value of $\\frac {x^3 + 1}{x^2 - 1}$ is taken to be: \n$\\textbf{(A)}\\ - 2 \\qquad \\textbf{(B)}\\ 0 \\qquad \\textbf{(C)}\\ \\frac {3}{2} \\qquad \\textbf{(D)}\\ \\infty \\qquad \\textbf{(E)}\\ -\\frac{3}{2}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Factoring the numerator using the sum of cubes identity and denominator using the difference of squares identity gives \\[\\dfrac{(x+1)(x^{2}-x+1)}{(x+1)(x-1)}\\]\nCancelling out a factor of $x+1$ from the numerator and denominator gives \\[\\dfrac{(x^{2}-x+1)}{(x-1)}\\]\nPlugging in $x= -1$ gives $\\dfrac{3}{-2}$ or $\\fbox{E}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1973_AHSME_Problems_27", "eval_func": null }, "input": { "text": "Cars A and B travel the same distance. Car A travels half that distance at $u$ miles per hour and half at $v$ miles per hour. Car B travels half the time at $u$ miles per hour and half at $v$ miles per hour. The average speed of Car A is $x$ miles per hour and that of Car B is $y$ miles per hour. Then we always have \n$\\textbf{(A)}\\ x \\leq y\\qquad \\textbf{(B)}\\ x \\geq y \\qquad \\textbf{(C)}\\ x=y \\qquad \\textbf{(D)}\\ xy$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $t$ be the total number of time in hours that Car B took to drive the distance. This means that for have the time Car B traveled $\\tfrac{ut}{2}$ miles and for half the time Car B traveled $\\tfrac{vt}{2}$ miles. That means the total distance traveled is $\\tfrac{ut+vt}{2}$ miles, so\n\\[y = \\frac{\\frac{ut+vt}{2}}{t}\\]\n\\[y = \\frac{u+v}{2}\\]\nBecause both cars traveled the same distance, for half the distance Car A took $\\tfrac{ut+vt}{4u}$ hours and for half the distance Car A took $\\tfrac{ut+vt}{4v}$ hours. That means\n\\[x = \\frac{\\frac{ut+vt}{2}}{\\frac{ut+vt}{4u} + \\tfrac{ut+vt}{4v}}\\]\n\\[x = \\frac{2}{\\frac{1}{u} + \\frac{1}{v}}\\]\nIn order to compare $x$ and $y$, we need to compare the values that $x$ and $y$ are equal to.\n\\[\\frac{2}{\\frac{1}{u} + \\frac{1}{v}} \\bigcirc \\frac{u+v}{2}\\]\nSince the denominators of both numbers are positive, cross-multiplying won\u2019t change the comparison sign.\n\\[4 \\bigcirc 2 + \\frac{u}{v} + \\frac{v}{u}\\]\n\\[2 \\bigcirc \\frac{u}{v} + \\frac{v}{u}\\]\nBecause $uv$ is positive, the comparison sign does not need to be changed either.\n\\[2uv \\bigcirc u^2 + v^2\\]\n\\[0 \\bigcirc (u-v)^2\\]\nBy the , $0 \\le (u-v)^2$. All of the steps are reversible, so $\\boxed{\\textbf{(A)}\\ x \\leq y}$. This can be confirmed by testing values of $u$ and $v$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1995_AHSME_Problems_5", "eval_func": null }, "input": { "text": "A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is \n\n$\\mathrm{(A) \\ \\text{500 thousand} } \\qquad \\mathrm{(B) \\ \\text{5 million} } \\qquad \\mathrm{(C) \\ \\text{50 million} } \\qquad \\mathrm{(D) \\ \\text{500 million} } \\qquad \\mathrm{(E) \\ \\text{5 billion} }$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The rectangular field is $300 \\text{ feet} \\cdot \\frac{12 \\text{ inches}}{1 \\text{ foot}} = 3600 \\text{ inches}$ wide and $400 \\text{ feet} \\cdot \\frac{12 \\text{ inches}}{1 \\text{ foot}} = 4800 \\text{ inches}$ inches long.\nIt has an area of $3.6\\cdot 10^3 \\cdot 4.8 \\cdot 10^3 = 17.28 \\cdot 10^6$ square inches. \nWe multiply by $3$ to account for the ants to get approximately $50 \\cdot 10^6$, which is $50$ million $\\Rightarrow \\mathrm{(C)}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1962_AHSME_Problems_20", "eval_func": null }, "input": { "text": "The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be: \n$\\textbf{(A)}\\ 108\\qquad\\textbf{(B)}\\ 90\\qquad\\textbf{(C)}\\ 72\\qquad\\textbf{(D)}\\ 54\\qquad\\textbf{(E)}\\ 36$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "If the angles are in an arithmetic progression, they can be expressed as\n$a$, $a+n$, $a+2n$, $a+3n$, and $a+4n$ for some real numbers $a$ and $n$.\nNow we know that the sum of the degree measures of the angles of a pentagon is $180(5-2)=540$.\nAdding our expressions for the five angles together, we get $5a+10n=540$.\nWe now divide by 5 to get $a+2n=108$. It so happens that $a+2n$ is one of the angles we defined earlier, so that angle must have a measure of $\\boxed{108\\textbf{ (A)}}$.\n(In fact, for any arithmetic progression with an odd number of terms,\nthe middle term is equal to the average of all the terms.)", "If we write the five terms as $a$, $a - n$, $a - 2n$, $a + n$ and $a + 2n$, we can see that adding them up, we get $5a = 540$ through this, we can see that $a = 108$, $\\fbox{\\textbf{(A)}}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1959_AHSME_Problems_42", "eval_func": null }, "input": { "text": "Given three positive integers $a,b,$ and $c$. Their greatest common divisor is $D$; their least common multiple is $m$. \nThen, which two of the following statements are true?\n$\\text{(1)}\\ \\text{the product MD cannot be less than abc} \\qquad \\\\ \\text{(2)}\\ \\text{the product MD cannot be greater than abc}\\qquad \\\\ \\text{(3)}\\ \\text{MD equals abc if and only if a,b,c are each prime}\\qquad \\\\ \\text{(4)}\\ \\text{MD equals abc if and only if a,b,c are each relatively prime in pairs} \\text{ (This means: no two have a common factor greater than 1.)}$\n$\\textbf{(A)}\\ 1,2 \\qquad\\textbf{(B)}\\ 1,3\\qquad\\textbf{(C)}\\ 1,4\\qquad\\textbf{(D)}\\ 2,3\\qquad\\textbf{(E)}\\ 2,4$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Because $1\\times2\\times4>1\\times4$, 1 is false. Because $1\\times1\\times1=1\\times1$, 3 is false. It follows that the answer is $\\boxed{\\textbf{E}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1982_AHSME_Problems_16", "eval_func": null }, "input": { "text": "A wooden cube has edges of length $3$ meters. Square holes, of side one meter, centered in each face are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is\n$\\text {(A)} 54 \\qquad \\text {(B)} 72 \\qquad \\text {(C)} 76 \\qquad \\text {(D)} 84\\qquad \\text {(E)} 86$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Each exterior unit square which is removed exposes 4 faces of the unit interior squares, so the entire surface area in square meters is $6 \\cdot 3^2 - 6 + 24=72.$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1952_AHSME_Problems_12", "eval_func": null }, "input": { "text": "The sum to infinity of the terms of an infinite geometric progression is $6$. The sum of the first two terms is $4\\frac{1}{2}$. The first term of the progression is:\n$\\textbf{(A) \\ }3 \\text{ or } 1\\frac{1}{2} \\qquad \\textbf{(B) \\ }1 \\qquad \\textbf{(C) \\ }2\\frac{1}{2} \\qquad \\textbf{(D) \\ }6 \\qquad \\textbf{(E) \\ }9\\text{ or }3$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "This geometric sequence can be written as $a+ar+ar^2+ar^3+\\cdots$. We are given that $a+ar=4\\frac{1}{2}$. Using the for the sum of an infinite geometric series, we know that $\\frac{a}{1-r}=6$. Solving for $r$ in the second equation, we find that $r=\\frac{6-a}{6}$. Plugging this into the first equation results in $a^2-12a+27=0$, which can be factored as $(a-3)(a-9)=0$. Hence, $a$ equals $\\boxed{\\textbf{(E)}\\ 9 \\text{ or }3}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1960_AHSME_Problems_15", "eval_func": null }, "input": { "text": "Triangle $I$ is equilateral with side $A$, perimeter $P$, area $K$, and circumradius $R$ (radius of the circumscribed circle). \nTriangle $II$ is equilateral with side $a$, perimeter $p$, area $k$, and circumradius $r$. If $A$ is different from $a$, then: \n$\\textbf{(A)}\\ P:p = R:r \\text{ } \\text{only sometimes} \\qquad \\\\ \\textbf{(B)}\\ P:p = R:r \\text{ } \\text{always}\\qquad \\\\ \\textbf{(C)}\\ P:p = K:k \\text{ } \\text{only sometimes} \\qquad \\\\ \\textbf{(D)}\\ R:r = K:k \\text{ } \\text{always}\\qquad \\\\ \\textbf{(E)}\\ R:r = K:k \\text{ } \\text{only sometimes}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "[asy] pair A=(0,50),O=(0,0),B=(43.301,-25),C=(-43.301,-25); draw(A--B--C--A); draw(circle(O,50)); draw(B--O--C); draw(anglemark(C,O,B,200)); draw((0,0)--(0,-25)); label(\"$60^{\\circ}$\",(-7,-12)); [/asy]\nFirst, find $P$, $K$, and $R$ in terms of $A$. Since all sides of an are the same, $P=3A$. From the area formula, $K=\\frac{A^2\\sqrt{3}}{4}$. By using 30-60-90 triangles, $R=\\frac{A\\sqrt{3}}{3}$.\nUsing the same steps, $p=3a$, $k=\\frac{a^2\\sqrt{3}}{4}$, and $r=\\frac{a\\sqrt{3}}{3}$.\nNote that $P/p = 3A/3a = A/a$ and $R/r = \\frac{A\\sqrt{3}}{3} \\div \\frac{a\\sqrt{3}}{3} = A/a$. That means $P/p = R/r$, so the answer is $\\boxed{\\textbf{(B)}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1979_AHSME_Problems_13", "eval_func": null }, "input": { "text": "The inequality $y-x<\\sqrt{x^2}$ is satisfied if and only if\n$\\textbf{(A) }y<0\\text{ or }y<2x\\text{ (or both inequalities hold)}\\qquad \\textbf{(B) }y>0\\text{ or }y<2x\\text{ (or both inequalities hold)}\\qquad \\textbf{(C) }y^2<2xy\\qquad \\textbf{(D) }y<0\\qquad \\textbf{(E) }x>0\\text{ and }y<2x$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Solution by e_power_pi_times_i\n$\\sqrt{x^2} = \\pm x$, so the inequality is just $y-x<\\pm x$. Therefore we get the two inequalities $y<0$ and $y<2x$. Checking the answer choices, we find that $\\boxed{\\textbf{(A) } y<0\\text{ or }y<2x\\text{ (or both inequalities hold)}}$ is the answer." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1965_AHSME_Problems_9", "eval_func": null }, "input": { "text": "The vertex of the parabola $y = x^2 - 8x + c$ will be a point on the $x$-axis if the value of $c$ is: \n$\\textbf{(A)}\\ - 16 \\qquad \\textbf{(B) }\\ - 4 \\qquad \\textbf{(C) }\\ 4 \\qquad \\textbf{(D) }\\ 8 \\qquad \\textbf{(E) }\\ 16$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Notice that if the vertex of a parabola is on the x-axis, then the x-coordinate of the vertex must be a solution to the quadratic. Since the quadratic is strictly increasing on either side of the vertex, the solution must have double multiplicity, or the quadratic is a perfect square trinomial. This means that for the vertex of $y = x^2 - 8x + c$ to be on the x-axis, the trinomial must be a perfect square, and have discriminant of zero. So,\n\\begin{align*} 0 &= b^2-4ac\\\\ 0 &= (-8)^2-4c\\\\ c &= 64\\\\ c &= 16\\\\ \\end{align*}\nTherefore $c=16$, and our answer is $\\boxed{\\textbf{(E)}}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1977_AHSME_Problems_8", "eval_func": null }, "input": { "text": "For every triple $(a,b,c)$ of non-zero real numbers, form the number $\\frac{a}{|a|}+\\frac{b}{|b|}+\\frac{c}{|c|}+\\frac{abc}{|abc|}$. \nThe set of all numbers formed is\n$\\textbf{(A)}\\ {0} \\qquad \\textbf{(B)}\\ \\{-4,0,4\\} \\qquad \\textbf{(C)}\\ \\{-4,-2,0,2,4\\} \\qquad \\textbf{(D)}\\ \\{-4,-2,2,4\\}\\qquad \\textbf{(E)}\\ \\text{none of these}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Solution by e_power_pi_times_i\n$\\dfrac{x}{|x|} = 1$ or $-1$ depending whether $x$ is positive or negative. If $a$, $b$, and $c$ are positive, then the entire thing amounts to $4$. If one of the three is negative and the other two positive, the answer is $0$. If two of the three is negative and one is positive, the answer is $0$. If all three are negative, the answer is $-4$. Therefore the set is $\\boxed{\\textbf{(B)}\\ \\{-4,0,4\\}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1972_AHSME_Problems_13", "eval_func": null }, "input": { "text": "[asy] draw(unitsquare);draw((0,0)--(.4,1)^^(0,.6)--(1,.2)); label(\"D\",(0,1),NW);label(\"E\",(.4,1),N);label(\"C\",(1,1),NE); label(\"P\",(0,.6),W);label(\"M\",(.25,.55),E);label(\"Q\",(1,.2),E); label(\"A\",(0,0),SW);label(\"B\",(1,0),SE); //Credit to Zimbalono for the diagram [/asy]\nInside square $ABCD$ (See figure) with sides of length $12$ inches, segment $AE$ is drawn where $E$ is the point on $DC$ which is $5$ inches from $D$. \nThe perpendicular bisector of $AE$ is drawn and intersects $AE, AD$, and $BC$ at points $M, P$, and $Q$ respectively. The ratio of segment $PM$ to $MQ$ is\n$\\textbf{(A) }5:12\\qquad \\textbf{(B) }5:13\\qquad \\textbf{(C) }5:19\\qquad \\textbf{(D) }1:4\\qquad \\textbf{(E) }5:21$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let the line passing through $M$ parallel to $AB$ intersect $AD$ and $BC$ and $S$ and $T$ respectively. Since $M$ is the midpoint of $AE$, $SM=\\frac{5}{2}$ and $TM=12-\\frac{5}{2}=\\frac{19}{2}$. Since $\\triangle PSM\\sim \\triangle QTM$, $PM:MQ=SM:MT=5:9$, hence our answer is $\\fbox{C}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1977_AHSME_Problems_30", "eval_func": null }, "input": { "text": "[asy] for (int i=0; i<9; ++i) { draw(dir(10+40*i)--dir(50+40*i)); } draw(dir(50) -- dir(90)); label(\"$a$\", dir(50) -- dir(90), N); draw(dir(10) -- dir(90)); label(\"$b$\", dir(10) -- dir(90), SW); draw(dir(-70) -- dir(90)); label(\"$d$\", dir(-70) -- dir(90), E); //Credit to MSTang for the diagram [/asy]\nIf $a,b$, and $d$ are the lengths of a side, a shortest diagonal and a longest diagonal, respectively, \nof a regular nonagon (see adjoining figure), then\n$\\textbf{(A) }d=a+b\\qquad \\textbf{(B) }d^2=a^2+b^2\\qquad \\textbf{(C) }d^2=a^2+ab+b^2\\qquad\\\\ \\textbf{(D) }b=\\frac{a+d}{2}\\qquad \\textbf{(E) }b^2=ad$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "By the law of cosines we can get the following expressions for $d^{2}$ and $b^{2}$:\n\\[d^{2}=2b^{2}(1-\\cos (100^\\circ))\\] \\[b^{2}=2a^{2}(1-\\cos (140^\\circ))\\]\nWe can substitute what we got for $b^2$ into the expression for $d^{2}$:\n\\[d^{2}=4a^{2}(1-\\cos (140^\\circ))(1-\\cos (100^\\circ))=4a^{2}(1-(\\cos (100^\\circ)+\\cos (140^\\circ))+\\cos (100^\\circ)\\cos (140^\\circ))\\]\nNow apply sum-to-product and product-to-sum identities:\n\\[d^{2}=4a^{2}(1-(2\\cos (120^\\circ)\\cos (120^\\circ))+\\frac{1}{2}(\\cos (240^\\circ)+\\cos (40^\\circ)))\\]\nSimplifying further gives us: \\[d^{2}=4a^{2}(1+\\cos (20^\\circ)-\\frac{1}{4}+\\frac{1}{2}\\cos (40^\\circ))\\]\nAfter using the fact that $\\cos (40^\\circ)=2\\cos^2 (20^\\circ)-1$, it's not hard to see that the expression in the parentheses is equal to $(\\cos (20^\\circ)+\\frac{1}{2})^{2}$. So we can square-root both sides to find the expression for $d$:\n\\[d=2a(\\cos (20^\\circ)+\\frac{1}{2})\\]\nNow let's look at the expression for $b^2$. We can apply the reverse of the double angle identity to show that $1-\\cos (140^\\circ)$ equals $2\\sin^2 (70^\\circ))$. So if we square root the entire expression we get that \\[b=2a\\sin (70^\\circ)=2a\\cos (20^\\circ)\\]\nWe now have everything in terms of $a$. Luckily when we consider choice $\\textbf{A}$ we can verify without much work that this must be the answer.", "Using Van Schooten's Theorem, we draw an equilateral triangle in the nonagon, and immediately see that $a+b=d$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1998_AHSME_Problems_12", "eval_func": null }, "input": { "text": "How many different prime numbers are factors of $N$ if\n$\\mathrm{(A) \\ }1 \\qquad \\mathrm{(B) \\ }2 \\qquad \\mathrm{(C) \\ }3 \\qquad \\mathrm{(D) \\ } 4\\qquad \\mathrm{(E) \\ }7$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Re-writing as exponents, we have $\\log_3 ( \\log_5 (\\log_ 7 N)) = 2^{11}$, and so forth, such that $N = 7^{5^{3^{2^{11}}}}$, which only has $7$ as a prime factor $\\mathbf{(A)}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1953_AHSME_Problems_39", "eval_func": null }, "input": { "text": "The product, $\\log_a b \\cdot \\log_b a$ is equal to: \n$\\textbf{(A)}\\ 1 \\qquad \\textbf{(B)}\\ a \\qquad \\textbf{(C)}\\ b \\qquad \\textbf{(D)}\\ ab \\qquad \\textbf{(E)}\\ \\text{none of these}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "\\[a^x=b\\]\n\\[b^y=a\\]\n\\[{a^x}^y=a\\]\n\\[xy=1\\]\n\\[\\log_a b\\log_b a=1\\]\nAs a result, the answer should be $\\boxed{\\textbf{(A) }1}$.", "Apply the change of base formula to $\\log_a b$ and $\\log_b a$. For simplicity, let us convert to base-10 log.\nBy change of base, the expression becomes $\\frac{\\log b}{\\log a} * \\frac{\\log a}{\\log b} = \\boxed{\\textbf{(A) }1}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1979_AHSME_Problems_26", "eval_func": null }, "input": { "text": "The function $f$ satisfies the functional equation $f(x) +f(y) = f(x + y ) - xy - 1$ for every pair $x,~ y$\nof real numbers. If $f( 1) = 1$, then the number of integers $n \\neq 1$ for which $f ( n ) = n$ is\n$\\textbf{(A) }0\\qquad \\textbf{(B) }1\\qquad \\textbf{(C) }2\\qquad \\textbf{(D) }3\\qquad \\textbf{(E) }\\infty$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We are given that $f(x) +f(y) = f(x + y ) - xy - 1$ and $f( 1) = 1$, so we can let $y = 1$. Thus we have:\n\\[f(x) + 1 = f(x + 1) - x - 1\\]\nRearranging gives a recursive formula for $f$:\n\\[f(x + 1) = f(x) + x + 2\\]\nWe notice that this is the recursive form for a quadratic, so f(x) must be of the form $ax^{2} + bx + c$. To solve for $a, b,$ and $c$, we can first work backwards to solve for the values of f(0) and f(-1):\n\\[f(1) = f(0) + 0 + 2 \\Rightarrow 1 = f(0) + 2 \\Rightarrow f(0) = -1\\]\n\\[f(0) = f(-1) -1 + 2 \\Rightarrow -1 = f(-1) + 1 \\Rightarrow f(-1) = -2\\]\nSince $f(0) = -1$:\n\\[f(0) = a(0)^{2} + b(0) + c = -1 \\Rightarrow c = -1\\] \nSince $f(1) = 1$: \n\\[f(1) = a(1)^{2} + b(1)+ c = a + b - 1 = 1 \\Rightarrow a + b = 2\\] \nSimilarly, since $f(-1) = -2$:\n\\[f(-1) = a(-1)^{2} + b(-1)+ c = a - b - 1 = -2 \\Rightarrow a - b = -1\\]\nThus we have the system of equations:\n\\[a + b = 2\\]\n\\[a - b = -1\\]\nWhich can be solved to yield $a = \\frac{1}{2}$, $b = \\frac{3}{2}$. Therefore, $f(x) = \\frac{1}{2} x^{2} + \\frac{3}{2} x - 1$.\nSince we are searching for values for which $f(x) = x$, we have the equation $\\frac{1}{2} x^{2} + \\frac{3}{2} x - 1 = x$. Subtracting $x$ yields $\\frac{1}{2} x^{2} + \\frac{1}{2} x - 1 = 0$, which we can simplify by dividing both sides by $\\frac{1}{2}$: $x^{2} + x - 2 = 0$. This factors into $(x + 2) (x - 1) = 0$, so therefore there are two solutions to $f(x) = x$: $-2$ and $1$. Since the problem asks only for solutions that do not equal $1$, the answer is $\\fbox{(B) 1}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1968_AHSME_Problems_29", "eval_func": null }, "input": { "text": "Given the three numbers $x,y=x^x,z=x^{x^x}$ with $.9x$ and $x+y0$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "From $x-y>x$, we get that $-y>0\\implies y<0$.\nFrom $x+yb$. Writing this out, we get $10a+b-(10b+a)=9a-9b=9(a-b)$. Therefore, the difference must be a multiple of $9$, and the only perfect cube with less than $3$ digits and is multiple of $9$ is $3^3=27$. Also, that means $a-b=3$, and there are $7$ possibilities of that, so our answer is \n$\\boxed{\\textbf{(D)}}$ There are exactly $7$ values of $N$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1958_AHSME_Problems_49", "eval_func": null }, "input": { "text": "In the expansion of $(a + b)^n$ there are $n + 1$ dissimilar terms. The number of dissimilar terms in the expansion of $(a + b + c)^{10}$ is:\n$\\textbf{(A)}\\ 11\\qquad \\textbf{(B)}\\ 33\\qquad \\textbf{(C)}\\ 55\\qquad \\textbf{(D)}\\ 66\\qquad \\textbf{(E)}\\ 132$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Expand the binomial $((a+b)+c)^n$ with the binomial theorem. We have:\n\\[\\sum\\limits_{k=0}^{10} \\binom{10}{k} (a+b)^k c^{10-k}\\]\nSo for each iteration of the summation operator, we add k+1 dissimilar terms. Therefore our answer is:\n\\[\\sum\\limits_{k=0}^{10} k+1 = \\frac{11(1+11)}{2} = 66 \\to \\boxed{\\textbf{D}}\\]", "Each term in the expansion of $(a+b+c)^{10}$ will have the form $a^i \\times b^j \\times c^k$, where $0\\le i, j, k\\le 10$ and $a+b+c=10$. So, we need to find the number of triplets of nonnegative integers $(a, b, c)$ such that $a+b+c=10$. Using Stars and Bars, this value is $\\binom{12}{2}=66$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1979_AHSME_Problems_8", "eval_func": null }, "input": { "text": "Find the area of the smallest region bounded by the graphs of $y=|x|$ and $x^2+y^2=4$.\n$\\textbf{(A) }\\frac{\\pi}{4}\\qquad \\textbf{(B) }\\frac{3\\pi}{4}\\qquad \\textbf{(C) }\\pi\\qquad \\textbf{(D) }\\frac{3\\pi}{2}\\qquad \\textbf{(E) }2\\pi$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Solution by e_power_pi_times_i\nThe graph of $x^2+y^2 = 4$ is a circle with radius $2$ centered at the origin. The graph of $y=|x|$ is the combined graphs of $y=x$ and $y=-x$ with a nonnegative y. Because the arguments of $y=x$ and $y=-x$ are $135^\\circ$ and $45^\\circ$ respectively, the angle between the graphs of $y=x$ and $y=-x$ is $90^\\circ$. Thus, the smallest region bounded by the graphs is $\\frac{1}{4}\\cdot2^2\\cdot\\pi = \\boxed{\\textbf{(C) } \\pi}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1951_AHSME_Problems_16", "eval_func": null }, "input": { "text": "If in applying the quadratic formula to a quadratic equation \n\\[f(x) \\equiv ax^2 + bx + c = 0,\\]\nit happens that $c = \\frac{b^2}{4a}$, then the graph of $y = f(x)$ will certainly:\n$\\mathrm{(A) \\ have\\ a\\ maximum } \\qquad \\mathrm{(B) \\ have\\ a\\ minimum} \\qquad$ $\\mathrm{(C) \\ be\\ tangent\\ to\\ the\\ x-axis} \\qquad$ $\\mathrm{(D) \\ be\\ tangent\\ to\\ the\\ y-axis} \\qquad$ $\\mathrm{(E) \\ lie\\ in\\ one\\ quadrant\\ only}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The discriminant of the quadratic equation is $b^2 - 4ac = b^2 - 4a\\left(\\frac{b^2}{4a}\\right) = 0$. This indicates that the equation has only one root (applying the quadratic formula, we get $x = \\frac{-b + \\sqrt{0}}{2a} = -b/2a$). Thus it follows that $f(x)$ touches the x-axis exactly once, and hence is tangent to the x-axis $\\Rightarrow \\mathrm{(C)}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1954_AHSME_Problems_13", "eval_func": null }, "input": { "text": "A quadrilateral is inscribed in a circle. If angles are inscribed in the four arcs cut off \nby the sides of the quadrilateral, their sum will be: \n$\\textbf{(A)}\\ 180^\\circ \\qquad \\textbf{(B)}\\ 540^\\circ \\qquad \\textbf{(C)}\\ 360^\\circ \\qquad \\textbf{(D)}\\ 450^\\circ\\qquad\\textbf{(E)}\\ 1080^\\circ$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The arcs created by all sides of the quadrilateral will sum to 360 because they cover the entire circle. The inscribed angles created by each side of the trapezoid will all be half the measure of each arc created by their respective side. Thus, the sum of the all the inscribed angles is half the sum of all the arcs. $360/2 = 180\\ \\fbox{(A)}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1973_AHSME_Problems_30", "eval_func": null }, "input": { "text": "Let $[t]$ denote the greatest integer $\\leq t$ where $t \\geq 0$ and $S = \\{(x,y): (x-T)^2 + y^2 \\leq T^2 \\text{ where } T = t - [t]\\}$. Then we have \n$\\textbf{(A)}\\ \\text{the point } (0,0) \\text{ does not belong to } S \\text{ for any } t \\qquad$\n$\\textbf{(B)}\\ 0 \\leq \\text{Area } S \\leq \\pi \\text{ for all } t \\qquad$\n$\\textbf{(C)}\\ S \\text{ is contained in the first quadrant for all } t \\geq 5 \\qquad$\n$\\textbf{(D)}\\ \\text{the center of } S \\text{ for any } t \\text{ is on the line } y=x \\qquad$\n$\\textbf{(E)}\\ \\text{none of the other statements is true}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The region $S$ is a circle radius $T$ and center $(T,0)$. Since $T = t-[t]$, $0 \\le T < 1$. That means the area of the circle is less than $\\pi$, and since the region can also be just a dot (achieved when $t$ is integer), the answer is $\\boxed{\\textbf{(B)}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1962_AHSME_Problems_11", "eval_func": null }, "input": { "text": "The difference between the larger root and the smaller root of $x^2 - px + (p^2 - 1)/4 = 0$ is: \n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ 1\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ p\\qquad\\textbf{(E)}\\ p+1$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Call the two roots $r$ and $s$, with $r \\ge s$. \nBy Vieta's formulas, $p=r+s$ and $(p^2-1)/4=rs.$\n(Multiplying both sides of the second equation by 4 gives $p^2-1=4rs$.)\nThe value we need to find, then, is $r-s$.\nSince $p=r+s$, $p^2=r^2+2rs+s^2$.\nSubtracting $p^2-1=4rs$ from both sides gives $1=r^2-2rs+s^2$.\nTaking square roots, $r-s=1 \\Rightarrow \\boxed{\\textbf{(B)}}$.\n(Another solution is to use the quadratic formula and see that the\nroots are $\\frac{p\\pm 1}2$, and their difference is 1.)" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1982_AHSME_Problems_25", "eval_func": null }, "input": { "text": "The adjacent map is part of a city: the small rectangles are blocks, and the paths in between are streets. \nEach morning, a student walks from intersection $A$ to intersection $B$, always walking along streets shown, \nand always going east or south. For variety, at each intersection where he has a choice, he chooses with \nprobability $\\frac{1}{2}$ whether to go east or south. Find the probability that through any given morning, he goes through $C$. \n[asy] defaultpen(linewidth(0.7)+fontsize(8)); size(250); path p=origin--(5,0)--(5,3)--(0,3)--cycle; path q=(5,19)--(6,19)--(6,20)--(5,20)--cycle; int i,j; for(i=0; i<5; i=i+1) { for(j=0; j<6; j=j+1) { draw(shift(6*i, 4*j)*p); }} clip((4,2)--(25,2)--(25,21)--(4,21)--cycle); fill(q^^shift(18,-16)*q^^shift(18,-12)*q, black); label(\"A\", (6,19), SE); label(\"B\", (23,4), NW); label(\"C\", (23,8), NW); draw((26,11.5)--(30,11.5), Arrows(5)); draw((28,9.5)--(28,13.5), Arrows(5)); label(\"N\", (28,13.5), N); label(\"W\", (26,11.5), W); label(\"E\", (30,11.5), E); label(\"S\", (28,9.5), S);[/asy]\n$\\text{(A)}\\frac{11}{32}\\qquad \\text{(B)}\\frac{1}{2}\\qquad \\text{(C)}\\frac{4}{7}\\qquad \\text{(D)}\\frac{21}{32}\\qquad \\text{(E)}\\frac{3}{4}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "[asy] defaultpen(linewidth(0.7)+fontsize(8)); size(250); path p=origin--(5,0)--(5,3)--(0,3)--cycle; path q=(5,19)--(6,19)--(6,20)--(5,20)--cycle; int i,j; for(i=0; i<5; i=i+1) { for(j=0; j<6; j=j+1) { draw(shift(6*i, 4*j)*p); }} clip((4,2)--(25,2)--(25,21)--(4,21)--cycle); fill(q^^shift(18,-16)*q^^shift(18,-12)*q, black); label(\"$A$\", (6,19), SE); label(\"$B$\", (23,4), NW); label(\"$C$\", (23,8), NW); label(\"$C_3$\", (17,7), SW); label(\"$C_2$\", (17,11), SW); label(\"$C_1$\", (17,15), SW); label(\"$C_0$\", (17,19), SW); filldraw((17,7)--(18,7)--(18,8)--(17,8)--cycle, black); filldraw((17,11)--(18,11)--(18,12)--(17,12)--cycle, black); filldraw((17,15)--(18,15)--(18,16)--(17,16)--cycle, black); filldraw((17,19)--(18,19)--(18,20)--(17,20)--cycle, black); draw((26,11.5)--(30,11.5), Arrows(5)); draw((28,9.5)--(28,13.5), Arrows(5)); label(\"N\", (28,13.5), N); label(\"W\", (26,11.5), W); label(\"E\", (30,11.5), E); label(\"S\", (28,9.5), S);[/asy]\nThe probability that the student passes through $C$ is the sum from $i=0$ to $3$ of the probabilities that he enters intersection $C_i$ in the adjoining figure and goes east. The number of paths from $A$ to $C_i$ is ${{2+i} \\choose 2}$, because each such path has $2$ eastward block segments and they can occur in any order. The probability of taking any one of these paths to $C_i$ and then going east is $(\\tfrac{1}{2})^{3+i}$ because there are $3+i$ intersections along the way (including $A$ and $C_i$) where an independent choice with probability $\\tfrac{1}{2}$ is made. So the answer is \\[\\sum\\limits_{i=0}^3 {{2+i} \\choose 2} \\left( \\frac{1}{2} \\right)^{3+i} = \\frac{1}{8} + \\frac{3}{16} + \\frac{6}{32} + \\frac{10}{64} = \\boxed{\\frac{21}{32}}.\\]" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1988_AHSME_Problems_4", "eval_func": null }, "input": { "text": "The slope of the line $\\frac{x}{3} + \\frac{y}{2} = 1$ is\n$\\textbf{(A)}\\ -\\frac{3}{2}\\qquad \\textbf{(B)}\\ -\\frac{2}{3}\\qquad \\textbf{(C)}\\ \\frac{1}{3}\\qquad \\textbf{(D)}\\ \\frac{2}{3}\\qquad \\textbf{(E)}\\ \\frac{3}{2}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "To find the slope, all we have to do is put the equation into slope-intercept form. We subtract $\\frac{x}{3}$ from both sides and then multiple all \nterms by $2$. This yields $y=-\\frac{2}{3}x+1$, so the slope is $-\\frac{2}{3} \\implies \\boxed{\\text{B}}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1968_AHSME_Problems_9", "eval_func": null }, "input": { "text": "The sum of the real values of $x$ satisfying the equality $|x+2|=2|x-2|$ is:\n$\\text{(A) } \\frac{1}{3}\\quad \\text{(B) } \\frac{2}{3}\\quad \\text{(C) } 6\\quad \\text{(D) } 6\\tfrac{1}{3}\\quad \\text{(E) } 6\\tfrac{2}{3}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "$\\fbox{E}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1953_AHSME_Problems_47", "eval_func": null }, "input": { "text": "If $x>0$, then the correct relationship is: \n$\\textbf{(A)}\\ \\log (1+x) = \\frac{x}{1+x} \\qquad \\textbf{(B)}\\ \\log (1+x) < \\frac{x}{1+x} \\\\ \\textbf{(C)}\\ \\log(1+x) > x\\qquad \\textbf{(D)}\\ \\log (1+x) < x\\qquad \\textbf{(E)}\\ \\text{none of these}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Plug in $x=9$. Then, you can see that the answer is $\\fbox{D}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1966_AHSME_Problems_30", "eval_func": null }, "input": { "text": "If three of the roots of $x^4+ax^2+bx+c=0$ are $1$, $2$, and $3$ then the value of $a+c$ is:\n$\\text{(A) } 35 \\quad \\text{(B) } 24 \\quad \\text{(C) } -12 \\quad \\text{(D) } -61 \\quad \\text{(E) } -63$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Since this is a quartic equation, there are going to be $4$ solutions. \nBy Vieta's formulas, since the $x^3$ term is $0$, the sum of the roots is also $0$. Therefore, the 4th root of this polynomial is $-6$. \nLastly, by Vieta's Formulas, $a+c=-(1\\cdot2+1\\cdot3+1\\cdot-6+2\\cdot3+2\\cdot-6+3\\cdot-6)+1\\cdot2\\cdot3\\cdot-6=-25-36=-61= \\fbox{D}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1993_AHSME_Problems_15", "eval_func": null }, "input": { "text": "For how many values of $n$ will an $n$-sided regular polygon have interior angles with integral measures?\n$\\text{(A) } 16\\quad \\text{(B) } 18\\quad \\text{(C) } 20\\quad \\text{(D) } 22\\quad \\text{(E) } 24$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Start with the facts that all polygons have their exterior angles sum to 360 and the exterior and interior angles make a linear pair of angles. So our goal is to find the number of divisors of 360 to make both the interior and exterior angles integers. The prime factorization of 360 is $2^3 * 3^2 * 5$. That means the number of divisors is 4*3*2 = 24. But we're not done yet. We cannot have a 1 or 2 sided polygon so we subtract off two bringing us to our final answer of 22 $\\fbox{D}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1964_AHSME_Problems_40", "eval_func": null }, "input": { "text": "A watch loses $2\\frac{1}{2}$ minutes per day. It is set right at $1$ P.M. on March 15. Let $n$ be the positive correction, in minutes, to be added to the time shown by the watch at a given time. When the watch shows $9$ A.M. on March 21, $n$ equals:\n$\\textbf{(A) }14\\frac{14}{23}\\qquad\\textbf{(B) }14\\frac{1}{14}\\qquad\\textbf{(C) }13\\frac{101}{115}\\qquad\\textbf{(D) }13\\frac{83}{115}\\qquad \\textbf{(E) }13\\frac{13}{23}$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "From March 15 $1$ P.M. on the watch to March 21 $9$ A.M. on the watch, the watch passed $20 + 5 \\times 24 = 140$ hours. \nSince $1$ watch hour equals $\\frac{24}{23 + \\frac{57.5}{60}} = \\frac{576}{575}$ real hour, the difference between the watch time and the actual time passed is $140 \\times \\left( \\frac{576}{575} - 1 \\right) = \\frac{28}{115}$ hour $=14\\frac{14}{23}$ minutes." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1996_AHSME_Problems_4", "eval_func": null }, "input": { "text": "Six numbers from a list of nine integers are $7,8,3,5,9$ and $5$. The largest possible value of the median of all nine numbers in this list is\n$\\text{(A)}\\ 5\\qquad\\text{(B)}\\ 6\\qquad\\text{(C)}\\ 7\\qquad\\text{(D)}\\ 8\\qquad\\text{(E)}\\ 9$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "First, put the six numbers we have in order, since we are concerned with the median: $3, 5, 5, 7, 8, 9$.\nWe have three more numbers to insert into the list, and the median will be the $5^{th}$ highest (and $5^{th}$ lowest) number on the list. If we top-load the list by making all three of the numbers greater than $9$, the median will be the highest it can possibly be. Thus, the maximum median is the fifth piece of data in the list, which is $8$, giving an answer of $\\boxed{D}$.\n(In fact, as long as the three new integers are greater than $8$, the median will be $8$.)\nThis illustrates one important fact about medians: no matter how high the three \"outlier\" numbers are, the median will never be greater than $8$. The arithmetic mean is, generally speaking, more sensitive to such outliers, while the median is resistant to a small number of data that are either too high or too low." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "1985_AHSME_Problems_12", "eval_func": null }, "input": { "text": "Let $p$, $q$ and $r$ be distinct prime numbers, where $1$ is not considered a prime. Which of the following is the smallest positive perfect cube having $n = pq^2r^4$ as a divisor?\n$\\mathrm{(A)\\ } p^8q^8r^8 \\qquad \\mathrm{(B) }\\left(pq^2r^2\\right)^3 \\qquad \\mathrm{(C) } \\left(p^2q^2r^2\\right)^3 \\qquad \\mathrm{(D) } \\left(pqr^2\\right)^3 \\qquad \\mathrm{(E) \\ }4p^3q^3r^3$", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "A number of the form $p^aq^br^c$ will be a perfect cube precisely when $a$, $b$, and $c$ are multiples of 3. (Clearly, since we are looking for the possible perfect cube, we can assume that it has no other prime factors.) Furthermore, for it to be a multiple of $pq^2r^4$, we must have $a \\geq 1$, $b \\geq 2$, and $c \\geq 4$. The smallest multiple of $3$ that is at least $1$ is $3$, the smallest that is at least $2$ is again $3$, and the smallest that is at least $4$ is $6$. Hence the smallest possible perfect cube with $n$ as a divisor is $p^3q^3r^6 = \\boxed{\\text{(D)} \\left(pqr^2\\right)^3}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2019_USAJMO_Problems_2", "eval_func": null }, "input": { "text": "Let $\\mathbb Z$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f:\\mathbb Z\\rightarrow\\mathbb Z$ and $g:\\mathbb Z\\rightarrow\\mathbb Z$ satisfying \\[f(g(x))=x+a\\quad\\text{and}\\quad g(f(x))=x+b\\] for all integers $x$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We claim that the answer is $|a|=|b|$.\nProof:\n$f$ and $g$ are surjective because $x+a$ and $x+b$ can take on any integral value, and by evaluating the parentheses in different order, we find $f(g(f(x)))=f(x+b)=f(x)+a$ and $g(f(g(x)))=g(x+a)=g(x)+b$. We see that if $a=0$ then $g(x)=g(x)+b$ to $b=0$ as well, so similarly if $b=0$ then $a=0$, so now assume $a, b\\ne 0$. \nWe see that if $x=|b|n$ then $f(x)\\equiv f(0) \\pmod{|a|}$, if $x=|b|n+1$ then $f(x)\\equiv f(1)\\pmod{|a|}$, if $x=|b|n+2$ then $f(x)\\equiv f(2)\\pmod{|a|}$... if $x=|b|(n+1)-1$ then $f(x)\\equiv f(|b|-1)\\pmod{|a|}$. This means that the $b$-element collection $\\left\\{f(0), f(1), f(2), ... ,f(|b|-1)\\right\\}$ contains all $|a|$ residues mod $|a|$ since $f$ is surjective, so $|b|\\ge |a|$. Doing the same to $g$ yields that $|a|\\ge |b|$, so this means that only $|a|=|b|$ can work.\nFor $a=b$ let $f(x)=x$ and $g(x)=x+a$, and for $a=-b$ let $f(x)=-x$ and $g(x)=-x-a$, so $|a|=|b|$ does work and are the only solutions, as desired.", "We claim that $f$ and $g$ exist if and only if $|a|=|b|$.\n\nFor some fixed $j$, let $f(j)=k$.\nIf $b=0$, then $g(k)=j$. Suppose $a\\ne 0$. Then $f(j)=f(g(k))=k+a\\ne k$, a contradiction. Thus, $a=0$. Similarly, if $a=0$, then $b=0$, satisfying $|a|=|b|$.\nOtherwise, $a,b\\ne 0$. We know that $g(k)=g(f(j))=j+b$, $f(j+b)=f(g(k))=k+a$, $g(k+a)=j+2b$, and so on: $f(j+nb)=k+na$ and $g(k+na)=j+(n+1)b$ for $n\\ge 0$.\nConsider the value of $g(k-a)$. Suppose $g(k-a)=j'\\ne j$. Then $f(j')=k$ and $g(f(j'))=j+b\\ne j'+b$, a contradiction. Thus, $g(k-a)=j$. We repeat with $f(j-b)$. Suppose $f(j-b)=k'-b\\ne k-b$. Then $g(k'-b)=j$ and $f(g(k'-b))=k\\ne k'$, a contradition. Thus, $f(j-b)=k-b$. Continuing, $g(k-2a)=j-a$, and so on: $f(j+nb)=k+na$ and $g(k+na)=j+(n+1)b$ now for all $n$.\nThis defines $f(x)$ for all $x\\equiv j\\pmod{|b|}$ and $g(x)$ for all $x\\equiv f(j)\\pmod{|a|}$.\nThis means that $x\\equiv j\\pmod{|b|}\\implies f(x)\\equiv f(j)\\pmod{|a|}$, and $y\\equiv f(j)\\pmod{|a|}\\implies g(y)\\equiv j\\pmod{|b|}$ which implies $f(x)\\equiv f(j)\\pmod{|a|}\\implies x\\equiv j\\pmod{|b|}$.\nAs a result, $f(x)$ maps each residue mod $|b|$ to a unique residue mod $|a|$, so $|a|\\ge|b|$. Similarly, $g(x)$ maps each residue mod $|a|$ to a unique residue mod $|b|$, so $|b|\\ge|a|$. Therefore, $|a|=|b|$.\n\n$|a|=|b|$ means that either $a=b$ or $a=-b$. $f(x)=x,g(x)=x+a$ works for the former and $f(x)=-x,g(x)=-x-a$ works for the latter, and we are done." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2021_USAJMO_Problems_5", "eval_func": null }, "input": { "text": "A finite set $S$ of positive integers has the property that, for each $s \\in S,$ and each positive integer divisor $d$ of $s$, there exists a unique element $t \\in S$ satisfying $\\text{gcd}(s, t) = d$. (The elements $s$ and $t$ could be equal.)\n\n\nGiven this information, find all possible values for the number of elements of $S$.\n\n\n", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The answer is $0$ (left out by problem author) or $2^n$ for any non-negative integer $n$.\nTo construct $|S|=2^n$, take $2n$ distinct primes $p_1, q_1, p_2, q_2, \\dots p_n, q_n$ and construct $2^n$ elements for $S$ by taking the product over all $n$ indices $i$ of either $p_i$ or $q_i$ for every $i$.\nNote that (from the problem statement) each element of $S$ has $|S|$ divisors.\nNow it suffices to show that there is no element $x \\in S$ such that $x$ is divisible by more than one power of any prime.\nSuppose there does exist an element $x$ and a prime $p$ such that $p^2 \\mid x$.\nThis implies that there exists an element $c \\in S$ that is divisible by $p$ but not $p^2$ by $\\gcd (x,c)=p$.\nExactly half of $c$'s divisors are not divisible by $p$, so it follows that exactly half of the elements in $S$ are not divisible by $p$.\nHowever, if $p^0 , p^1 , \\dots p^k$ are the powers of $p$ dividing $x$ for some $k \\ge 2$, only $\\frac{1}{k+1}$ of the divisors of $x$ are not divisible by $p$.\nBut this means that $\\frac{1}{k+1}$ of the elements of $S$ are not divisible by $p$, contradiction.\nTherefore, $|S|$ must be some power of $2$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2016_USAJMO_Problems_4", "eval_func": null }, "input": { "text": "Find, with proof, the least integer $N$ such that if any $2016$ elements are removed from the set $\\{1, 2,...,N\\}$, one can still find $2016$ distinct numbers among the remaining elements with sum $N$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Since any $2016$ elements are removed, suppose we remove the integers from $1$ to $2016$. Then the smallest possible sum of $2016$ of the remaining elements is \\[2017+2018+\\cdots + 4032 = 1008 \\cdot 6049 = 6097392\\]\nso clearly $N\\ge 6097392$. We will show that $N=6097392$ works.\n$\\vspace{0.2 in}$\n$\\{1,2\\cdots 6097392\\}$ contain the integers from $1$ to $6048$, so pair these numbers as follows:\n\\[1, 6048\\]\n\\[2, 6047\\]\n\\[3, 6046\\]\n\\[\\cdots\\]\n\\[3024, 3025\\]\nWhen we remove any $2016$ integers from the set $\\{1,2,\\cdots N\\}$, clearly we can remove numbers from at most $2016$ of the $3024$ pairs above, leaving at least $1008$ complete pairs. To get a sum of $N$, simply take these $1008$ pairs, all of which sum to $6049$. The sum of these $2016$ elements is $1008 \\cdot 6049 = 6097392$, as desired.\n$\\vspace{0.2 in}$\nWe have shown that $N$ must be at least $6097392$, and that this value is attainable. Therefore our answer is $\\boxed{6097392}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2016_USAJMO_Problems_6", "eval_func": null }, "input": { "text": "Find all functions $f:\\mathbb{R}\\rightarrow \\mathbb{R}$ such that for all real numbers $x$ and $y$, \\[(f(x)+xy)\\cdot f(x-3y)+(f(y)+xy)\\cdot f(3x-y)=(f(x+y))^2.\\]", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Set $x = y = 0$ to obtain $f(0) = 0.$\n Set $x = 0$ to obtain $f(y)f(-y) = f(y)^2.$\n$\\indent$ In particular, if $f(y) \\ne 0$ then $f(y) = f(-y).$\n$\\indent$ In addition, replacing $y \\to -t$, it follows that $f(t) = 0 \\implies f(-t) = 0$ for all $t \\in \\mathbb{R}.$\n Set $x = 3y$ to obtain $\\left[f(y) + 3y^2\\right]f(8y) = f(4y)^2.$\n$\\indent$ In particular, replacing $y \\to t/8$, it follows that $f(t) = 0 \\implies f(t/2) = 0$ for all $t \\in \\mathbb{R}.$\n Set $y = -x$ to obtain $f(4x)\\left[f(x) + f(-x) - 2x^2\\right] = 0.$\n$\\indent$ In particular, if $f(x) \\ne 0$, then $f(4x) \\ne 0$ by the observation from Step 3, because $f(4x) = 0 \\implies f(2x) = 0 \\implies f(x) = 0.$ Hence, the above equation implies that $2x^2 = f(x) + f(-x) = 2f(x)$, where the last step follows from the first observation from Step 2.\n$\\indent$ Therefore, either $f(x) = 0$ or $f(x) = x^2$ for each $x.$\n$\\indent$ Looking back on the equation from Step 3, it follows that $f(y) + 3y^2 \\ne 0$ for any nonzero $y.$ Therefore, replacing $y \\to t/4$ in this equation, it follows that $f(t) = 0 \\implies f(2t) = 0.$\n If $f(a) = f(b) = 0$, then $f(b - a) = 0.$\n$\\indent$ This follows by choosing $x, y$ such that $x - 3y = a$ and $3x - y = b.$ Then $x + y = \\tfrac{b - a}{2}$, so plugging $x, y$ into the given equation, we deduce that $f\\left(\\tfrac{b - a}{2}\\right) = 0.$ Therefore, by the third observation from Step 4, we obtain $f(b - a) = 0$, as desired.\n If $f \\not\\equiv 0$, then $f(t) = 0 \\implies t = 0.$\n$\\indent$ Suppose by way of contradiction that there exists an nonzero $t$ with $f(t) = 0.$ Choose $x, y$ such that $f(x) \\ne 0$ and $x + y = t.$ The following three facts are crucial:\n$\\indent$ 1. $f(y) \\ne 0.$ This is because $(x + y) - y = x$, so by Step 5, $f(y) = 0 \\implies f(x) = 0$, impossible.\n$\\indent$ 2. $f(x - 3y) \\ne 0.$ This is because $(x + y) - (x - 3y) = 4y$, so by Step 5 and the observation from Step 3, $f(x - 3y) = 0 \\implies f(4y) = 0 \\implies f(2y) = 0 \\implies f(y) = 0$, impossible.\n$\\indent$ 3. $f(3x - y) \\ne 0.$ This is because by the second observation from Step 2, $f(3x - y) = 0 \\implies f(y - 3x) = 0.$ Then because $(x + y) - (y - 3x) = 4x$, Step 5 together with the observation from Step 3 yield $f(3x - y) = 0 \\implies f(4x) = 0 \\implies f(2x) = 0 \\implies f(x) = 0$, impossible.\n$\\indent$ By the second observation from Step 4, these three facts imply that $f(y) = y^2$ and $f(x - 3y) = \\left(x - 3y\\right)^2$ and $f(3x - y) = \\left(3x - y\\right)^2.$ By plugging into the given equation, it follows that\n\\begin{align*} \\left(x^2 + xy\\right)\\left(x - 3y\\right)^2 + \\left(y^2 + xy\\right)\\left(3x - y\\right)^2 = 0. \\end{align*} But the above expression miraculously factors into $\\left(x + y\\right)^4$! This is clearly a contradiction, since $t = x + y \\ne 0$ by assumption. This completes Step 6.\n By Step 6 and the second observation from Step 4, the only possible solutions are $f \\equiv 0$ and $f(x) = x^2$ for all $x \\in \\mathbb{R}.$ It's easy to check that both of these work, so we're done.", "Step 1: $x=y=0 \\implies f(0)=0$\nStep 2: $x=0 \\implies f(y)f(-y)=f(y)^{2}$. Now, assume $y \\not = 0$. Then, if $f(y)=0$, we substitute in $-y$ to get $f(y)f(-y)=f(-y)^{2}$, or $f(y)=f(-y)=0$. Otherwise, we divide both sides by $f(y)$ to get $f(y)=f(-y)$. If $y=0$, we obviously have $f(0)=f(0)$. Thus, the function is even.\n.\nStep 3: $y=-x \\implies 2f(4x)(f(x)-x^{2})=0$. Thus, $\\forall x$, we have $f(4x)=0$ or $f(x)=x^{2}$.\nStep 4: We now assume $f(x) \\not = 0$, $x\\not = 0$. We have $f(\\frac{x}{4})=\\frac{x^{2}}{16}$. Now, setting $x=y=\\frac{x}{4}$, we have $f(\\frac{x}{2})=\\frac{x^{2}}{4}$ or $f(\\frac{x}{2})=0$. The former implies that $f(x)=0$ or $x^{2}$. The latter implies that $f(x)=0$ or $f(x)=\\frac{x^{2}}{2}$. Assume the latter. $y=-2x \\implies -\\frac{3y^{2}}{2}f(7y)-{2y^{2}}f(5y)=\\frac{x^{2}}{2}$. Clearly, this implies that $f(x)$ is negative for some $m$. Now, we have $f(\\frac{m}{4})=\\frac{m^{2}}{16} \\implies f(\\frac{m}{2})=0,\\frac{m^{2}}{4} \\implies f(m) \\geq 0$, which is a contradiction. Thus, $\\forall x$$f(x)=0$ or $f(x)=x^{2}$.\nStep 5: We now assume $f(x)=0$, $f(y)=y^{2}$ for some $x,y \\not = 0$. Let $m$ be sufficiently large integer, let $z=|4^{m}x|$ and take the absolute value of $y$(since the function is even). Choose $c$ such that $3z-c=y$. Note that we have $\\frac{c}{z}$~$3$ and $\\frac{y}{z}$~$0$. Note that $f(z)=0$. Now, $x=z, y=c \\implies$ LHS is positive, as the second term is positive and the first term is nonnegative and thus the right term is equal to $(z+c)^{4}$~$256z^{4}$. Now if $f(z-3c)=0$, the second term of the LHS/RHS clearly ~0 as $m \\to \\infty$. if $f(z-3c)=0$, then we have LHS/RHS ~ $0$, otherwise, we have LHS/RHS~$\\frac{8^{2}\\cdot 3z^{4}}{256z^{4}}$~$\\frac{3}{4}$, a contradiction, as we're clearly not dividing by $0$, and we should have LHS/RHS=1." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2023_USAJMO_Problems_4", "eval_func": null }, "input": { "text": "Two players, $B$ and $R$, play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with $B$. On $B$'s turn, $B$ selects one white unit square and colors it blue. On $R$'s turn, $R$ selects two white unit squares and colors them red. The players alternate until $B$ decides to end the game. At this point, $B$ gets a score, given by the number of unit squares in the largest (in terms of area) simple polygon containing only blue unit squares. What is the largest score $B$ can guarantee?\n(A simple polygon is a polygon (not necessarily convex) that does not intersect itself and has no holes.)", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "It is clear that $B$ can guarantee a score of $4$ squares. We will show that $R$ has a strategy to limit blue to $4$ squares, thus solving the problem.\nPartition the grid into 2x2 squares. Red's strategy is as follows: \n- If $B$ plays in a 2x2 square, play the two adjacent squares to $B$'s square that are not in the 2x2 square.\n- If one (or both) of these moves are blocked, instead play a square a megaparsec away from the rest of the moves. This move can only benefit you and will not change the outcome of the game.\nBy induction, it is clear that no two blue squares that are adjacent are not in the same 2x2 square. Thus, we conclude that $R$ has limited blue to a maximum score of $2^2 = 4$, and the proof is complete. $\\square$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2015_USAJMO_Problems_6", "eval_func": null }, "input": { "text": "Steve is piling $m\\geq 1$ indistinguishable stones on the squares of an $n\\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions $(i, k), (i, l), (j, k), (j, l)$ for some $1\\leq i, j, k, l\\leq n$, such that $i0$ and place a stone at $(i, j)$, then we subtract $g_i$ and $h_j$ by $1$ each, until $g_i$ and $h_i$ become $0$ $\\forall i$, which will happen when $m$ stones are placed, because $\\sum_{i=1}^n g_i$ and $\\sum_{i=1}^n h_i$ are both initially $m$ and decrease by $1$ after each stone is placed. Note that in this process $r_i+g_i$ and $c_i+h_i$ remains invariant, thus, the final piling satisfies the conditions above.\nBy the above lemmas, the number of ways to pile is simply the number of ways to choose the sequences $r_i$ and $c_i$ such that $\\sum_{i=1}^n r_i=\\sum_{i=1}^n c_i=m$ and $r_i, c_i \\geq 0 \\forall i$. By stars and bars, the number of ways is $\\binom{n+m-1}{m}^{2}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2019_USAJMO_Problems_3", "eval_func": null }, "input": { "text": "Let $ABCD$ be a cyclic quadrilateral satisfying $AD^2+BC^2=AB^2$. The diagonals of $ABCD$ intersect at $E$. Let $P$ be a point on side $\\overline{AB}$ satisfying $\\angle APD=\\angle BPC$. Show that line $PE$ bisects $\\overline{CD}$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $PE \\cap DC = M$. Also, let $N$ be the midpoint of $AB$.\nNote that only one point $P$ satisfies the given angle condition. With this in mind, construct $P'$ with the following properties:\n$AP' \\cdot AB = AD^2 \\quad \\text{and} \\quad BP' \\cdot AB = CD^2$\nClaim: $P = P'$\nProof:\nThe conditions imply the similarities $ADP \\sim ABD$ and $BCP \\sim BAC$ whence $\\measuredangle APD = \\measuredangle BDA = \\measuredangle BCA = \\measuredangle CPB$ as desired. $\\square$\nClaim: $PE$ is a symmedian in $AEB$\nProof:\nWe have \n\\[AP \\cdot AB = AD^2 \\iff AB^2 \\cdot AP = AD^2 \\cdot AB\\]\n\\[\\iff \\left( \\frac{AB}{AD} \\right)^2 = \\frac{AB}{AP}\\]\n\\[\\iff \\left( \\frac{AB}{AD} \\right)^2 - 1 = \\frac{AB}{AP} - 1\\]\n\\[\\iff \\frac{AB^2 - AD^2}{AD^2} = \\frac{BP}{AP}\\]\n\\[\\iff \\left(\\frac{BC}{AD} \\right)^2 = \\left(\\frac{BE}{AE} \\right)^2 = \\frac{BP}{AP}\\]\nas desired. $\\blacksquare$\nSince $P$ is the isogonal conjugate of $N$, $\\measuredangle PEA = \\measuredangle MEC = \\measuredangle BEN$. However $\\measuredangle MEC = \\measuredangle BEN$ implies that $M$ is the midpoint of $CD$ from similar triangles, so we are done. $", "By monotonicity, we can see that the point $P$ is unique. Therefore, if we find another point $P'$ with all the same properties as $P$, then $P = P'$\nPart 1) Let $N$ be a point on $\\overline{AB}$ such that $AN\\cdot AB = AD^2$, and $BN \\cdot AB = BC^2$. Obviously $N$ exists because adding the two equations gives $AN\\cdot AB + BN \\cdot AB = AD^2 + BC^2 = AB^2$, which is the problem statement. Notice that converse PoP gives\\[AN\\cdot AB = AD^2 \\implies \\bigtriangleup ADN \\sim \\bigtriangleup ABD\\]\\[BN\\cdot AB = AD^2 \\implies \\bigtriangleup CBN \\sim \\bigtriangleup ABC\\]Therefore, $\\angle AND = \\angle ADB = \\angle ACB = \\angle CNB$, so $N$ does indeed satisfy all the conditions $P$ does, so $N = P$. Hence, $\\bigtriangleup ADP \\sim \\bigtriangleup ABD$ and $\\bigtriangleup CBP \\sim\\bigtriangleup ABC$.\nPart 2) Define $G$ as the midpoint of $CD$. Furthermore, create a point $X$ such that $DX || EC$ and $CX || ED$. Obviously $XCED$ must be a parallelogram. Now we set up for Jacobi's. The problem already gives us that $\\angle APD = \\angle CPB$, which is good for starters. Furthermore, $\\bigtriangleup ADP \\sim \\bigtriangleup ABD$ tells us that\\[\\angle ADP = \\angle ABD = \\angle ACD = \\angle XDC\\]This gives us our second needed angle equivalence. Lastly, $\\bigtriangleup CBP \\sim\\bigtriangleup ABC$ will give\\[\\angle BCP = \\angle BAC = \\angle BDC = \\angle XCD\\]which is our last necessary angle equivalence to apply Jacobi's. Finally, applying Jacobi's tells us that $AC$, $BD$, and $XP$ are concurrent $\\implies$ $X$, $E$, $P$ collinear. Additionally, since parallelogram diagonals bisect each other, $X$, $G$, and $E$ are collinear, so finally we obtain that $P$, $E$, and $G$ are collinear, as desired." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2024_USAJMO_Problems_3", "eval_func": null }, "input": { "text": "Let $a(n)$ be the sequence defined by $a(1)=2$ and $a(n+1)=(a(n))^{n+1}-1$ for each integer $n\\geq1$. Suppose that $p>2$ is prime and $k$ is a positive integer. Prove that some term of the sequence $a(n)$ is divisible by $p^k$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Lemma $1$:\nGiven a prime $p$, a positive integer $k$, and an even $m$ such that $p^k|a(m)$, we must have that $p^{k+1}|a(m+2)$.\n\nProof of Lemma $1$:\n$a(m+1)\\equiv (a(m))^{m+1}-1\\equiv -1 \\mod p^{k+1}$\nThen, $a(m+2)\\equiv (a(m+1))^{m+2}-1\\equiv (-1)^{m+2}-1\\equiv 0 \\mod p^{k+1}$\n\nTherefore, by induction, if there exists an even integer $m$ such that $p|a(m)$, then for all integers $k$, $p^k|a(m+2k-2)$, so we are done if there exists an even $m$ such that $p|a(m)$.\n\nNow, consider the case where there is some prime $p>2$ such that there are no even integers $m$ such that $p|a(m)$.\nLemma $2$:\nIn this case, we must have that $p|a(m)$ if $m\\equiv -1 \\mod p-1$ for all integers $m$.\nProof of Lemma $2$:\nSuppose for the sake of contradiction that there exists some $m$ such that $m\\equiv -1\\mod p-1$ and $p$ does not divide $a(m)$. Then, we have $a(m+1)\\equiv (a(m))^{m+1}-1\\equiv 1-1\\equiv 0\\mod p$, by Fermat's Little Theorem. Since for all $p>2$, $p-1$ is even, then $m+1$ would be even. However this results in a contradiction.\nThen, we get that if $m\\equiv -1\\mod p-1$, then $0\\equiv a(m)\\equiv (a(m-1))^m-1\\mod p\\implies (a(m-1))^{m+1}\\equiv 1\\equiv a(m-1)\\mod p$.\nThen, by LTE, $v_p(a(m))=v_p((a(m-1))^m-1)=v_p(a(m-1)-1)+v_p(m)>v_p(m)$. Since $\\gcd(p-1,p)=1$, then $\\gcd(p-1,p^k)=1$ for all positive integers $k$, so then by Chines Remainder Theorem there exists integers $m$ such that $m\\equiv -1\\mod p-1$ and $m\\equiv 0\\mod p^k$, so we are done" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2012_USAJMO_Problems_1", "eval_func": null }, "input": { "text": "Given a triangle $ABC$, let $P$ and $Q$ be points on segments $\\overline{AB}$ and $\\overline{AC}$, respectively, such that $AP = AQ$. Let $S$ and $R$ be distinct points on segment $\\overline{BC}$ such that $S$ lies between $B$ and $R$, $\\angle{BPS} = \\angle{PRS}$, and $\\angle{CQR} = \\angle{QSR}$. Prove that $P$, $Q$, $R$, $S$ are concyclic (in other words, these four points lie on a circle).", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Since $\\angle BPS = \\angle PRS$, the circumcircle of triangle $PRS$ is tangent to $AB$ at $P$. Similarly, since $\\angle CQR = \\angle QSR$, the circumcircle of triangle $QRS$ is tangent to $AC$ at $Q$.\n[asy] import markers; unitsize(0.5 cm); pair A, B, C, O, P, Q, R, S; A = (2,12); B = (0,0); C = (14,0); P = intersectionpoint(A--B,Circle(A,8)); Q = intersectionpoint(A--C,Circle(A,8)); O = extension(P, P + rotate(90)*(A - P), Q, Q + rotate(90)*(A - Q)); S = intersectionpoint(B--C,arc(O, abs(O - P), 180, 270)); R = intersectionpoint(B--C,arc(O, abs(O - P), 270, 360)); draw(A--B--C--cycle); draw(Circle(O, abs(O - P))); draw(S--P--R); draw(S--Q--R); label(\"$A$\", A, N); label(\"$B$\", B, SW); label(\"$C$\", C, SE); label(\"$P$\", P, W); label(\"$Q$\", Q, NE); label(\"$R$\", R, SE); label(\"$S$\", S, SW); markangle(1, B, P, S, radius=4mm, marker(markinterval(stickframe(n=1,2mm),true))); markangle(1, P, R, S, radius=4mm, marker(markinterval(stickframe(n=1,2mm),true))); markangle(1, R, Q, C, radius=4mm, marker(markinterval(stickframe(n=2,2mm),true))); markangle(1, R, S, Q, radius=4mm, marker(markinterval(stickframe(n=2,2mm),true))); [/asy]\nFor the sake of contradiction, suppose that the circumcircles of triangles $PRS$ and $QRS$ are not the same circle. Since $AP = AQ$, $AP^2 = AQ^2$, so the power of A with respect to both circumcircles is the same. Thus, $A$ lies on the of both circles. However, both circles pass through $R$ and $S$, so the radical axis of both circles is $RS$. Hence, $A$ lies on $RS$, which is a contradiction.\nTherefore, the two circumcircles are the same circle. In other words, $P$, $Q$, $R$, and $S$ all lie on the same circle.", "Note that (as in the first solution) the circumcircle of triangle $PRS$ is tangent to $AB$ at $P$. Similarly, since $\\angle CQR = \\angle QSR$, the circumcircle of triangle $QRS$ is tangent to $AC$ at $Q$.\nNow, suppose these circumcircles are not the same circle. They already intersect at $R$ and $S$, so they cannot intersect anymore. Thus, AS must touch the two circumcircles at points $M$ and $N$, with $M$ on the circumcircle of triangle $PRS$. By Power of a Point, $AQ^2 = AM \\cdot AS$ and $AP^2 = AN \\cdot AS$. Hence, because $AP = AQ$, $AM = AN$, a contradiction because then, as they lie on the same line segment, M and N must be the same point! (Note line segment, not line.) Hence, the two circumcircles are the same circle." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2010_USAJMO_Problems_5", "eval_func": null }, "input": { "text": "Two permutations $a_1, a_2, \\ldots, a_{2010}$ and\n$b_1, b_2, \\ldots, b_{2010}$ of the numbers $1, 2, \\ldots, 2010$\nare said to intersect if $a_k = b_k$ for some value of $k$ in the\nrange $1 \\le k\\le 2010$. Show that there exist $1006$ permutations\nof the numbers $1, 2, \\ldots, 2010$ such that any other such\npermutation is guaranteed to intersect at least one of these $1006$\npermutations.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $n$ be a positive integer. Let $m$ be the smallest positive integer with\n$m > n - m$. Since $n > n - n = 0$, $m \\le n$. Let $N = \\{1, \\ldots, n\\}$\nbe the set of positive integers from $1$ to $n$. Let $M \\subset N$,\nbe $M = \\{1, \\ldots, m\\}$.\nLet $P_n$ be the set of of permutations of $N$.\nLet $C_m$ be the set of cyclic permutations of $M$, there are $m$\ncyclic permutations in all, and $C_m$ acts transitively on $M$, i.e.\nfor every pair of elements $a,b \\in M$, there is an element of $C_m$\nthat maps $a$ to $b$.\nLet $C'_m \\subset P_n$ be the permutations in $P_n$ that leave $N\\setminus M$\nfixed, and restricted to $M$ yield one of the permutations in $C_m$.\nThere is a natural one-to-one correspondence between $C'_m$ and $C_m$.\nWe claim that the $m$ permutations $C'_m$ intersect every permutation in\n$P_n$.\nSuppose, to the contrary, that there exists a permutation $p \\in P_n$\nthat does not intersect any permutation in $C'_m$. Since $C'_m$ acts\ntransitively on $M \\subset N$ the permutation $p$ cannot send any element of\n$M$ to any other element of $M$, therefore it must send all the\nelements of $M$ to $N\\setminus M$, but since $N\\setminus M$ has $n - m$ elements and $m > n - m$, this is impossible\nby the pigeonhole principle. Therefore such a permutation cannot\nexist, and the permutations in $C'_m$ intersect every permutation in\n$P_n$.\nFor $n = 2010$ we get $m = 1006$, which is the required special\ncase of the general result above.", "I construct the following permutations by continuously rotating the first 1006 numbers):\n(1, 2, 3 ... 1005, 1006, 1007 ... 2009, 2010)\n(2, 3, 4 ... 1006, 1, 1007 ... 2009, 2010)\n...\n...\n...\n(1006, 1, 2 ... 1005, 1007, ... 2009, 2010)\n\nI claim that these permutations above satisfy the property that any other permutation will intersect with at least one of them.\n:\nAssume for the sake of contradiction that there exists a permutation that won't intersect with these, say $(a_1, a_2, a_3 ... a_{2010})$.\n:\nIf there exists a k such that $1 \\leq k \\leq 1006$ $a_k \\leq 1006$, we will get an intersection with the permutations. \nNote that for any 2 distinct permutations in our list, the numbers in kth index must be different, since each permutation is a rotation of an previous permutation. Also, the numbers can't exceed 1006, so, each number must occur exactly once in the kth index.\n\nUsing the lemma, in order to avoid intersections, we need $1007 \\leq a_k \\leq 2010$, for all $1 \\leq k \\leq 1006$.\nBut, there are 1004 numbers such that $1007 \\leq n \\leq 2010$, but we need to have 1004 numbers to fill in 1006 spots, and thus, by pigeonhole principle(the numbers are the holes, the spots are the pigeons), there must be 2 spots that have the same number! However, in a permutation, all numbers must be distinct, so we have a contradiction.\nSo, we have shown such a set of permutations exists satisfying that any permutation is guaranteed to intersect one of them, and the proof is complete." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2014_USAJMO_Problems_5", "eval_func": null }, "input": { "text": "Let $k$ be a positive integer. Two players $A$ and $B$ play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the players alternately take turns with $A$ moving first. In his move, $A$ may choose two adjacent hexagons in the grid which are empty and place a counter in both of them. In his move, $B$ may choose any counter on the board and remove it. If at any time there are $k$ consecutive grid cells in a line all of which contain a counter, $A$ wins. Find the minimum value of $k$ for which $A$ cannot win in a finite number of moves, or prove that no such minimum value exists.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The answer is $k=6$. We prove that $A$ can win for $k=5$ (which hence proves it for $k<5$ as well) and show that $B$ can thwart $A$ for $k\\geq 6$.\nArrange the board so that a pair of opposite sides are horizontal. Create a coordinate system on the board by setting the center of some hexagon as the origin and setting the hexagons directly above and above-and-right as $(0, 1)$ and $(1, 0)$, respectively. Then, for example, the below-and-right hexagon touching the origin is $(1, -1)$. So two hexagons touch if their coordinate difference is one of these.\nNow for $k=5$, person $A$ places his counters only in $\\{0, 1, 2, 3, 4\\}\\times\\{0, 1\\}$. Note that if, at $A$'s turn, there are 4 counters in either column, then $A$ can win immediately, so let us assume that in both columns there are at least 2 missing, meaning that at most $6$ counters are on the board. We would like to find when $A$ cannot play under these circumstances. If we look at the disjoint sets\n$\\{(0, 0), (0, 1), (1, 0)\\}, \\{(1, 1), (2, 0), (2, 1)\\}, \\{(3, 0), (3, 1), (4, 0)\\}, \\{(4, 1)\\}$\nwe see that either some set has at least $2$ hexagons without counters, in which case $A$ can move, or all four sets have exactly $1$ missing counter. Similarly for the sets\n$\\{(0, 0)\\}, \\{(0, 1), (1, 0), (1, 1)\\}, \\{(2, 0), (2, 1), (3, 0)\\}, \\{(3, 1), (4, 0), (4, 1)\\}$\nSo both $(0, 0), (4, 1)$ have no token on them. This means that $(0, 1), (1, 0), (3, 1), (4, 0)$ do. Thus $(3, 0)$ and $(1, 1)$ do not. So this is the only situation in which $A$ can neither win immediately nor play in only these 10 hexagons.\nSo $A$ plays only in these 10 hexagons until either he has a win or he can't anymore. If he wins, then we're done. Otherwise, $A$ plays in hexagons $(5, 0)$ and $(5, 1)$. Then $B$ either removes $(5, 1)$ so that $A$ can win at $(2, 0)$, or $B$ removes $(5, 0)$ and $A$ plays at $(5, 0)$ and $(4, 1)$ and then at either $(3, 0)$ or $(1, 1)$ the next turn, or $B$ removes one of $\\{0, 1, 2, 3, 4\\}\\times\\{0, 1\\}$ in which case $A$ can either win immediately at $(3, 0)$ or can play in both columns, and then win the next turn.\nNow if $k\\geq 6$, then if $A$ plays on anything in the lattice generated by $(2, -1)$ and $(1, 1)$, that is $(2a+b, b-a)$ for $a, b$ integers, then $B$ removes it. Otherwise, $B$ removes any of $A$'s counters. This works because in order for $A$ to win, there must be at least 2 counters in this lattice, but $A$ can only put a counter on $1$ at any time, so there's at most 1 on the lattice at any time.\nSo $A$ wins if $k\\leq 5$, and $B$ wins if $k\\geq 6$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2018_USAJMO_Problems_2", "eval_func": null }, "input": { "text": "Let $a,b,c$ be positive real numbers such that $a+b+c=4\\sqrt[3]{abc}$. Prove that \\[2(ab+bc+ca)+4\\min(a^2,b^2,c^2)\\ge a^2+b^2+c^2.\\]", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "WLOG let $a \\leq b \\leq c$. Add $2(ab+bc+ca)$ to both sides of the inequality and factor to get: \\[4(a(a+b+c)+bc) \\geq (a+b+c)^2\\]\nBy substituting $a+b+c=4\\sqrt[3]{abc}$, we get:\n\\[\\frac{4a\\sqrt[3]{abc}+bc}{2} \\geq 2\\sqrt[3]{a^2b^2c^2}\\]\nThe last inequality is true by AM-GM. Since all these steps are reversible, the proof is complete.", "WLOG let $a \\geq b \\geq c$. Note that the equations are homogeneous, so WLOG let $c=1$.\nThus, the inequality now becomes $2ab + 2a + 2b + 4 \\geq a^2 + b^2 + 1$, which simplifies to $2(a+b) + 3 \\geq (a-b)^2$.\nNow we will use the condition. Letting $x=a+b$ and $y=a-b$, we have\n$x+1=\\sqrt[3]{16(x^2-y^2)} \\implies 16y^2=-x^3+13x^2-3x-1$.\nPlugging this into the inequality, we have $2x+3 \\geq \\frac{1}{16}(-x^3+13x^2-3x-1) \\implies x^3-13x^2+35x+49 = (x-7)^2(x+1) \\geq 0$, which is true since $x \\geq 0$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2020_USAJMO_Problems_3", "eval_func": null }, "input": { "text": "An empty $2020 \\times 2020 \\times 2020$ cube is given, and a $2020 \\times 2020$ grid of square unit cells is drawn on each of its six faces. A is a $1 \\times 1 \\times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:\n- The two $1 \\times 1$ faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are $3 \\cdot {2020}^2$ possible positions for a beam.)\n- No two beams have intersecting interiors.\n- The interiors of each of the four $1 \\times 2020$ faces of each beam touch either a face of the cube or the interior of the face of another beam.\nWhat is the smallest positive number of beams that can be placed to satisfy these conditions?", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Place the cube in the xyz-coordinate, with the positive x-axis pointing forward, the positive y-axis pointing right, and the positive z-axis pointing up. Let the position of a unit cube be $(x, y, z)$ if it is centered at $(x, y, z)$. Place the $2020 \\times 2020 \\times 2020$ cube so that the edges are parallel to the axes, and two of its corners are at $(1, 1, 1)$ and $(2020, 2020, 2020)$. Now call a beam z-oriented if its endpoints differ only in z-coordinates, and similarly call it x-oriented or y-oriented if the endpoints differ in x- or y-coordinates, respectively.\nWe claim that the answer is $\\boxed{3030}$. First we will prove 3030 suffices. Place the first beam so its endpoints lie at $(1, 1, 1)$, and $(2020, 1, 1)$. Place the 2nd beam so its endpoints lie at $(1, 1, 2)$ and $(1, 2020, 2)$. Place the 3rd beam so its endpoints lie at $(2, 2, 1)$ and $(2, 2, 2020)$. Now it is clear that the only faces among these three beams that are not touching the faces of the $2020 \\times 2020 \\times 2020$ cube of the face of another beam are the top face of beam 2, and the front and right faces of beam 3. \nNow place the 4th beam from $(1, 3, 3)$ to $(2020, 3, 3)$, 2 units up and right of beam 1. Then place beam 5 from $(3, 1, 4)$ to $(3, 2020, 4)$, 2 units forward and up of beam 2. Place beam 6 from $(4, 4, 1)$ to $(4, 4, 2020)$, 2 unites forward and right of beam 3. Note that these 3 beams will touch the top face of beam 2, and the front and right faces of beam 3. Now the only faces that are not touching another face are the top of beam 5, and the front and right of beam 6. \nWe can repeat this process 1010 times, placing the beams in groups of 3 so that beam $3n+1$ is 2 units up and right of beam $3n-2$, beam $3n+2$ is 2 units forward and up of beam $3n-1$, and beam $3n+3$ is 2 units forward and right of beam $3n$, for all $1\\leq n \\leq 1009$. After placing each set of 3 as such, the faces of the previous beams will all satisfy condition 3. And for the last set of 3 (that is, beams 3028, 3029, and 3030), their faces will satisfy condition 3 as well, since the top face of beam 3029 and the front/right faces of beam 3030 will touch the cube (again, note that all other faces will be touching another beam).\nThus, we have shown that 3030 beams suffices, so we will now show that 3030 beams is the minimum. Let there be $A$ x-oriented beams, $B$ y-oriented beams, and $C$ z-oriented beams. Now look each $2020 \\times 1 \\times 2020$ slice of the cube, parallel to the xz-plane. Note there are 2020 such slices. There are two cases. Either $B=2020^2$, or at least one of A or C is nonzero. In the first case, we clearly have at least $2020^2>3030$ beams as desired. In the second case, we must have at least one beam in each $2020 \\times 1 \\times 2020$ slice, in order to satisfy condition 3. If there is some slice without a beam, then there will be some x- or z-oriented beam with no beams touching its top or bottom face. Additionally, notice that only x- or z-oriented beams can fit in a $2020 \\times 1 \\times 2020$ slice, so $A+C\\geq 2020$. Similarly, we must have $A=2020^2$, $C=2020^2$, or both $A+B\\geq 2020$ and $B+C\\geq 2020$. In every case, the number of beams, $A+B+C$, will be at least 3030 as desired." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2021_USAJMO_Problems_4", "eval_func": null }, "input": { "text": "Carina has three pins, labeled $A, B$, and $C$, respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance $1$ away. What is the least number of moves that Carina can make in order for triangle $ABC$ to have area 2021?\n(A lattice point is a point $(x, y)$ in the coordinate plane where $x$ and $y$ are both integers, not necessarily positive.)", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The answer is $128$, achievable by $A=(10,0), B=(0,-63), C=(-54,1)$. We now show the bound.\nWe first do the following optimizations:\n-if you have a point goes both left and right, we may obviously delete both of these moves and decrease the number of moves by $2$.\n-if all of $A,B,C$ lie on one side of the plane, for example $y>0$, we shift them all down, decreasing the number of moves by $3$, until one of the points is on $y=0$ for the first time.\nNow we may assume that $A=(a,d)$, $B=(b,-e)$, $C=(-c,f)$ where $a,b,c,d,e,f \\geq 0$. Note we may still shift all $A,B,C$ down by $1$ if $d,f>0$, decreasing the number of moves by $1$, until one of $d,f$ is on $y=0$ for the first time. So we may assume one of $(a,b)$ and $(d,f)$ is $0$, by symmetry. In particular, by shoelace the answer to 2021 JMO Problem 4 is the minimum of the answers to the following problems:\n\nCase 1 (where $a=d=0$) if $wx-yz=4042$, find the minimum possible value of $w+x+y+z$.\nCase 2 (else) $wy+xy+xz=(w+x)(y+z)-wz=4042$, find the minimum possible value of $w+x+y+z$.\n\nNote that $(m+n)^2=4mn+(m-n)^2$ so if $m+n$ is fixed then $mn$ is maximized exactly when $|m-n|$ is minimized. In particular, if $m+n \\leq 127$ then $mn-op \\leq mn \\leq 63*64 = 4032 <4042$ as desired.\n" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2022_USAJMO_Problems_5", "eval_func": null }, "input": { "text": "Find all pairs of primes $(p,q)$ for which $p-q$ and $pq-q$ are both perfect squares.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We first consider the case where one of $p,q$ is even. If $p=2$, $p-q=0$ and $pq-q=2$ which doesn't satisfy the problem restraints. If $q=2$, we can set $p-2=x^2$ and $2p-2=y^2$ giving us $p=y^2-x^2=(y+x)(y-x)$. This forces $y-x=1$ so $p=2x+1\\rightarrow 2x+1=x^2+2 \\rightarrow x=1$ giving us the solution $(p,q)=(3,2)$.\n\nNow assume that $p,q$ are both odd primes. Set $p-q=x^2$ and $pq-q=y^2$ so $(pq-q)-(p-q)=y^2-x^2 \\rightarrow p(q-1)$ $=(y+x)(y-x)$. Since $y+x>y-x$, $p | (x+y)$. Note that $q-1$ is an even integer and since $y+x$ and $y-x$ have the same parity, they both must be even. Therefore, $x+y=pk$ for some positive even integer $k$. On the other hand, $p>p-q=x^2 \\rightarrow p>x$ and $p^2-p>pq-q=y^2 \\rightarrow p>y$. Therefore, $2p>x+y$ so $x+y=p$, giving us a contradiction.\n\nTherefore, the only solution to this problem is $(p,q)=(3,2)$.\n" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2015_USAJMO_Problems_4", "eval_func": null }, "input": { "text": "Find all functions $f:\\mathbb{Q}\\rightarrow\\mathbb{Q}$ such that\\[f(x)+f(t)=f(y)+f(z)\\]for all rational numbers $x 2^{25} - 1$, a contradiction. Thus, $n = 25$ is a valid counterexample." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2021_USAJMO_Problems_2", "eval_func": null }, "input": { "text": "Rectangles $BCC_1B_2,$ $CAA_1C_2,$ and $ABB_1A_2$ are erected outside an acute triangle $ABC.$ Suppose that\\[\\angle BC_1C+\\angle CA_1A+\\angle AB_1B=180^{\\circ}.\\]Prove that lines $B_1C_2,$ $C_1A_2,$ and $A_1B_2$ are concurrent.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We first claim that the three circles $(BCC_1B_2),$ $(CAA_1C_2),$ and $(ABB_1A_2)$ share a common intersection.\nLet the second intersection of $(BCC_1B_2)$ and $(ABB_1A_2)$ be $X$. Then \n\\begin{align*} \\angle AXC &= 360^\\circ - \\angle BXA - \\angle CXB \\\\ &= 360^\\circ - (180^\\circ - \\angle AB_1B + 180^\\circ - \\angle BC_1C) \\\\& = 180^\\circ - \\angle CA_1A, \\end{align*}\nwhich implies that $AA_1C_2CX$ is cyclic as desired.\nNow we show that $X$ is the intersection of $B_1C_2,$ $C_1A_2,$ and $A_1B_2.$ Note that $\\angle C_1XB = \\angle BXA_2 = 90^\\circ,$ so $A_2, X, C_1$ are collinear. Similarly, $B_1, X, C_2$ and $A_1, X, B_2$ are collinear, so the three lines concur and we are done." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2015_USAJMO_Problems_2", "eval_func": null }, "input": { "text": "Solve in integers the equation\n\\[x^2+xy+y^2 = \\left(\\frac{x+y}{3}+1\\right)^3.\\]", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We first notice that both sides must be integers, so $\\frac{x+y}{3}$ must be an integer. \nWe can therefore perform the substitution $x+y = 3t$ where $t$ is an integer. \nThen:\n\\begin{align*} (3t)^2 - xy &= (t+1)^3 \\\\ 9t^2 + x (x - 3t) &= t^3 + 3t^2 + 3t + 1 \\\\ 4x^2 - 12xt + 9t^2 &= 4t^3 - 15t^2 + 12t + 4 \\\\ (2x - 3t)^2 &= (t - 2)^2(4t + 1) \\end{align*}\n$4t+1$ is therefore the square of an odd integer and can be replaced with $(2n+1)^2 = 4n^2 + 4n +1$.\nBy substituting using $t = n^2 + n$ we get:\n\\begin{align*} (2x - 3n^2 - 3n)^2 &= [(n^2 + n - 2)(2n+1)]^2 \\\\ 2x - 3n^2 - 3n &= \\pm (2n^3 + 3n^2 -3n -2) \\\\ x = n^3 + 3n^2 - 1 &\\text{ or } x = - n^3 + 3n + 1 \\end{align*}\nUsing substitution we get the solutions: $(n^3 + 3n^2 - 1, - n^3 + 3n + 1) \\cup ( - n^3 + 3n + 1, n^3 + 3n^2 - 1)$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2022_USAJMO_Problems_4", "eval_func": null }, "input": { "text": "Let $ABCD$ be a rhombus, and let $K$ and $L$ be points such that $K$ lies inside the rhombus, $L$ lies outside the rhombus, and $KA=KB=LC=LD$. Prove that there exist points $X$ and $Y$ on lines $AC$ and $BD$ such that $KXLY$ is also a rhombus.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let's draw ($\\ell$) perpendicular bisector of $\\overline{KL}$. Let $X, Y$ be intersections of $\\ell$ with $AC$ and $BD$, respectively. $KXLY$ is a kite. Let $O$ mid-point of $\\overline{KL}$. Let $M$ mid-point of $\\overline{BD}$ (and also $M$ is mid-point of $\\overline{AC}$). $X, O, Y$ are on the line $\\ell$.\n$BK=DL$, $BX=XD$, $XK=XL$ and so $\\triangle BXK \\cong \\triangle DXL$ (side-side-side). By spiral similarity, $\\triangle BXD \\sim\\triangle KXL$. Hence, we get\n\\[\\angle XBD = \\angle XDB = \\angle XKL = \\angle XLK = b .\\]\nSimilarly, $AK =CL$, $YK = YL$, $YA=YC$ and so $\\triangle AKY \\cong \\triangle CYL$ (side-side-side). From spiral similarity, $\\triangle YKL\\sim \\triangle YAC$. Thus,\n\\[\\angle YAC = \\angle YCA = \\angle YKL = \\angle YLK = a .\\]\nIf we can show that $a=b$, then the kite $KXLY$ will be a rhombus.\nBy spiral similarities, $\\dfrac{BX}{BD} = \\dfrac{XK}{KL}$ and $\\dfrac{YA}{AC} = \\dfrac{YK}{KL}$. Then, $KL = \\dfrac{BD \\cdot XK}{BX} = \\dfrac{AC \\cdot YK}{YA}$.\n$\\dfrac{XK}{YK} = \\dfrac{AC \\cdot BX}{AY \\cdot BD}$. Then, $\\dfrac{YK}{XK} = \\dfrac{(BD/2) \\cdot AY}{(AC/2) \\cdot BX} = \\dfrac{\\sin b}{\\sin a}$. Also, in the right triangles $\\triangle KXO$ and $\\triangle KYO$, $\\dfrac{YK}{XK} = \\dfrac{OK/\\cos a}{OK/\\cos b} = \\dfrac{\\cos b}{\\cos a}$. Therefore,\n\\[\\dfrac{\\sin b}{\\sin a} = \\dfrac{\\cos b}{\\cos a}.\\]\n$\\sin a \\cos b = \\sin b \\cos a \\implies \\sin a \\cos b - \\sin b \\cos a = 0 \\implies \\sin(a-b) = 0$ and we get $a=b$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2011_USAJMO_Problems_4", "eval_func": null }, "input": { "text": "A is defined as any finite string of letters. A word is a if it reads the same backwards as forwards. Let a sequence of words $W_0$, $W_1$, $W_2$, $\\dots$ be defined as follows: $W_0 = a$, $W_1 = b$, and for $n \\ge 2$, $W_n$ is the word formed by writing $W_{n - 2}$ followed by $W_{n - 1}$. Prove that for any $n \\ge 1$, the word formed by writing $W_1$, $W_2$, $\\dots$, $W_n$ in succession is a palindrome.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $r$ be the reflection function on the set of words, namely $r(a_1\\dots a_n) = a_n \\dots a_1$ for all words $a_1 \\dots a_n$, $n\\ge 1$. Then the following property is evident (e.g. by mathematical induction):\n$r(w_1 \\dots w_k) = r(w_k) \\dots r(w_1)$, for any words $w_1, \\dots, w_k$, $k \\ge 1$.\nWe use mathematical induction to prove the statement of the problem. First, $W_1 = b$, $W_1W_2 = bab$, $W_1W_2W_3 = babbab$ are palindromes. Second, suppose $n\\ge 3$, and that the words $W_1 W_2 \\dots W_k$ ($k = 1$, $2$, $\\dots$, $n$) are all palindromes, i.e. $r(W_1W_2\\dots W_k) = W_1W_2\\dots W_k$. Now, consider the word $W_1 W_2 \\dots W_{n+1}$:\n\\[r(W_1 W_2 \\dots W_{n+1}) = r(W_{n+1}) r(W_1 W_2 \\dots W_{n-2}W_{n-1}W_n)\\]\n\\[= r(W_{n-1}W_n) W_1 W_2 \\dots W_{n-2} W_{n-1} W_n\\]\n\\[= r(W_{n-1}W_n) r(W_1 W_2 \\dots W_{n-2}) W_{n+1}\\]\n\\[= r(W_1 W_2 \\dots W_{n-2} W_{n-1}W_n) W_{n+1}\\]\n\\[= W_1W_2\\dots W_n W_{n+1}.\\]\nBy the principle of mathematical induction, the statement of the problem is proved." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2021_USAJMO_Problems_3", "eval_func": null }, "input": { "text": "An equilateral triangle $\\Delta$ of side length $L>0$ is given. Suppose that $n$ equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside $\\Delta$, such that each unit equilateral triangle has sides parallel to $\\Delta$, but with opposite orientation. (An example with $n=2$ is drawn below.)\n[asy] draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle,linewidth(0.5)); filldraw((0.45,0.55)--(0.65,0.55)--(0.55,0.55-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); filldraw((0.54,0.3)--(0.34,0.3)--(0.44,0.3-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); [/asy]\nProve that\\[n \\leq \\frac{2}{3} L^{2}.\\]", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "I will use the word \"center\" to refer to the centroid of any equilateral triangle.\nConsider the center of each triangle we place and the surrounding area that is \"denied\" placement for any centers. With a little bit of testing and sliding the equilateral triangle along each side of the center triangle, it is easy to see that this region forms a hexagon of side length $1$ where any other center being in this hexagon would mean that the two equilateral triangles at those centers would overlap. \nLet us define a region for each center as its \"personal space\", where its personal space intersecting with any other center's personal space results in the two equilateral triangles intersecting. Given the above fact, it is easy to see that this personal space region encapsulates a hexagon of sidelength $\\frac{1}{2}$ centered at each center by simply scaling the denied region by a factor of $\\frac{1}{2}$. The area of each personal space would simply be $6(\\frac{\\sqrt{3}}{4}(\\frac{1}{2})^2)=\\frac{3\\sqrt{3}}{8}$. Since each personal space cannot intersect, the maximum amount of equilateral triangles would be bounded by $\\frac{\\text{Area of Big Triangle}}{\\text{Area of individual personal space}}=\\frac{L^2\\frac{\\sqrt{3}}{4}}{\\frac{3\\sqrt{3}}{8}}=\\frac{2}{3}L^2$, which completes the problem." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2017_USAJMO_Problems_4", "eval_func": null }, "input": { "text": "Are there any triples $(a,b,c)$ of positive integers such that $(a-2)(b-2)(c-2) + 12$ is prime that properly divides the positive number $a^2 + b^2 + c^2 + abc - 2017$?", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The answer is no. Substitute $x=a-2,y=b-2,z=c-2$. This means that $x,y,z\\geq -1$. Then \\[a^2+b^2+c^2+abc-2017=(x+y+z-41)(x+y+z+49)+xyz+12.\\] It is given in the problem that this is positive. Now, suppose for the sake of contradiction that $xyz+12$ is a prime. Clearly $x,y,z\\neq 0$. Then we have \\[\\frac{(x+y+z-41)(x+y+z+49)}{xyz+12}\\] is an integer greater than or equal to $1$. This also implies that $x+y+z > 41$. Since $xyz+12$ is prime, we must have \\[xyz+12\\mid x+y+z-41\\text{ or } xyz+12\\mid x+y+z+49.\\] Additionally, $x, y, z$ must be odd, so that $xyz+12$ is odd while $x+y+z-41,x+y+z+49$ are even. So, if \\[xyz+12\\mid x+y+z-41\\text{ or }xyz+12\\mid x+y+z+49,\\] we must have \\[2(xyz+12)\\leq x+y+z-41\\text{ or }2(xyz+12)\\leq x+y+z+49.\\] Now suppose WLOG that $x=-1$ and $y,z>0$. Then we must have $yz\\leq 10$, impossible since $x+y+z>41$. Again, suppose that $x,y=-1$ and $z>0$. Then we must have \\[2(z+12)\\leq z-43\\text{ or }2(z+12)\\leq z+47,\\] and since in this case we must have $z>43$, this is also impossible.\nThen the final case is when $x,y,z$ are positive odd numbers. Note that if $xyz>x+y+z$ for positive integers $x,y,z$, then $abc>a+b+c$ for positive integers $a,b,c$ where $a>x,b>y,c>z$. Then we only need to prove the case where $x+y+z=43$, since $x+y+z$ is odd. Then one of \\[2(xyz+12)\\leq 2\\text{ and/or }2(xyz+12)\\leq 92\\] is true, implying that $xyz\\leq -11$ or $xyz\\leq 34$. But if $x+y+z=43$, then $xyz$ is minimized when $x=1,y=1,z=41$, so that $xyz\\geq 41$. This is a contradiction, so we are done." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2015_USAJMO_Problems_5", "eval_func": null }, "input": { "text": "Let $ABCD$ be a cyclic quadrilateral. Prove that there exists a point $X$ on segment $\\overline{BD}$ such that $\\angle BAC=\\angle XAD$ and $\\angle BCA=\\angle XCD$ if and only if there exists a point $Y$ on segment $\\overline{AC}$ such that $\\angle CBD=\\angle YBA$ and $\\angle CDB=\\angle YDA$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Note that lines $AC, AX$ are isogonal in $\\triangle ABD$, so an inversion centered at $A$ with power $r^2=AB\\cdot AD$ composed with a reflection about the angle bisector of $\\angle DAB$ swaps the pairs $(D,B)$ and $(C,X)$. Thus, \\[\\frac{AD}{XD}\\cdot \\frac{XD}{CD}=\\frac{AC}{BC}\\cdot \\frac{AB}{CA}\\Longrightarrow (A,C;B,D)=-1\\]so that $ACBD$ is a harmonic quadrilateral. By symmetry, if $Y$ exists, then $(B,D;A,C)=-1$. We have shown the two conditions are equivalent, whence both directions follow.", "All angles are directed. Note that lines $AC, AX$ are isogonal in $\\triangle ABD$ and $CD, CE$ are isogonal in $\\triangle CDB$. From the law of sines it follows that \n\\[\\frac{DX}{XB}\\cdot \\frac{DE}{ED}=\\left(\\frac{AD}{DB}\\right)^2=\\left(\\frac{DC}{BC}\\right)^2.\\]\nTherefore, the ratio equals $\\frac{AD\\cdot DC}{DB\\cdot BC}.$\nNow let $Y$ be a point of $AC$ such that $\\angle{ABE}=\\angle{CBY}$. We apply the above identities for $Y$ to get that $\\frac{CY}{YA}\\cdot \\frac{CE}{EA}=\\left(\\frac{CD}{DA}\\right)^2$. So $\\angle{CDY}=\\angle{EDA}$, the converse follows since all our steps are reversible.\n\nBeware that directed angles, or angles $\\bmod$ $180$, are not standard olympiad material. If you use them, provide a definition." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2011_USAJMO_Problems_2", "eval_func": null }, "input": { "text": "Let $a$, $b$, $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \\le 4$. Prove that\n\\[\\frac{ab + 1}{(a + b)^2} + \\frac{bc + 1}{(b + c)^2} + \\frac{ca + 1}{(c + a)^2} \\ge 3.\\]", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Since\n\\begin{align*} (a+b)^2 + (b+c)^2 + (c+a)^2 &= 2(a^2 + b^2 + c^2 + ab + bc + ca) \\\\ \t&= a^2 + b^2 + c^2 + (a + b + c)^2, \\end{align*}\nit is natural to consider a change of variables:\n\\begin{align*} \\alpha &= b + c \\\\ \\beta &= c + a \\\\ \\gamma &= a + b \\end{align*}\nwith the inverse mapping given by:\n\\begin{align*} a &= \\frac{\\beta + \\gamma - \\alpha}2 \\\\ b &= \\frac{\\alpha + \\gamma - \\beta}2 \\\\ c &= \\frac{\\alpha + \\beta - \\gamma}2 \\end{align*}\nWith this change of variables, the constraint becomes\n\\[\\alpha^2 + \\beta^2 + \\gamma^2 \\le 4,\\]\nwhile the left side of the inequality we need to prove is now\n\\begin{align*} & \\frac{\\gamma^2 - (\\alpha - \\beta)^2 + 4}{4\\gamma^2} + \\frac{\\alpha^2 - (\\beta - \\gamma)^2 + 4}{4\\alpha^2} + \\frac{\\beta^2 - (\\gamma - \\alpha)^2 + 4}{4\\beta^2} \\ge \\\\ & \\frac{\\gamma^2 - (\\alpha - \\beta)^2 + \\alpha^2 + \\beta^2 + \\gamma^2}{4\\gamma^2} + \\frac{\\alpha^2 - (\\beta - \\gamma)^2 + \\alpha^2 + \\beta^2 + \\gamma^2}{4\\alpha^2} + \\frac{\\beta^2 - (\\gamma - \\alpha)^2 + \\alpha^2 + \\beta^2 + \\gamma^2}{4\\beta^2} = \\\\ & \\frac{2\\gamma^2 + 2\\alpha\\beta}{4\\gamma^2} + \\frac{2\\alpha^2 + 2\\beta\\gamma}{4\\alpha^2} + \\frac{2\\beta^2 + 2\\gamma\\alpha}{4\\beta^2} = \\\\ & \\frac32 + \\frac{\\alpha\\beta}{2\\gamma^2} + \\frac{\\beta\\gamma}{2\\alpha^2} + \\frac{\\gamma\\alpha}{2\\beta^2}. \\end{align*}\nTherefore it remains to prove that\n\\[\\frac{\\alpha\\beta}{2\\gamma^2} + \\frac{\\beta\\gamma}{2\\alpha^2} + \\frac{\\gamma\\alpha}{2\\beta^2} \\ge \\frac32.\\]\nWe note that the product of the three (positive) terms is 1/8, therefore by AM-GM their mean is at least 1/2, and thus their sum is at least 3/2 and we are done.", "Rearranging the condition yields that\n\\[a^2 + b^2 + c^2 +ab+bc+ac \\le 2\\]\nNow note that\n\\[\\frac{2ab+2}{(a+b)^2} \\ge \\frac{2ab+a^2 + b^2 + c^2 +ab+bc+ac}{(a+b)^2}=\\frac{(a+b)^2 + (c+a)(c+b)}{(a+b)^2}\\]\nSumming this for all pairs of $\\{ a,b,c \\}$ gives that\n\\[\\sum_{cyc} \\frac{2ab+2}{(a+b)^2} \\ge 3+ \\sum_{cyc}\\frac{(c+a)(c+b)}{(a+b)^2} \\ge 6\\]\nBy AM-GM. Dividing by $2$ gives the desired inequality." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2023_USAJMO_Problems_1", "eval_func": null }, "input": { "text": "Find all triples of positive integers $(x,y,z)$ that satisfy the equation\n\\begin{align*} 2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023 \\end{align*}", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We claim that the only solutions are $(2,3,3)$ and its permutations.\nFactoring the above squares and canceling the terms gives you:\n$8(xyz)^2 + 2(x^2 +y^2 + z^2) = 4((xy)^2 + (yz)^2 + (zx)^2) + 2024$\nJumping on the coefficients in front of the $x^2$, $y^2$, $z^2$ terms, we factor into: \n$(2x^2 - 1)(2y^2 - 1)(2z^2 - 1) = 2023$\nRealizing that the only factors of 2023 that could be expressed as $(2x^2 - 1)$ are $1$, $7$, and $17$, we simply find that the only solutions are $(2,3,3)$ by inspection.\n-Max\n\nAlternatively, a more obvious factorization is:\n$2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023$\n$(\\sqrt{2}x+\\sqrt{2}y+\\sqrt{2}z+2\\sqrt{2}xyz)^2-(2xy+2yz+2zx+1)^2=2023$\n$(2\\sqrt{2}xyz+2xy+2yz+2zx+\\sqrt{2}x+\\sqrt{2}y+\\sqrt{2}z+1)(2\\sqrt{2}xyz-2xy-2yz-2zx+\\sqrt{2}x+\\sqrt{2}y+\\sqrt{2}z-1)=2023$\n$(\\sqrt{2}x+1)(\\sqrt{2}y+1)(\\sqrt{2}z+1)(\\sqrt{2}x-1)(\\sqrt{2}y-1)(\\sqrt{2}z-1)=2023$\n$(2x^2-1)(2y^2-1)(2z^2-1)=2023$\nProceed as above. " ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2014_USAJMO_Problems_4", "eval_func": null }, "input": { "text": "Let $b\\geq 2$ be an integer, and let $s_b(n)$ denote the sum of the digits of $n$ when it is written in base $b$. Show that there are infinitely many positive integers that cannot be represented in the form $n+s_b(n)$, where $n$ is a positive integer.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Define $S(n) = n + s_b(n)$, and call a number if it cannot equal $S(n)$ for a positive integer $n$.\nWe claim that in the interval $(b^p, b^{p+1}]$ there exists an unrepresentable number, for every positive integer $p$.\nIf $b^{p+1}$ is unrepresentable, we're done. Otherwise, time for our lemma:\nLemma: Define the function $f(p)$ to equal the number of integers x less than $b^p$ such that $S(x) \\ge b^p$. If $b^{p+1} = S(y)$ for some y, then $f(p+1) > f(p)$.\nProof: Let $F(p)$ be the set of integers x less than $b^p$ such that $S(x) \\ge b^p$. Then for every integer in $F(p)$, append the digit $(b-1)$ to the front of it to create a valid integer in $F(p+1)$. Also, notice that $(b-1) \\cdot b^p \\le y < b^{p+1}$. Removing the digit $(b-1)$ from the front of y creates a number that is not in $F(p)$. Hence, $F(p) \\rightarrow F(p+1)$, but there exists an element of $F(p+1)$ not corresponding with $F(p)$, so $f(p+1) > f(p)$.\nNote that our lemma combined with the Pigeonhole Principle essentially proves the claim. Therefore, because there are infinitely many disjoint intervals containing an unrepresentable number, there are infinitely many unrepresentable numbers.", "Let $f(n) = n + s_b(n)$. It is easy to see that $f(b^2) = b^2 + 1$, and $f(b^2 - b + 1) = b^2 - b + 1 + b - 1 + 1 = b^2 + 1$, because the digits of $b^2 - b + 1$ are $b-1$ and $1$. Additionally, if we take any number, let's say $x$, and multiply it by $b^3$, and then add it to $b^2$ and $b^2 - b + 1$, we still obtain $f(b^3x + b^2) = f(b^3x + b^2 - b + 1)$, because we are simply adding $b^3x + x$ to each value. \nThus, we conclude that there are infinitely many numbers that are counted twice in $f(n)$, and that these numbers come $x^3$ apart. Finally, it is clear that for a large $N$, the number of numbers that are counted at least twice is at least $\\left\\lfloor\\frac{N}{x^3}\\right\\rfloor$. Because $f(n) > n$ for all $n$, this means that there are at least $\\left\\lfloor\\frac{N}{x^3}\\right\\rfloor$ numbers uncounted, and because this is unbounded, the proof is complete." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2013_USAJMO_Problems_2", "eval_func": null }, "input": { "text": "Each cell of an $m\\times n$ board is filled with some nonnegative integer. Two numbers in the filling are said to be if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a if it satisfies the following two conditions:\n(i) The difference between any two adjacent numbers is either $0$ or $1$. \n(ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to $0$ .\nDetermine the number of distinct gardens in terms of $m$ and $n$ .", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We claim that any configuration of $0$'s produces a distinct garden. To verify this claim, we show that, for any cell that is nonzero, the value of that cell is its distance away from the nearest zero, where distance means the shortest chain of adjacent cells connecting two cells. Now, since we know that any cell with a nonzero value must have a cell adjacent to it that is less than its value, there is a path that goes from this cell to the $0$ that is decreasing, which means that the value of the cell must be its distance from the $0 \\rightarrow$ as the path must end. From this, we realize that, for any configuration of $0$'s, the value of each of the cells is simply its distance from the nearest $0$, and therefore one garden is produced for every configuration of $0$'s. \n\nHowever, we also note that there must be at least one $0$ in the garden, as otherwise the smallest number in the garden, which is less than or equal to all of its neighbors, is $>0$, which violates condition $(ii)$. There are $2^{mn}$ possible configurations of $0$ and not $0$ in the garden, one of which has no $0$'s, so our total amount of configurations is $\\boxed{2^{mn} -1}$" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2018_USAJMO_Problems_6", "eval_func": null }, "input": { "text": "Karl starts with $n$ cards labeled $1,2,3,\\dots,n$ lined up in a random order on his desk. He calls a pair $(a,b)$ of these cards swapped if $a>b$ and the card labeled $a$ is to the left of the card labeled $b$. For instance, in the sequence of cards $3,1,4,2$, there are three swapped pairs of cards, $(3,1)$, $(3,2)$, and $(4,2)$.\nHe picks up the card labeled 1 and inserts it back into the sequence in the opposite position: if the card labeled 1 had $i$ card to its left, then it now has $i$ cards to its right. He then picks up the card labeled $2$ and reinserts it in the same manner, and so on until he has picked up and put back each of the cards $1,2,\\dots,n$ exactly once in that order. (For example, the process starting at $3,1,4,2$ would be $3,1,4,2\\to 3,4,1,2\\to 2,3,4,1\\to 2,4,3,1\\to 2,3,4,1$.)\nShow that no matter what lineup of cards Karl started with, his final lineup has the same number of swapped pairs as the starting lineup.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Note that in this solution, the term \"inversions\" is used synonymously with \"swapped pairs.\"\nWe define a new process $P'$ where, when re-inserting card $i$, we additionally change its label from $i$ to $n+i$. For example, an example of $P'$ also starting with $3142$ is:\n\\[3142 \\longrightarrow 3452 \\longrightarrow 6345 \\longrightarrow 6475 \\longrightarrow 6785.\\]\nNote that now, each step of $P'$ preserves the number of inversions. Moreover, the final configuration of $P'$ is the same as the final configuration of $P$ with all cards incremented by $n$, and thus, of course, has the same number of inversions." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2012_USAJMO_Problems_2", "eval_func": null }, "input": { "text": "Find all integers $n \\ge 3$ such that among any $n$ positive real numbers $a_1$, $a_2$, $\\dots$, $a_n$ with\n\\[\\max(a_1, a_2, \\dots, a_n) \\le n \\cdot \\min(a_1, a_2, \\dots, a_n),\\]\nthere exist three that are the side lengths of an acute triangle.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Without loss of generality, assume that the set $\\{a\\}$ is ordered from least to greatest so that the bounding condition becomes $a_n \\le n \\cdot a_1.$ Now set $b_i \\equiv \\frac{a_i}{a_1},$ and since a triangle with sidelengths from $\\{a\\}$ will be similar to the corresponding triangle from $\\{b\\},$ we simply have to show the existence of acute triangles in $\\{b\\}.$ Note that $b_1 = 1$ and for all $i$, $b_i \\le n.$\nNow three arbitrary sidelengths $x$, $y$, and $z$, with $x \\le y \\le z,$ will form a valid triangle if and only if $x+y>z.$ Furthermore, this triangle will be acute if and only if $x^2 + y^2 > z^2.$ However, the first inequality can actually be inferred from the second, since $x+y>z \\longrightarrow x^2 + y^2 +2xy > z^2$ and $2xy$ is trivially greater than $0.$ So we just need to find all $n$ such that there is necessarily a triplet of $b$'s for which $b_i^2 + b_j^2 > b_k^2$ (where $b_i < b_j < b_k$).\nWe now make another substitution: $c_i \\equiv b_i ^2.$ So $c_1 = 1$ and for all $i$, $c_i \\le n^2.$ Now we examine the smallest possible sets $\\{c\\}$ for small $n$ for which the conditions of the problem are not met. Note that by smallest, we mean the set whose greatest element is as small as possible. If $n=3$, then the smallest possible set, call it $\\{s_3\\},$ is trivially $\\{1,1,2\\}$, since $c_1$ and $c_2$ are obviously minimized and $c_3$ follows as minimal. Using this as the base case, we see inductively that in general $\\{s_n\\}$ is the set of the first $n$ Fibonacci numbers. To show this note that if $\\{s_n\\} = \\{F_0, F_1, ... F_n\\}$, then $\\{s_{n+1}\\} = \\{F_0, F_1, ... F_n, c_{n+1}\\}.$ The smallest possible value for $c_{n+1}$ is the sum of the two greatest values of $\\{s_n\\}$ which are $F_{n-1}$ and $F_n$. But these sum to $F_{n+1}$ so $\\{s_{n+1}\\} = \\{F_0, F_1, ... F_{n+1}\\}$ and our induction is complete.\nNow since we know that the Fibonacci set is the smallest possible set which does not satisfy the conditions of the problem, then any set $\\{c\\}$ whose greatest term is less than $F_{n-1}$ must satisfy the conditions. And since $\\{c\\}$ is bounded between $1$ and $n^2$, then the conditions of the problem are met if and only if $F_{n-1} > n^2$. The first $n$ for which this restriction is satisfied is $n=13$ and the exponential behavior of the Fibonacci numbers ensure that every $n$ greater than $13$ will also satisfy this restriction. So the final solution set is $\\boxed{\\{n \\ge 13\\}}$.", "1. Define the Fibonacci numbers to be $F_1 = F_2 = 1$ and $F_k = F_{k-1} + F_{k-2}$ for $k \\ge 3$.\n2. If the chosen $n$ is such that $F_n \\le n^2$, then choose the sequence $a_n$ such that $a_k = \\sqrt{F_k}$ for $1 \\le k \\le n$. It is easy to verify that such a sequence satisfies the condition that the largest term is less than or equal to $n$ times the smallest term. Also, because for any three terms $x = \\sqrt{F_a}, y = \\sqrt{F_b}, z = \\sqrt{F_c}$ with $a n^2$ produce an acute triangle. (If, without loss of generality, $a_n$ is an increasing sequence, such that no three (in particular, consecutive) terms form an acute triangle, then $a_2^2 \\ge F_1a_1^2, a_3^2 \\ge a_2^2 + a_1^2 \\ge F_2a^2$, and by induction $a_n^2 > F_na_1^2$, a contradiction to the condition's inequality.)\n4. Note that $F_{12} = 144 = 12^2$ and $F_{13} = 233 > 169 = 13^2$. It is easily to verify through strong induction that all $n$ greater than 12 make $F_n > n^2$. Thus, $\\boxed{n \\ge 13}$ is the desired solution set." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2014_USAJMO_Problems_1", "eval_func": null }, "input": { "text": "Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \\[\\min{\\left (\\frac{10a^2-5a+1}{b^2-5b+10},\\frac{10b^2-5b+1}{c^2-5c+10},\\frac{10c^2-5c+1}{a^2-5a+10}\\right )}\\leq abc\\]", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Since $(a-1)^5\\ge 0$,\n\\[a^5-5a^4+10a^3-10a^2+5a-1\\ge 0\\]\nor\n\\[10a^2-5a+1\\le a^3(a^2-5a+10)\\]\nSince $a^2-5a+10=\\left( a-\\dfrac{5}{2}\\right)^2 +\\dfrac{15}{4}>0$,\n\\[\\frac{10a^2-5a+1}{a^2-5a+10}\\le a^3\\]\nAlso note that $10a^2-5a+1=10\\left( a-\\dfrac{1}{4}\\right)^2+\\dfrac{3}{8}> 0$,\nWe conclude\n\\[0\\le \\frac{10a^2-5a+1}{a^2-5a+10}\\le a^3\\]\nSimilarly, \n\\[0\\le \\frac{10b^2-5b+1}{b^2-5b+10}\\le b^3\\]\n\\[0\\le \\frac{10c^2-5c+1}{c^2-5c+10}\\le c^3\\]\nSo \\[\\left(\\frac{10a^2-5a+1}{a^2-5a+10}\\right)\\left(\\frac{10b^2-5b+1}{b^2-5b+10}\\right)\\left(\\frac{10c^2-5c+1}{c^2-5c+10}\\right)\\le a^3b^3c^3\\]\nor\n\\[\\left(\\frac{10a^2-5a+1}{b^2-5b+10}\\right)\\left(\\frac{10b^2-5b+1}{c^2-5c+10}\\right)\\left(\\frac{10c^2-5c+1}{a^2-5a+10}\\right) \\le(abc)^3\\]\nTherefore,\n\\[\\min\\left(\\frac{10a^2-5a+1}{b^2-5b+10},\\frac{10b^2-5b+1}{c^2-5c+10},\\frac{10c^2-5c+1}{a^2-5a+10}\\right )\\le abc.\\]" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2012_USAJMO_Problems_6", "eval_func": null }, "input": { "text": "Let $P$ be a point in the plane of triangle $ABC$, and $\\gamma$ a line passing through $P$. Let $A'$, $B'$, $C'$ be the points where the reflections of lines $PA$, $PB$, $PC$ with respect to $\\gamma$ intersect lines $BC$, $AC$, $AB$, respectively. Prove that $A'$, $B'$, $C'$ are collinear.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "By the on triangle $AB'P$,\n\\[\\frac{AB'}{\\sin \\angle APB'} = \\frac{AP}{\\sin \\angle AB'P},\\]\nso\n\\[AB' = AP \\cdot \\frac{\\sin \\angle APB'}{\\sin \\angle AB'P}.\\]\n[asy] import graph; import geometry; unitsize(0.5 cm); pair[] A, B, C; pair P, R; A[0] = (2,12); B[0] = (0,0); C[0] = (14,0); P = (4,5); R = 5*dir(70); A[1] = extension(B[0],C[0],P,reflect(P + R,P - R)*(A[0])); B[1] = extension(C[0],A[0],P,reflect(P + R,P - R)*(B[0])); C[1] = extension(A[0],B[0],P,reflect(P + R,P - R)*(C[0])); draw((P - R)--(P + R),red); draw(A[1]--B[1]--C[1]--cycle,blue); draw(A[0]--B[0]--C[0]--cycle); draw(A[0]--P); draw(B[0]--P); draw(C[0]--P); draw(P--A[1]); draw(P--B[1]); draw(P--C[1]); draw(A[1]--B[0]); draw(A[1]--B[0]); label(\"$A$\", A[0], N); label(\"$B$\", B[0], S); label(\"$C$\", C[0], SE); dot(\"$A'$\", A[1], SW); dot(\"$B'$\", B[1], NE); dot(\"$C'$\", C[1], W); dot(\"$P$\", P, SE); label(\"$\\gamma$\", P + R, N); [/asy]\nSimilarly,\n\\begin{align*} B'C &= CP \\cdot \\frac{\\sin \\angle CPB'}{\\sin \\angle CB'P}, \\\\ CA' &= CP \\cdot \\frac{\\sin \\angle CPA'}{\\sin \\angle CA'P}, \\\\ A'B &= BP \\cdot \\frac{\\sin \\angle BPA'}{\\sin \\angle BA'P}, \\\\ BC' &= BP \\cdot \\frac{\\sin \\angle BPC'}{\\sin \\angle BC'P}, \\\\ C'A &= AP \\cdot \\frac{\\sin \\angle APC'}{\\sin \\angle AC'P}. \\end{align*}\nHence,\n\\begin{align*} &\\frac{AB'}{B'C} \\cdot \\frac{CA'}{A'B} \\cdot \\frac{BC'}{C'A} \\\\ &= \\frac{\\sin \\angle APB'}{\\sin \\angle AB'P} \\cdot \\frac{\\sin \\angle CB'P}{\\sin \\angle CPB'} \\cdot \\frac{\\sin \\angle CPA'}{\\sin \\angle CA'P} \\cdot \\frac{\\sin \\angle BA'P}{\\sin \\angle BPA'} \\cdot \\frac{\\sin \\angle BPC'}{\\sin \\angle BC'P} \\cdot \\frac{\\sin \\angle AC'P}{\\sin \\angle APC'}. \\end{align*}\nSince angles $\\angle AB'P$ and $\\angle CB'P$ are supplementary or equal, depending on the position of $B'$ on $AC$,\n\\[\\sin \\angle AB'P = \\sin \\angle CB'P.\\]\nSimilarly,\n\\begin{align*} \\sin \\angle CA'P &= \\sin \\angle BA'P, \\\\ \\sin \\angle BC'P &= \\sin \\angle AC'P. \\end{align*}\nBy the reflective property, $\\angle APB'$ and $\\angle BPA'$ are supplementary or equal, so\n\\[\\sin \\angle APB' = \\sin \\angle BPA'.\\]\nSimilarly,\n\\begin{align*} \\sin \\angle CPA' &= \\sin \\angle APC', \\\\ \\sin \\angle BPC' &= \\sin \\angle CPB'. \\end{align*}\nTherefore,\n\\[\\frac{AB'}{B'C} \\cdot \\frac{CA'}{A'B} \\cdot \\frac{BC'}{C'A} = 1,\\]\nso by , $A'$, $B'$, and $C'$ are collinear.", "We will perform barycentric coordinates on the triangle $PCC'$, with $P=(1,0,0)$, $C'=(0,1,0)$, and $C=(0,0,1)$. Set $a = CC'$, $b = CP$, $c = C'P$ as usual. Since $A$, $B$, $C'$ are collinear, we will define $A = (p\u00a0: k\u00a0: q)$ and $B = (p\u00a0: \\ell\u00a0: q)$.\nClaim: Line $\\gamma$ is the angle bisector of $\\angle APA'$, $\\angle BPB'$, and $\\angle CPC'$. \nThis is proved by observing that since $A'P$ is the reflection of $AP$ across $\\gamma$, etc.\nThus $B'$ is the intersection of the isogonal of $B$ with respect to $\\angle P$\nwith the line $CA$; that is,\n\\[B' = \\left( \\frac pk \\frac{b^2}{\\ell}: \\frac{b^2}{\\ell}\u00a0: \\frac{c^2}{q} \\right).\\]\nAnalogously, $A'$ is the intersection of the isogonal of $A$ with respect to $\\angle P$\nwith the line $CB$; that is,\n\\[A' = \\left( \\frac{p}{\\ell} \\frac{b^2}{k}\u00a0: \\frac{b^2}{k}\u00a0: \\frac{c^2}{q} \\right).\\]\nThe ratio of the first to third coordinate in these two points\nis both $b^2pq\u00a0: c^2k\\ell$, so it follows $A'$, $B'$, and $C'$ are collinear." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2024_USAJMO_Problems_4", "eval_func": null }, "input": { "text": "Let $n \\ge 3$ be an integer. Rowan and Colin play a game on an $n \\times n$ grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid, and Colin is allowed to permute the columns of the grid. A grid coloring is $orderly$ if:\n1. no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and\n2. no matter how Colin permutes the column of the coloring, Rowan can then permute the rows to restore the original grid coloring;\nIn terms of $n$, how many orderly colorings are there?", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We focus on the leftmost column for simplicity. Let $m$ be the number of red squares in this column. We then have five cases:\n\n1. $m=1$\nWhen Rowan permutes the rows of the coloring, we consider only the first column, which by the above contains $m=1$ red colors, so there are ${n \\choose 1}=n$ ways to permute the first column\u2019s rows. Thus every other column will have to contain one different permutation of the first column; otherwise, there will be at least one permutation of which there is no corresponding column.\n\nFurthermore, each permutation will be different, so each row will contain one and only one red square, which also fulfills the case of if Colin permutes the coloring first. Thus there are $n\\cdot (n-1)\\cdot(n-2)\\cdot\\cdot\\cdot2\\cdot1=n!$ different colorings for this case (the same as choosing squares such as no square is in the same row or column as any other square).\n\n2. $m=n-1$\nThis is essentially the same as case 1 except for the coloring; now there is one blue square and the rest are red squares. Thus there are also $n!$ different colorings for this case.\n\n3. $m=0$\nSince we have an entirely blue column, we are unable to have a column with $1$ red square only as doing so would leave one permutation that is not covered by at least one column (that space is being taken for the blank column). We are also unable to have a completely blue column as doing so would allow for Colin to shift the columns and in doing so fail for Rowan to shift back the columns. We also cannot have a column with any other number of red squares other than $0$ as will be shown below, so there is $1$ case here in which the entire coloring is red.\n\n4. $m=n$\nThis is the same is an entire blue column, and, similar to above, we have $1$ coloring.\n\n5. $13$ (when $n=3$, there is no such positive integer $m$ that satisfies the conditions). Note that if we have any column with a different number of red squares, it is an unattainable column and is thus not optimal.\n\nLemma: Given that $m$ and $n$ are positive integers such that $13$, it is true for all $m$ and $n$ that ${n \\choose m}>n$.\nProof: Assume that $m<\\frac{n-1}{2}$.\n$\\Leftrightarrow$ $m+1{n \\choose m+1}$ for $m>\\frac{n-1}{2}$.\nNow we split our proof into two cases.\nCase 1: $n$ is even.\nThe largest integer less than $\\frac{n-1}{2}$ is $\\frac{n}{2}-1$, so we know that:\n${n \\choose \\frac{n}{2}}>{n \\choose \\frac{n}{2}-1}>\\cdot\\cdot\\cdot>{n \\choose 2}$\nby induction. On the other hand, the smallest integer greater than $\\frac{n-1}{2}$ is $\\frac{n}{2}$, so we know that:\n${n \\choose \\frac{n}{2}}>{n \\choose \\frac{n}{2}+1}>\\cdot\\cdot\\cdot>{n \\choose n-2}$\nalso by induction. Thus out of the given range for $m$ we know that ${n \\choose 2}$ and ${n \\choose n-2}$ are the minimum values, and all that is left is to prove that they are both greater than $n$. Furthermore, since ${n \\choose 2}={n \\choose n-2}$, we only have to prove that ${n \\choose 2}>n$.\nWe start with the given: $n>3$\n$\\Leftrightarrow$ $\\frac{n-1}{2}>1$\n$\\Leftrightarrow$ $\\frac{n(n-1)}{2}>n$\n$\\Leftrightarrow$ $\\frac{n!}{2!(n-2)!}>n$\n$\\Leftrightarrow$ ${n \\choose 2}>n$\nThus we have proven the inequality for all even $n$.\nCase 2: $n$ is odd.\nThe greatest integer less than $\\frac{n-1}{2}$ is $\\frac{n-3}{2}$, so we know that:\n${n \\choose \\frac{n-1}{2}}>{n \\choose \\frac{n-3}{2}}>\\cdot\\cdot\\cdot>{n \\choose 2}$\nby induction. On the other hand, the smallest integer greater than $\\frac{n-1}{2}$ is $\\frac{n+1}{2}$, so we know that:\n${n \\choose \\frac{n+1}{2}}>{n \\choose \\frac{n+3}{2}}>\\cdot\\cdot\\cdot>{n \\choose n-2}$\nalso by induction. Since ${n \\choose \\frac{n+1}{2}}={n \\choose \\frac{n-1}{2}}$, we know that once again, ${n \\choose n-2}={n \\choose 2}$ is the minimum of the given range for $m$, and the same proof applies. Thus, the inequality holds true for odd and in turn all positive integers $n>3$.\n\nAs a result, due to our lemma, there are always more permutations of the columns than the number of columns itself, so there will always exist a permutation of the column such that there are no corresponding original columns of which to match with. Thus there are no solutions for this case.\n\nIn conclusion, there are a total of $2\\cdot n!+2$ different colorings for which the above apply.\n" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2023_USAJMO_Problems_3", "eval_func": null }, "input": { "text": "Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a maximal grid-aligned configuration on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find the maximum value of $k(C)$ as a function of $n$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "To start off, we put the initial non-covered square in a corner (marked by the shaded square). Let's consider what happens when our first domino slides over the empty square. We will call such a move where we slide a domino over the uncovered square a \"step\":\n[asy] size(5cm); draw((0,0)--(0,3.2)); draw((1,0)--(1,3.2)); draw((0,0)--(1.2,0)); draw((0,1)--(1.2,1)); draw((0,2)--(1.2,2)); draw((0,3)--(1.2,3)); fill(origin--(1,0)--(1,1)--(0,1)--cycle, grey); draw((0.1,1.1)--(0.1,2.9)--(0.9,2.9)--(0.9,1.1)--cycle,dashed); draw((0.1,0.1)--(0.1,1.9)--(0.9,1.9)--(0.9,0.1)--cycle); [/asy]\nWhen the vertically-oriented domino above the shaded square moved down to cover the shaded square, it had uncovered a new square two squares above the original one. Next, we consider the second another direction the uncovered square can move to:\n\n[asy] size(5cm); draw((0,0)--(0,3.2)); draw((1,0)--(1,3.2)); draw((0,0)--(1.2,0)); draw((0,1)--(1.2,1)); draw((0,2)--(1.2,2)); draw((0,3)--(1.2,3)); fill((0,2)--(1,2)--(1,3)--(0,3)--cycle, grey); draw((0.1,2.9)--(0.1,2.1)--(1.9,2.1)--(1.9,2.9)--cycle); draw((1.1,2.9)--(1.1,2.1)--(2.9,2.1)--(2.9,2.9)--cycle,dashed); draw((0.1,0.1)--(0.1,1.9)--(0.9,1.9)--(0.9,0.1)--cycle); [/asy]\nIn this case, the horizontally-oriented domino besides the shaded square moved to cover the previous shaded square. The new uncovered square, as a result, would be either two squares to the left, or to the right of the previous shaded square.\nSo, to summarize, during each \"move\", the uncovered square can move either two steps up, down, left, or right. We can make a path for which the shaded square takes:\n[asy] size(5cm); draw((0,0)--(0,5)); draw((1,0)--(1,5)); draw((2,0)--(2,5)); draw((3,0)--(3,5)); draw((4,0)--(4,5)); draw((5,0)--(5,5)); draw((0,0)--(5,0)); draw((0,1)--(5,1)); draw((0,2)--(5,2)); draw((0,3)--(5,3)); draw((0,4)--(5,4)); draw((0,5)--(5,5)); draw((0.5,0.5)--(0.5,2.5)--(2.5,2.5)--(2.5,0.5)--(4.5,0.5)--(4.5,2.5)--(4.5,4.5)--(2.5,4.5)--(0.5,4.5)); dot((0.5,0.5)); dot((0.5,2.5)); dot((2.5,2.5)); dot((2.5,0.5)); dot((4.5,0.5)); dot((4.5,2.5)); dot((4.5,4.5)); dot((2.5,4.5)); dot((0.5,4.5)); [/asy]\nWe put the original shaded square was in the corner, so, the possible number of squares that cound be uncovered is $\\left(\\dfrac{n+1}{2}\\right)^2 = \\frac{n^2+2n+1}{4}$, all of which can be connected by a single path. Let those $\\frac{n^2+2n+1}{4}$ belong to a set $S$ of squares.\nIf the initial uncovered square was not an element of $S$, then it could either be $1$ or $\\sqrt{2}$ away from a square in $S$. If the initial square was a distance of $1$ from a element of $S$, then there would be $\\left(\\dfrac{n+1}{2}\\right)\\left(\\dfrac{n-1}{2}\\right)$ possible uncovered squares:\n[asy] draw((0,0)--(0,5)); draw((1,0)--(1,5)); draw((2,0)--(2,5)); draw((3,0)--(3,5)); draw((4,0)--(4,5)); draw((5,0)--(5,5)); draw((0,0)--(5,0)); draw((0,1)--(5,1)); draw((0,2)--(5,2)); draw((0,3)--(5,3)); draw((0,4)--(5,4)); draw((0,5)--(5,5)); dot((0.5,1.5)); dot((0.5,3.5)); dot((2.5,1.5)); dot((2.5,3.5)); dot((4.5,1.5)); dot((4.5,3.5)); [/asy]\nIf the initial square was a distance of $\\sqrt{2}$ away from a square in $S$, then there would be $\\left(\\dfrac{n-1}{2}\\right)^2$ possible squares that could be uncovered:\n[asy] draw((0,0)--(0,5)); draw((1,0)--(1,5)); draw((2,0)--(2,5)); draw((3,0)--(3,5)); draw((4,0)--(4,5)); draw((5,0)--(5,5)); draw((0,0)--(5,0)); draw((0,1)--(5,1)); draw((0,2)--(5,2)); draw((0,3)--(5,3)); draw((0,4)--(5,4)); draw((0,5)--(5,5)); dot((1.5,1.5)); dot((1.5,3.5)); dot((3.5,1.5)); dot((3.5,3.5)); [/asy]\nSo, that means the maximal number of squares that can be uncovered is $\\frac{n^2+2n+1}{4}$, thus making a maxmial of $k(C)=\\boxed{\\frac{n^2+2n+1}{4}}$ achievable configurations from $C$ ($C$'s uncovered square must belong in $S$)." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2017_USAJMO_Problems_5", "eval_func": null }, "input": { "text": "Let $O$ and $H$ be the circumcenter and the orthocenter of an acute triangle $ABC$. Points $M$ and $D$ lie on side $BC$ such that $BM = CM$ and $\\angle BAD = \\angle CAD$. Ray $MO$ intersects the circumcircle of triangle $BHC$ in point $N$. Prove that $\\angle ADO = \\angle HAN$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "It's well known that the reflection of $H$ across $\\overline{BC}$, $H'$, lies on $(ABC)$. Then $(BHC)$ is just the reflection of $(BH'C)$ across $\\overline{BC}$, which is equivalent to the reflection of $(ABC)$ across $\\overline{BC}$. Reflect points $A$ and $N$ across $\\overline{BC}$ to points $A'$ and $N'$, respectively. Then $N'$ is the midpoint of minor arc $\\overarc{BC}$, so $A, D, N'$ are collinear in that order. It suffices to show that $\\angle AA'N'=\\angle ADO$.\n $\\triangle AA'N' \\sim \\triangle ADO$. The proof easily follows.\n Note that $\\angle BAA'=\\angle CAO=90^{\\circ}-\\angle ABC$. Then we have $\\angle A'AN'=\\angle BAD-\\angle BAA'=\\angle CAD-\\angle CAO=\\angle DAO$. So, it suffices to show that \\[\\frac{AA'}{AN'}=\\frac{AD}{AO}\\rightarrow AA'\\cdot AO=AN'\\cdot AD.\\] Notice that $\\triangle ABA' \\sim \\triangle AOC$, so that \\[\\frac{AB}{AA'}=\\frac{AO}{AC}\\rightarrow AA'\\cdot AO=AB\\cdot AC.\\] Therefore, it suffices to show that \\[AB\\cdot AC=AN'\\cdot AD\\rightarrow \\frac{AB}{AN'}=\\frac{AD}{AC}.\\] But it is easy to show that $\\triangle BAN'\\sim \\triangle DAC$, implying the result. $\\blacksquare$", "[asy] size(9cm); pair A = dir(130); pair B = dir(220); pair C = dir(320); draw(unitcircle, lightblue); pair P = dir(-90); pair Q = dir(90); pair D = extension(A, P, B, C); pair O = origin; pair M = extension(B, C, O, P); pair N = 2*M-P; draw(A--B--C--cycle, lightblue); draw(A--P--Q, lightblue); draw(A--N--D--O--A, lightblue); draw(A--D--N--O--cycle, red); dot(\"$A$\", A, dir(A)); dot(\"$B$\", B, dir(B)); dot(\"$C$\", C, dir(C)); dot(\"$P$\", P, dir(P)); dot(\"$Q$\", Q, dir(Q)); dot(\"$D$\", D, dir(225)); dot(\"$O$\", O, dir(315)); dot(\"$M$\", M, dir(315)); dot(\"$N$\", N, dir(315)); [/asy]\nSuppose ray $OM$ intersects the circumcircle of $BHC$ at $N'$, and let the foot of the A-altitude of $ABC$ be $E$. Note that $\\angle BHE=90-\\angle HBE=90-90+\\angle C=\\angle C$. Likewise, $\\angle CHE=\\angle B$. So, $\\angle BHC=\\angle BHE+\\angle CHE=\\angle B+\\angle C$.\n$BHCN'$ is cyclic, so $\\angle BN'C=180-\\angle BHC=180-\\angle B-\\angle C=\\angle A$. Also, $\\angle BAC=\\angle A$. These two angles are on different circles and have the same measure, but they point to the same line $BC$! Hence, the two circles must be congruent. (This is also a well-known result)\nWe know, since $M$ is the midpoint of $BC$, that $OM$ is perpendicular to $BC$. $AH$ is also perpendicular to $BC$, so the two lines are parallel. $AN$ is a transversal, so $\\angle HAN=\\angle ANO$. We wish to prove that $\\angle ANO=\\angle ADO$, which is equivalent to $AOND$ being cyclic.\nNow, assume that ray $OM$ intersects the circumcircle of $ABC$ at a point $P$. Point $P$ must be the midpoint of $\\stackrel{\\frown}{BC}$. Also, since $AD$ is an angle bisector, it must also hit the circle at the point $P$. The two circles are congruent, which implies $MN=MP\\implies ND=DP\\implies$ NDP is isosceles. Angle ADN is an exterior angle, so $\\angle ADN=\\angle DNP+\\angle DPO=2\\angle DPO$. \nAssume WLOG that $\\angle B>\\angle C$. So, $\\angle DPO=\\angle APO=\\frac{\\angle B+\\angle C}{2}-\\angle C=\\frac{\\angle B-\\angle C}{2}$.\nIn addition, $\\angle AON=\\angle AOP=\\angle AOB+\\angle BOP=2\\angle C+\\angle A$. Combining these two equations, $\\angle AON+\\angle ADN=\\angle B-\\angle C+2\\angle C+\\angle A=\\angle A+\\angle B+\\angle C=180$. \nOpposite angles sum to $180$, so quadrilateral $AOND$ is cyclic, and the condition is proved." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2020_USAJMO_Problems_2", "eval_func": null }, "input": { "text": "Let $\\omega$ be the incircle of a fixed equilateral triangle $ABC$. Let $\\ell$ be a variable line that is tangent to $\\omega$ and meets the interior of segments $BC$ and $CA$ at points $P$ and $Q$, respectively. A point $R$ is chosen such that $PR = PA$ and $QR = QB$. Find all possible locations of the point $R$, over all choices of $\\ell$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Call a point if it is a possible location for $R$.\nLet the incircle of $\\triangle ABC$ touch $BC$ at $D$, $AC$ at $E$, and $\\ell$ at $T$. Also, let the center of the incircle be $I$. Clearly, a point is good iff it lies on the circle containing $A$ with center $P$ as well as the circle containing $B$ with center $Q$. Call these circles $\\omega_1$ and $\\omega_2$, respectively.\nNote that point $T$ can only lie on minor arc $\\overarc{DE}$ (excluding the endpoints). \n A point $X$ is good iff $XT\\perp PQ$ and $XT=AD=BE$.\n WLOG, let $X$ be on the same side of $PQ$ as $I$. Then we have that $T, I, X$ are collinear. In particular, we have $\\angle XTP=90^{\\circ}$. Then $TX=AD$, $PT=PD$, $\\angle XTP=\\angle ADP$, so that $\\triangle ADP\\cong \\triangle XTP\\rightarrow PA=PX$. Similarly, $QB=QX$, so $X$ is good. Then $X'$, the reflection of $X$ across $T$, is also good. But $\\omega_1$ and $\\omega_2$ have at most two intersections, so $X$ and $X'$ must be these intersections, and since a point is good iff it lies on both circles, we are done.\nNow, we know that $R, I, T$ are collinear. Then we have two cases:\n $R$ and $I$ lie on the same side of $PQ$. Then we have $RI+IT=AI+ID=BI+IE$, so that $RI=AI=BI$. Then we have that $R, A, B$ lie on a circle with center $I$. Note that because $T$ lies on $\\overarc{DE}$, $R$ must lie on $\\overarc{AB}$. So, one of the solutions is $\\boxed{R \\text{ in arc } AB\\text{ (excluding the endpoints)}}$.\n $R$ and $I$ lie on opposite sides of $PQ$. Then extend $ID$ out from $D$ to $X$ such that $DX=TR$, and extend $IE$ out from $E$ to $Y$ such that $EY=TR$. Then we see that $IR=IX=IY$, so that $\\boxed{R \\text{ lies on arc } XY\\text{ (excluding the endpoints)}}$.", "We claim that $R$ can lie on minor arc $AB$ of the circumcircle of triangle ABC, and it can also lie on the dilation of this arc about the center of triangle $ABC$ with a factor of $-2.$\n\nLet $D, E,$ and $F$ be the feet of the angle bisectors from points $A, B,$ and $C$ respectively. Trivially, $DEF$ is also the medial triangle, orthic triangle, and contact triangle (ABC is equilateral).\n\nLet $I$ be the incenter of ABC. Trivially, $I$ is the centroid, orthocenter, and circumcenter of ABC (ABC is equilateral). Also, $AD=BE=CF=3r$ where $r$ is the radius of circle $\\omega$ (This is trivial). $T$ is the point of tangency of $\\omega$ and segment $\\overline{PQ}$. \n\nR has to lie on the intersection of circles $\\omega1$(center P, radius PA) and $\\omega2$(center Q, radius QB), and for each choice of P, there exist two locations for R. The location that we claim to lie on the minor arc AB of the circumcircle of ABC shall be denoted M, and the other location shall be denoted N. \n\nDefine triangle XYZ to be the homothety of triangle ABC about I with a factor of -2.\n\nCritical claim: M, T, I, and N are collinear.\n\nProof:\nFirst we shall prove that T lies on MN using phantom points.\n\nLet the intersection of MN and PQ be denoted as K. We shall prove that K and T are the same point.\nLet $PT = p$ and $QT = q$. Because of the equal tangent theorem, $PD=PT=p$ and $QE=QT=q$. Hence, by the pythagorean theorem (recall $AD=BE=3r$), $PA^2 = 9r^2 + p^2$ and $QB^2 = 9r^2 + q^2$. Since PN = PA and QN = QB, then $PN^2 = 9r^2 + p^2$ and $QN^2 = 9r^2 + q^2$. \n\nPQ is the perpendicular bisector of MN because MN is the radical axis of $\\omega1$ and $\\omega2$. Hence, M is the reflection of N across K. Also, NK is the altitude of triangle PNQ, so $PK^2-QK^2 = PN^2-QN^2 = p^2 - q^2$ by using the pythagorean theorem and earlier expressions for $PN^2$ and $QN^2$. However, $PK+QK=PQ=p+q$. Now, we have a system of equations to solve for PK and QK in terms of p and q.\n\nDividing the first equation by the second (we can do this because p+q is always nonzero), we get $PK-QK=p-q$. Combining this with our PK+QK result, we get $PK = p$ and $QK = q$. However, $PT = p$ and $QT = q$, and only one point can exist on PQ for which this result holds true. As a result, K and T are the same point, otherwise it is a contradiction. Hence M, T, and N are collinear.\n\n$IT \\parallel MN$. This is because both MN and IT are perpendicular to PQ (IT is perpendicular to PQ because PQ is a tangent with point of tangency T). However, both lines share point T, as discussed earlier. Hence, IT and MN are the same line, and M, T, I, and N are collinear.\n\nIn fact, from our earlier results from the lengths of PN and PT, we can use the pythagorean theorem to get that $NT = 3r$, a result that is always true and independent of P and Q! Also, because M is the reflection of N over K (which is the same as T), $MT = 3r$ also. However, T varies based on P and Q. On the other hand, $IT = r$ and M, T, I and N are collinear. Remembering our earlier definitions of M and N, we get that $MI = 2r$ and $IN = 4r$, with M on the opposite side of N and T from I. Hence, M can be taken to N with a homothety about I with a factor of -2, and T can be taken to M with a homothety about I with a factor of -2. Since, trivially, the circumradius of ABC is 2r (ABC is equilateral), it seems like M can lie anywhere on the circumcircle of ABC.\n\nHowever, we must take into account the restrictions on P and Q. This limits T to only minor arc DE on the incircle of ABC, hence, because of our earlier homothety statement, M is restricted to minor arc AB on the circumcircle of ABC. Because of our homothety statement about N, N has to lie on minor arc XY on the circumcircle of triangle XYZ.\n\nBecause we defined both M and N to be possible locations for R, $\\fbox{R can only lie on minor arc AB of the circumcircle of triangle ABC, and also on minor arc XY of the circumcircle of triangle XYZ}$.\n\n" ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2012_USAJMO_Problems_4", "eval_func": null }, "input": { "text": "Let $\\alpha$ be an irrational number with $0 < \\alpha < 1$, and draw a circle in the plane whose circumference has length 1. Given any integer $n \\ge 3$, define a sequence of points $P_1$, $P_2$, $\\dots$, $P_n$ as follows. First select any point $P_1$ on the circle, and for $2 \\le k \\le n$ define $P_k$ as the point on the circle for which the length of arc $P_{k - 1} P_k$ is $\\alpha$, when travelling counterclockwise around the circle from $P_{k - 1}$ to $P_k$. Supose that $P_a$ and $P_b$ are the nearest adjacent points on either side of $P_n$. Prove that $a + b \\le n$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Use mathematical induction. For $n=3$ it is true because one point can't be closest to $P_3$ in both ways, and that $1+2\\le 3$. Suppose that for some $n$, the nearest adjacent points $P_a$ and $P_b$ on either side of $P_n$ satisfy $a+b \\le n$. Then consider the nearest adjacent points $P_c$ and $P_d$ on either side of $P_{n+1}$. It is by the assumption of the nearness we can see that either $(c,d)=(a+1,b+1)$ still holds, or $P_1$ jumps into the interior of the arc $P_{a+1}P_{n}P_{b+1}$, so that $c$ or $d$ equals to $1$. Let's consider the following two cases.\n(i) Suppose $a+b=n$.\nSince the length of the arc $P_nP_a$ is $\\{(a-n)\\alpha\\}$ (where $\\{x\\}$ equals to $x$ subtracted by the greatest integer not exceeding $x$) and length of the arc $P_bP_n$ is $\\{(n-b)\\alpha\\} = \\{a\\alpha\\}$, we now consider a point $P_0$ which is defined by $P_1$ traveling clockwise on the circle such that the length of arc $P_0P_1$ is $\\alpha$. We claim that $P_0$ is in the interior of the arc $P_bP_nP_a$. Algebraically, it is equivalent to either $\\{0-n\\alpha\\} < \\{a\\alpha-n\\alpha\\}$ or $\\{n\\alpha -0 \\} < \\{n\\alpha - b\\alpha\\} = \\{a\\alpha\\}$.\nSuppose the latter fails, i.e. $\\{n\\alpha\\} \\ge \\{a\\alpha\\}$. Then suppose $n\\alpha = m_1 + r_1$ and $a\\alpha = m_2 + r_2$, where $m_1$, $m_2$ are integers and $0< r_2\\le r_1 <1$ ($r_2$ is not zero because $a\\alpha$ is irrational). We now have \n\\[\\{0-n\\alpha\\} = \\{-m_1-1 + (1 -r_1)\\}=1-r_1\\] and \\[\\{a\\alpha -n\\alpha\\} = \\{m_2-m_1-1 + (1+r_2-r_1)\\} = 1+r_2-r_1>1-r_1\\]\nTherefore $P_0$ is either closer to $P_n$ than $P_a$ on the $P_a$ side, or closer to $P_n$ than $P_b$ on the $P_b$ side. In other words, $P_1$ is the closest adjacent point of $P_{n+1}$ on the $P_{a+1}$ side, or the closest adjacent point of $P_{n+1}$ on the $P_{b+1}$ side. Hence $P_c$ or $P_d$ is $P_1$, therefore $c+d \\le n+1$.\n(ii) Suppose $a+b\\le n-1$\nThen either $c+d = (a+1)+(b+1) \\le n+1$ when $c=a+1$ and $d=b+1$, or $c+d \\le 1+n$ when one of $P_c$ or $P_d$ is $P_1$.\nIn either case, $c+d\\le n+1$ is true." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2022_USAJMO_Problems_1", "eval_func": null }, "input": { "text": "For which positive integers $m$ does there exist an infinite arithmetic sequence of integers $a_1,a_2,\\cdots$ and an infinite geometric sequence of integers $g_1,g_2,\\cdots$ satisfying the following properties?\n$\\bullet$ $a_n-g_n$ is divisible by $m$ for all integers $n>1$;\n$\\bullet$ $a_2-a_1$ is not divisible by $m$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We claim that $m$ satisfies the given conditions if and only if $m$ is a perfect square.\nTo begin, we let the common difference of $\\{a_n\\}$ be $d$ and the common ratio of $\\{g_n\\}$ be $r$. Then, rewriting the conditions modulo $m$ gives:\n\\[a_2-a_1=d\\not\\equiv 0\\pmod{m}\\text{ (1)}\\]\n\\[a_n\\equiv g_n\\pmod{m}\\text{ (2)}\\]\nCondition $(1)$ holds if no consecutive terms in $a_i$ are equivalent modulo $m$, which is the same thing as never having consecutive, equal, terms, in $a_i\\pmod{m}$. By Condition $(2)$, this is also the same as never having equal, consecutive, terms in $g_i\\pmod{m}$:\n\\[(1)\\iff g_l\\not\\equiv g_{l-1}\\pmod{m}\\text{ for any integer }l>1\\]\n\\[\\iff g_{l-1}(r-1)\\not\\equiv 0\\pmod{m}.\\text{ (3)}\\]\n\nAlso, Condition $(2)$ holds if\n\\[g_{l+1}-g_l\\equiv g_l-g_{l-1}\\pmod{m}\\]\n\\[g_{l-1}(r-1)^2\\equiv0\\pmod{m}\\text{ (4)}.\\]\nRestating, $(1),(2)\\quad \\textrm{if} \\quad(3),(4)$, and the conditions $g_{l-1}(r-1)\\not\\equiv 0\\pmod{m}$ and $g_{l-1}(r-1)^2\\equiv0\\pmod{m}$ hold if and only if $m$ is a perfect square.\n[will finish that step here]\nNote: This shouldn't work since we see that m = 12 is a solution. Let the initials for both series by 1, then let the ratio be 7 and the common difference to be 6. We see multiplying by 7 mod 12 that the geometric sequence is alternating from 1 to 7 to 1 to 7 and so on, which is the same as adding 6. Therefore, this solution is wrong. My counter-conjecture is that all non square-free m (4, 8, 9, 16, 18, 25...) should all work, but I don't have a proof. However, if you edit the one above, you can see non square-free m will work. In order to construct a ratio, we could us (4) and find a square multiple of m, take the square root and add 1 to get the ratio. Let $m = at^2$ then $at + 1 \\not\\equiv 1 \\pmod{at^2}$ or $at$ is not divisble by $at^2$. If $t = 1$, this is false and this is not possible. But if it isn't, if $m$ isn't square free, then it should work." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2015_USAJMO_Problems_3", "eval_func": null }, "input": { "text": "Quadrilateral $APBQ$ is inscribed in circle $\\omega$ with $\\angle P = \\angle Q = 90^{\\circ}$ and $AP = AQ < BP$. Let $X$ be a variable point on segment $\\overline{PQ}$. Line $AX$ meets $\\omega$ again at $S$ (other than $A$). Point $T$ lies on arc $AQB$ of $\\omega$ such that $\\overline{XT}$ is perpendicular to $\\overline{AX}$. Let $M$ denote the midpoint of chord $\\overline{ST}$. As $X$ varies on segment $\\overline{PQ}$, show that $M$ moves along a circle.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "[asy] size(8cm); pair A=(1,0); pair B=(-1,0); pair P=dir(70); pair Q=dir(-70); pair O=(0,0); pair X=0.3*P + 0.7*Q; pair Y=5*X-4*A; pair S=intersectionpoints(A--Y,circle(O,1))[1]; pair Z=(A-X)*dir(-90) + X; pair T=intersectionpoint(X--Z,circle(O,1)); pair M=(S+T)/2; draw(circle(O,1)); draw(B--A--P--B--Q--A--S--T--X); draw(P--Q); dot(\"$A$\",A,dir(A)); dot(\"$B$\",B,dir(B)); dot(\"$P$\",P,dir(P)); dot(\"$Q$\",Q,dir(Q)); dot(\"$X$\", X, SE); dot(\"$S$\",S,dir(S)); dot(\"$T$\",T,dir(T)); dot(\"$M$\",M,dir(M)); dot((0,0)); [/asy]\n\nWe will use coordinate geometry.\nWithout loss of generality,\nlet the circle be the unit circle centered at the origin, \n\\[A=(1,0) P=(1-a,b), Q=(1-a,-b)\\],\nwhere $(1-a)^2+b^2=1$.\nLet angle $\\angle XAB=A$, which is an acute angle, $\\tan{A}=t$, then $X=(1-a,at)$.\nAngle $\\angle BOS=2A$, $S=(-\\cos(2A),\\sin(2A))$.\nLet $M=(u,v)$, then $T=(2u+\\cos(2A), 2v-\\sin(2A))$.\nThe condition $TX \\perp AX$ yields: $(2v-\\sin(2A)-at)/(2u+\\cos(2A)+a-1)=\\cot A.$ (E1)\nUse identities $(\\cos A)^2=1/(1+t^2)$, $\\cos(2A)=2(\\cos A)^2-1= 2/(1+t^2) -1$, $\\sin(2A)=2\\sin A\\cos A=2t^2/(1+t^2)$, we obtain $2vt-at^2=2u+a$. (E1')\nThe condition that $T$ is on the circle yields $(2u+\\cos(2A))^2+ (2v-\\sin(2A))^2=1$, namely $v\\sin(2A)-u\\cos(2A)=u^2+v^2$. (E2)\n$M$ is the mid-point on the hypotenuse of triangle $STX$, hence $MS=MX$, yielding $(u+\\cos(2A))^2+(v-\\sin(2A))^2=(u+a-1)^2+(v-at)^2$. (E3)\nExpand (E3), using (E2) to replace $2(v\\sin(2A)-u\\cos(2A))$ with $2(u^2+v^2)$, and using (E1') to replace $a(-2vt+at^2)$ with $-a(2u+a)$, and we obtain\n$u^2-u-a+v^2=0$, namely $(u-\\frac{1}{2})^2+v^2=a+\\frac{1}{4}$, which is a circle centered at $(\\frac{1}{2},0)$ with radius $r=\\sqrt{a+\\frac{1}{4}}$.", "Let the midpoint of $AO$ be $K$. We claim that $M$ moves along a circle with radius $KP$.\nWe will show that $KM^2 = KP^2$, which implies that $KM = KP$, and as $KP$ is fixed, this implies the claim.\n$KM^2 = \\frac{AM^2+OM^2}{2}-\\frac{AO^2}{4}$ by the median formula on $\\triangle AMO$.\n$KP^2 = \\frac{AP^2+OP^2}{2}-\\frac{AO^2}{4}$ by the median formula on $\\triangle APO$.\n$KM^2-KP^2 = \\frac{1}{2}(AM^2+OM^2-AP^2-OP^2)$.\nAs $OP = OT$, $OP^2-OM^2 = MT^2$ from right triangle $OMT$. $(1)$\nBy $(1)$, $KM^2-KP^2 = \\frac{1}{2}(AM^2-MT^2-AP^2)$.\nSince $M$ is the circumcenter of $\\triangle XTS$, and $MT$ is the circumradius, the expression $AM^2-MT^2$ is the power of point $A$ with respect to $(XTS)$. However, as $AX*AS$ is also the power of point $A$ with respect to $(XTS)$, this implies that $AM^2-MT^2=AX*AS$. $(2)$\nBy $(2)$, $KM^2-KP^2 = \\frac{1}{2}(AX*AS-AP^2)$\nFinally, $\\triangle APX \\sim \\triangle ASP$ by AA similarity ($\\angle XAP = \\angle SAP$ and $\\angle APX = \\angle AQP = \\angle ASP$), so $AX*AS = AP^2$. $(3)$\nBy $(3)$, $KM^2-KP^2=0$, so $KM^2=KP^2$, as desired." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2019_USAJMO_Problems_6", "eval_func": null }, "input": { "text": "Two rational numbers $\frac{m}{n}$ and $\frac{n}{m}$ are written on a blackboard, where $m$ and $n$ are relatively prime positive integers. At any point, Evan may pick two of the numbers $x$ and $y$ written on the board and write either their arithmetic mean $\frac{x+y}{2}$ or their harmonic mean $\frac{2xy}{x+y}$ on the board as well. Find all pairs $(m,n)$ such that Evan can write $1$ on the board in finitely many steps.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We claim that all odd $m, n$ work if $m+n$ is a positive power of 2.\nProof:\nWe first prove that $m+n=2^k$ works. By weighted averages we have that $\\frac{n(\\frac{m}{n})+(2^k-n)\\frac{n}{m}}{2^k}=\\frac{m+n}{2^k}=1$ can be written, so the solution set does indeed work. We will now prove these are the only solutions.\nAssume that $m+n\\ne 2^k$, so then $m+n\\equiv 0\\pmod{p}$ for some odd prime $p$. Then $m\\equiv -n\\pmod{p}$, so $\\frac{m}{n}\\equiv \\frac{n}{m}\\equiv -1\\pmod{p}$. We see that the arithmetic mean is $\\frac{-1+(-1)}{2}\\equiv -1\\pmod{p}$ and the harmonic mean is $\\frac{2(-1)(-1)}{-1+(-1)}\\equiv -1\\pmod{p}$, so if 1 can be written then $1\\equiv -1\\pmod{p}$ and $2\\equiv 0\\pmod{p}$ which is obviously impossible, and we are done." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2016_USAJMO_Problems_2", "eval_func": null }, "input": { "text": "Prove that there exists a positive integer $n < 10^6$ such that $5^n$ has six consecutive zeros in its decimal representation.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let digit $1$ of a number be the units digit, digit $2$ be the tens digit, and so on. Let the 6 consecutive zeroes be at digits $k-5$ through digit $k$. The criterion is then obviously equivalent to \n\\[5^n \\bmod 10^k < 10^{k-6}\\]\nWe will prove that $n = 20+2^{19}, k = 20$ satisfies this, thus proving the problem statement (since $n = 20+2^{19} < 10^6$). \nWe want \n\\[5^{20+2^{19}}\\pmod{10^{20}}\\]\nWe can split this into its prime factors. Obviously, it is a multiple of $5^{20}$, so we only need to consider it $\\mod2^{20}$.\n\\[5^{20}\\cdot5^{2^{19}}\\pmod{2^{20}}\\]\n\\[5^{20}\\cdot5^{\\varphi(2^{20})}\\pmod{2^{20}}\\]\n($\\varphi$ is the Euler Totient Function.) By Euler's Theorem, since $\\bigl(5, 2^{20}\\bigr)=1$, \n\\[5^{\\varphi(2^{20})}\\equiv1\\pmod{2^{20}}\\]\nso\n\\[5^{20}\\cdot5^{\\varphi (2^{20})}\\equiv5^{20}\\pmod{2^{20}}\\]\nSince $2^{20} > 10^6, 5^{20} < 10^{14} = 10^{20-6}$,\n\\[5^n \\bmod 10^k < 10^{k-6}\\]\nfor $n = 20+2^{19}$ and $k = 20$, and thus the problem statement has been proven." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2018_USAJMO_Problems_5", "eval_func": null }, "input": { "text": "Let $p$ be a prime, and let $a_1, \\dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers \\[a_1 + k, a_2 + 2k, \\dots, a_p + pk\\]produce at least $\\tfrac{1}{2} p$ distinct remainders upon division by $p$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "$\\textbf{Lemma: }$ For fixed $i\\neq j,$ where $i, j\\in\\{1, 2, ..., p\\},$ the statement $a_i + ik\\equiv a_j + jk\\text{ (mod } p\\text{)}$ holds for exactly one $k\\in {1, 2, ..., p}.$\n$\\textbf{Proof: }$ Notice that the left side minus the right side is congruent to $(a_i - a_j) + (i - j)k$ modulo $p.$ For this difference to equal $0,$ there is a unique solution for $k$ modulo $p$ given by $k\\equiv (a_j - a_i)(i - j)^{-1}\\text{ (mod } p\\text{)},$ where we have used the fact that every nonzero residue modulo $p$ has a unique multiplicative inverse. Therefore, there is exactly one $k\\in {1, 2, ..., p}$ that satisfies $a_i + ik\\equiv a_j + jk\\text{ (mod } p\\text{)}$ for any fixed $i\\neq j. \\textbf{ End Lemma}$\nSuppose that you have $p$ graphs $G_1, G_2, ..., G_p,$ and graph $G_k$ consists of the vertices $(i, k)$ for all $1\\le i\\le p.$ Within any graph $G_k,$ vertices $(i_1, k)$ and $(i_2, k)$ are connected by an edge if and only if $a_{i_1} + i_1k\\equiv a_{i_2} + i_2k\\text{ (mod } p\\text{)}.$ Notice that the number of disconnected components of any graph $G_k$ equals the number of distinct remainders when divided by $p$ given by the numbers $a_1 + k, a_2 + 2k, ..., a_p + pk.$\nThese $p$ graphs together have exactly one edge for every unordered pair of elements of $\\{1, 2, ..., p\\},$ so they have a total of exactly $\\frac{p(p-1)}{2}$ edges. Therefore, there exists at least one graph $G_k$ that has strictly fewer than $\\frac{p}{2}$ edges, meaning that it has more than $\\frac{p}{2}$ disconnected components. Therefore, the collection of numbers $\\{a_i + ik: 1\\le i\\le p\\}$ for this particular value of $k$ has at least $\\frac{p}{2}$ distinct remainders modulo $p.$ This completes the proof." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2013_USAJMO_Problems_4", "eval_func": null }, "input": { "text": "Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$, $2+2$, $2+1+1$, $1+2+1$, $1+1+2$, and $1+1+1+1$. Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "First of all, note that $f(n)$ = $\\sum_{i=0}^{k} f(n-2^{i})$ where $k$ is the largest integer such that $2^k \\le n$. We let $f(0) = 1$ for convenience. \nFrom here, we proceed by induction, with our claim being that the only $n$ such that $f(n)$ is odd are $n$ representable of the form $2^{a} - 1, a \\in \\mathbb{Z}$\nWe induct on $a$. It is trivially true for $a = 0$ and $a = 1$. From here, we show that, if the only numbers $n \\le 2^{a-1} - 1$ where $f(n)$ is odd are of the form described above, then the only numbers $n \\le 2^{a} -1$ that are odd are of that form. We first consider all numbers $b$, such that $2^{a-1} \\le b \\le 2^{a} - 2$, going from the lower bound to the upper bound (a mini induction, you might say). We know that $f(b) = \\sum_{i=0}^{a-1} f(b-2^{i})$. For a number in this summation to be odd, $b - 2^i = 2^m -1 \\rightarrow b = 2^i + 2^m - 1$. However, we know that $b > 2^{a-1}$, so $m$ must be equal to $a-1$, or else $b$ cannot be in that interval. Now, from this, we know that $i < a-1$, as $b<2^{a} - 1$. Therefore, $i$ and $m$ are distinct, and thus $f(b - 2^i)$ and $f(b- 2^{a-1})$ are odd; since there are just two odd numbers, the ending sum for any $b$ is even. Finally, considering $2^{a} - 1$, the only odd number is $f(2^{a} - 1 - 2^{a-1})$, so the ending sum is odd. $\\Box$\nThe smallest $n$ greater than $2013$ expressible as $2^d - 1, d \\in \\mathbb{N}$ is $2^{11} -1 = \\boxed{2047}$", "Of course, as with any number theory problem, use actual numbers to start, not variables! By plotting out the first few sums (do it!) and looking for patterns, we observe that $f(n)=\\sum_{\\textrm{power}=0}^{\\textrm{pow}_{\\textrm{larg}}} f(n-2^{\\textrm{power}})$, where $\\textrm{pow}_{\\textrm{larg}}$ represents the largest power of $2$ that is smaller than $n$. I will call this sum the Divine Sign, or DS.\nBut wait a minute... we are trying to determine odd/even of $f(n)$. Why not call all the evens 0 and odds 1, basically using mod 2? Sounds so simple. Draw a small table for the values: as $n$ goes up from $0$, you get: $1,1,0,1,0,0,0,1,0...$. We have to set $f(0)=1$ for this to work. Already it looks like $f(n)$ is only odd if $n=2^{\\textrm{power}}-1$.\nThe only tool here is induction. The base case is clearly established. Then let's assume we successfully made our claim up to $2^n-1$. We need to visit numbers from $2^n$ to $2^{n+1}-1$. Realize that $2^n$ has $0$ for $f$ because there will be two numbers in DS that give a $f$ of one: $2^{n-1}$ and $1$. \nBut to look at whether a value of $f(\\textrm{number})$ is 1 or 0, we need to revisit our first equation. We can answer this rather natural question: When will a number to be inducted upon, say $2^n+k$, ever have a 1 as $f(\\textrm{number})$ in the DS equation? Well- because by our assumption of the claim up to $2^n-1$, we know that the only way for that to happen is if $2^n+k-2^{\\textrm{power}}$ in the DS is equal to $2^{\\textrm{Some power}} - 1$. Clearly $1 \\leq k \\leq 2^n - 1$.\nFinally, we can simplify. Using our last equation, $2^n+k-2^{\\textrm{power}}=2^{\\textrm{Some power}}-1$, regrouping gives $2^n+k=2^{\\textrm{power}}+2^{\\textrm{Some power}}-1$. \nMost importantly, realize that $\\textrm{power}$ can be from $0$ to $n$, because of the restraints on $k$ mentioned earlier. Same with $\\textrm{Some power}$. Immediately at least one of $\\textrm{power}$ and $\\textrm{Some power}$ has to be $n$. If both were smaller, LHS is greater, contradiction. If both were greater, RHS is greater, contradiction.\nTherefore, by setting one of $\\textrm{power}$ or $\\textrm{Some power}$ to $n$, we realize $k=2^{\\textrm{A certain power}}-1$.\nThe conclusion is clear, right? Each $k$ from $1$ to $2^{n-1}-1$ yields two distinct cases: one of $\\textrm{power}$ and $\\textrm{Some power}$ is equal to $n$, while the other is LESS THAN $n$. But for $k=2^n-1$, there is ONE CASE: BOTH values have to equal $n$. Therefore, the only $k$ that has $f(2^n+k)$ as odd must only be $2^n-1$, because the other ones yield a $f$ of 1+1=0 in our mod. That proves our induction for a new power of 2, namely $n+1$, meaning that $f(\\textrm{number})$ is only odd if $\\textrm{number} = 2^{\\textrm{Power of two}} - 1$, and we are almost done...\nThus, the answer is $2^{11}-1=\\boxed{2047}$." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2010_USAJMO_Problems_3", "eval_func": null }, "input": { "text": "Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter\n$AB$. Denote by $P, Q, R, S$ the feet of the perpendiculars from $Y$ onto\nlines $AX, BX, AZ, BZ$, respectively. Prove that the acute angle\nformed by lines $PQ$ and $RS$ is half the size of $\\angle XOZ$, where\n$O$ is the midpoint of segment $AB$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Let $\\alpha = \\angle BAZ$, $\\beta = \\angle ABX$.\nSince $XY$ is a chord of the circle with diameter $AB$,\n$\\angle XAY = \\angle XBY = \\gamma$. From the chord $YZ$,\nwe conclude $\\angle YAZ = \\angle YBZ = \\delta$.\nTriangles $BQY$ and $APY$ are both right-triangles, and share the\nangle $\\gamma$, therefore they are similar, and so the ratio $PY\u00a0: YQ = AY\u00a0: YB$. Now by the angles $\\angle AXB = \\angle AYB = \\angle AZB$ are all right-angles. Also, $\\angle PYQ$,\nbeing the fourth angle in a quadrilateral with 3 right-angles is\nagain a right-angle. Therefore $\\triangle PYQ \\sim \\triangle AYB$ and\n$\\angle YQP = \\angle YBA = \\gamma + \\beta$.\nSimilarly, $RY\u00a0: YS = AY\u00a0: YB$, and so $\\angle YRS = \\angle YAB = \\alpha + \\delta$.\nNow $RY$ is perpendicular to $AZ$ so the direction $RY$ is $\\alpha$ counterclockwise from the vertical, and since $\\angle YRS = \\alpha + \\delta$ we see that $SR$ is $\\delta$ clockwise from the vertical. (Draw an actual vertical line segment if necessary.)\nSimilarly, $QY$ is perpendicular to $BX$ so the direction $QY$ is $\\beta$ clockwise from the vertical, and since $\\angle YQP$ is $\\gamma + \\beta$ we see that $QY$ is $\\gamma$ counterclockwise from the vertical.\nTherefore the lines $PQ$ and $RS$ intersect at an angle $\\chi = \\gamma + \\delta$. Now by the central angle theorem $2\\gamma = \\angle XOY$\nand $2\\delta = \\angle YOZ$, and so $2(\\gamma + \\delta) = \\angle XOZ$,\nand we are done.", "[asy] currentpicture=new picture; size(12cm); pair O, A, B, X, Y, Z, P, Q, R, SS, T; O=(0, 0); A=(-1, 0); B=(1, 0); X=(Cos(144), Sin(144)); Y=(Cos(105), Sin(105)); Z=(Cos(27), Sin(27)); P=foot(Y, A, X); Q=foot(Y, B, X); R=foot(Y, A, Z); SS=foot(Y, B, Z); T=foot(Y, A, B); dot(O); dot(A); dot(B); dot(X); dot(Y); dot(Z); dot(P); dot(Q); dot(R); dot(SS); dot(T); draw(arc(O, 1, 0, 180)); draw(circumcircle(T, A, Y), dotted); draw(circumcircle(T, B, Y), dotted); draw(A -- B); draw(Z -- O -- X -- A -- Z -- B -- X); draw(A -- Y -- B); draw(P -- T -- SS); draw(P -- Y -- Q); draw(R -- Y -- SS); draw(X -- P); draw(Z -- SS); draw(Y -- T); draw(rightanglemark(Y, T, B, 1.25)); draw(rightanglemark(Y, P, A, 1.25)); draw(rightanglemark(Y, Q, X, 1.25)); draw(rightanglemark(Y, R, Z, 1.25)); draw(rightanglemark(Y, SS, B, 1.25)); draw(rightanglemark(A, X, B, 1.25)); draw(rightanglemark(A, Y, B, 1.25)); draw(rightanglemark(A, Z, B, 1.25)); label(\"$O$\", O, S); label(\"$A$\", A, SW); label(\"$B$\", B, SE); label(\"$X$\", X, (X-B)/length(X-B)); label(\"$Y$\", Y, Y); label(\"$Z$\", Z, (Z-A)/length(Z-A)); label(\"$P$\", P, (P-T)/length(P-T)); label(\"$Q$\", Q, SW); label(\"$R$\", R, SE); label(\"$S$\", SS, (SS-T)/length(SS-T)); label(\"$T$\", T, S); [/asy]\nLet $T$ be the foot of the perpendicular from $Y$ to $\\overline{AB}$, let $O$ be the center of the semi-circle.\nSince we have a semi-circle, if we were to reflect it over $\\overline {AB}$, we would have a full circle, with $\\triangle{AXB}$ and $\\triangle{AZB}$ inscribed in it. Now, notice that $Y$ is a point on that full circle, so we can say that $T$ lies on the Simson Line $\\overline{PQ}$ from $Y$ to $\\triangle AXB$ and that it also lies on the Simson line $\\overline {RS}$ from $Y$ to $\\triangle AZB$. Thus, $T$ lies on two distinct lines in a plane, which means that $T=\\overline{PQ}\\cap\\overline{RS}$. Therefore, it suffices to show that $\\angle PTS=\\tfrac{1}{2}\\angle XOZ$. \nSince $m\\angle YTA + m\\angle YPA = 90^\\circ + 90^\\circ = 180^\\circ$ and $m \\angle YTB + m \\angle YSB = 90^\\circ + 90^\\circ = 180^\\circ$, we know that $TAPY$ and $TBSY$ are cyclic quadrilaterals. \nWe use this fact to get \\[\\angle PTS=\\angle PTY+\\angle YTS=\\angle PAY+\\angle YBS=\\angle XAY+\\angle YBZ. \\space \\space (1)\\] \\\\\nNow note that $\\angle XAY$ is the inscribed angle of minor arc $\\overset{\\huge\\frown}{PY}$, and $\\angle XOY$ is the central angle of minor arc $\\overset{\\huge\\frown}{AB}$, so $\\angle XAY = \\frac{\\overset{\\huge\\frown}{PY}}{2} = \\frac{\\angle XOY}{2}$. Similarly, $\\angle YBZ = \\frac{\\overset{\\huge\\frown}{YZ}}{2}=\\frac{\\angle YOZ}{2}$. Thus we can say \\[\\angle XAY + \\angle YBZ = \\frac{\\angle XOY}{2} + \\frac{\\angle YOZ}{2}=\\frac{\\angle XOY + \\angle YOZ}{2} = \\frac{\\angle XOZ}{2}. \\space \\space (2)\\]\n\nCombining statements $(1)$ and $(2)$, we can say that $\\angle PTS = \\frac{\\angle XOZ}{2}$, as desired." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2013_USAJMO_Problems_6", "eval_func": null }, "input": { "text": "Find all real numbers $x,y,z\\geq 1$ satisfying \\[\\min(\\sqrt{x+xyz},\\sqrt{y+xyz},\\sqrt{z+xyz})=\\sqrt{x-1}+\\sqrt{y-1}+\\sqrt{z-1}.\\]", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Without loss of generality, let $1 \\le x \\le y \\le z$. Then $\\sqrt{x + xyz} = \\sqrt{x - 1} + \\sqrt{y - 1} + \\sqrt{z - 1}$.\nSuppose x = y = z. Then $\\sqrt{x + x^3} = 3\\sqrt{x-1}$, so $x + x^3 = 9x - 9$. It is easily verified that $x^3 - 8x + 9 = 0$ has no solution in positive numbers greater than 1. Thus, $\\sqrt{x + xyz} \\ge \\sqrt{x - 1} + \\sqrt{y - 1} + \\sqrt{z - 1}$ for x = y = z. We suspect if the inequality always holds.\nLet x = 1. Then we have $\\sqrt{1 + yz} \\ge \\sqrt{y-1} + \\sqrt{z-1}$, which simplifies to \\[1 + yz \\ge y + z - 2 + 2\\sqrt{(y-1)(z-1)}\\] and hence \\[yz - y - z + 3 \\ge 2\\sqrt{(y-1)(z-1)}\\] Let us try a few examples: if y = z = 2, we have $3 > 2$; if y = z, we have $y^2 - 2y + 3 \\ge 2(y-1)$, which reduces to $y^2 - 4y + 5 \\ge 0$. The discriminant (16 - 20) is negative, so in fact the inequality is strict. Now notice that yz - y - z + 3 = (y-1)(z-1) + 2. Now we see we can let $u = \\sqrt{(y-1)(z-1)}$! Thus, \\[u^2 - 2u + 2 = (u-1)^2 + 1 > 0\\] and the claim holds for x = 1.\nIf x > 1, we see the $\\sqrt{x - 1}$ will provide a huge obstacle when squaring. But, using the identity $(x+y+z)^2 = x^2 + y^2 + z^2 + xy + yz + xz$:\n\\[x + xyz \\ge x - 1 + y - 1 + z - 1 + 2\\sqrt{(x-1)(y-1)} + 2\\sqrt{(y-1)(z-1)} + 2\\sqrt{(x-1)(y-1)}\\]\nwhich leads to\n\\[xyz \\ge y + z - 3 + 2\\sqrt{(x-1)(y-1)} + 2\\sqrt{(y-1)(z-1)} + 2\\sqrt{(x-1)(z-1)}\\]\nAgain, we experiment. If x = 2, y = 3, and z = 3, then $18 > 7 + 4\\sqrt{6}$.\nNow, we see the finish: setting $u = \\sqrt{x-1}$ gives $x = u^2 + 1$. We can solve a quadratic in u! Because this problem is a #6, the crown jewel of USAJMO problems, we do not hesitate in computing the messy computations:\n\\[u^2(yz) - u(2\\sqrt{y-1} + 2\\sqrt{z-1}) + (3 + yz - y - z - 2\\sqrt{(y-1)(z-1)}) \\ge 0\\]\nBecause the coefficient of $u^2$ is positive, all we need to do is to verify that the discriminant is nonpositive:\n\\[b^2 - 4ac = 4(y-1) + 4(z-1) - 8\\sqrt{(y-1)(z-1)} - yz(12 + 4yz - 4y - 4z - 8\\sqrt{(y-1)(z-1)})\\]\nLet us try a few examples. If y = z, then the discriminant D = $8(y-1) - 8(y-1) - yz(12 + 4y^2 - 8y - 8(y-1)) = -yz(4y^2 - 16y + 20) = -4yz(y^2 - 4y + 5) < 0$.\nWe are almost done, but we need to find the correct argument. (How frustrating!)\nSuccess! The discriminant is negative. Thus, we can replace our claim with a strict one, and there are no real solutions to the original equation in the hypothesis." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2023_USAJMO_Problems_6", "eval_func": null }, "input": { "text": "Isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\\omega$. Let $D$ be an arbitrary point inside $BC$ such that $BD\\neq DC$. Ray $AD$ intersects $\\omega$ again at $E$ (other than $A$). Point $F$ (other than $E$) is chosen on $\\omega$ such that $\\angle DFE = 90^\\circ$. Line $FE$ intersects rays $AB$ and $AC$ at points $X$ and $Y$, respectively. Prove that $\\angle XDE = \\angle EDY$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "All angle and side length names are defined as in the figures below. Figure 1 is the diagram of the problem while Figure 2 is the diagram of the Ratio Lemma. Do note that the point names defined in the Ratio Lemma are not necessarily the same defined points on Figure 1.\nFirst, we claim the Ratio Lemma:\n\\[\\frac{\\sin{\\theta_2}}{\\sin{\\theta_2\u2019}}=\\frac{PB}{PC}\\cdot\\frac{b}{c}.\\]\nWe prove this as follows:\nBy the Law of Sines of $\\triangle{ABP},$ we get:\n\\[\\frac{\\sin\\theta_2}{BP}=\\frac{\\sin\\theta_1}{c}=\\frac{\\sin B}{AP}.\\]\nBy the Law of Sines of $\\triangle{ACP},$ we get:\n\\[\\frac{\\sin\\theta_2\u2019}{PC}=\\frac{\\sin\\theta_1\u2019}{b}=\\frac{\\sin C}{AP}.\\]\nDividing each of the corresponding equations, we get:\n\\[\\frac{PC}{PB}\\cdot\\frac{\\sin\\theta_2}{\\sin\\theta_1}=\\frac{b}{c}\\cdot\\frac{\\sin\\theta_1}{\\sin\\theta_1\u2019}=\\frac{\\sin B}{\\sin C}.\\]\nNoting that $\\sin \\theta_1= \\sin(180^\\circ - \\theta_1\u2019)=\\sin \\theta_1\u2019,$ we have:\n\\[\\frac{\\sin\\theta_2}{\\sin\\theta_1}=\\frac{PB}{PC}\\cdot\\frac{b}{c}.\\]\nWe transition into our problem.\nBy Power of a Point on circle $\\omega,$ we have:\n\\[XK\\cdot XD=XF\\cdot XE\\]\n\\[YL\\cdot YD=YF\\cdot YE.\\]\nBy power of a point on circle $\\omega_1,$ we have:\n\\[XB\\cdot XA=XF\\cdot XE\\]\n\\[YC\\cdot YA=YF\\cdot YE.\\]\nUsing the transitive property, we have:\n\\[XK\\cdot XD=XB\\cdot XA\\]\n\\[YC\\cdot YA=YL\\cdot YD.\\]\nThus, by reverse Power of a Point, we have that quadrilaterals $ADKB$ and $ACLD$ are cyclic.\nNow, we move to angle chasing.\nIt is from here in which we focus on quadrilateral $AKEL.$\nLet $\\angle KDE=\\beta$ and $\\angle EDL=\\gamma.$ Now realize that all we want to show, as given by the problem statement, is $\\beta=\\gamma.$\nThus, $\\angle{KED}=90^\\circ-\\beta, \\angle{DEL}=90^\\circ-\\gamma.$\nWe also have:\n\\begin{align*}\n\\angle LAD&=180^\\circ-(\\angle ADL+\\alpha)\\\\\n&=180^\\circ-((180^\\circ-\\gamma)+\\alpha))\\\\\n&=\\gamma-\\alpha.\n\\end{align*}\nSimilarly, $\\angle{KAD}=\\beta-\\alpha.$\nCalling back to what we want to show - if we want to show that $\\beta=\\gamma,$ it suffices to show that $\\angle KAD=\\angle LAD$ and $\\angle KED=\\angle LED.$ But this is equivalent to showing $\\triangle{AKE}\\sim\\triangle{ALE}.$\nHowever, applying our Ratio Lemma to $\\triangle{AKE},$ we have:\n\\[\\frac{\\sin\\alpha}{\\sin 90^\\circ}=\\frac{AD}{DE}\\cdot\\frac{KE}{KA}.\\]\nApplying our Ratio Lemma to $\\triangle{ALE},$ we have:\n\\[\\frac{\\sin\\alpha}{\\sin 90^\\circ}=\\frac{AD}{DE}\\cdot\\frac{LE}{LA}.\\]\nFrom the transitive property, we have:\n\\[\\frac{KE}{KA}=\\frac{LE}{LA}.\\]\nNow, realize that we already have a shared side $AE$ and equivalent angles $\\angle{ALE}=90^\\circ+\\alpha=\\angle{AKE}.$ Thus, we establish a stronger bound between these two triangles - a congruence.\nThus, these two triangles are similar, and our proof is complete." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2023_USAJMO_Problems_5", "eval_func": null }, "input": { "text": "A positive integer $a$ is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer $n$ on the board with $n+a$, and on Bob's turn he must replace some even integer $n$ on the board with $n/2$. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends.\nAfter analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of $a$ and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "We claim that the game will always end if and only if $\\nu_{2}(n) < \\nu_{2} (a)$ for all $n$ that are in the list of positive integers.\n\nFirst, we will prove the if direction. Notice that if Alice adds $a$ to $n,$ since we have \\[\\nu_{2} (ka+n) = \\nu_{2} (n) < \\nu_{2} (a)\\] for all integers $k>0,$ eventually Bob will decrease $\\nu_{2} (n)$ by $1,$ and Alice will not be able to change $\\nu_{2} (n),$ and so $\\nu_{2}(n)$ will eventually become $0.$\nSimilarly, all of the other numbers on the list will have the same fate as $n,$ and so, no matter what Bob or Alice do, the game will end.\n\nNow, we complete the only if direction, i.e. if $\\nu_{2} (n) \\ge \\nu_{2} (a),$ where $n$ is one of the numbers in the list, we will prove that Alice can keep the game going forever. \n\nNotice that Bob can only decrease $\\nu_{2} (n)$ by $1$ at a time, and so if $\\nu_{2} (n) \\le \\nu_{2} (a),$ we need $\\nu_{2} (n) = \\nu_{2} (a)$ at some point. But, then, if this is true, take $n = 2^{k}(2m+1),$ and $a = 2^{k}(2\\ell+1),$ and notice that \n\\[a+n = 2^{k}(2m+1)+2^{k}(2\\ell+1) = 2^{k}(2m+2\\ell+2) = 2^{k+1}(m+\\ell+1),\\] so\n$\\nu_{2}(a+n) > \\nu_{2} (n).$\nThus, if Bob gets $\\nu_{2} (n)$ equal to $\\nu_{2} (a),$ Alice can simply add $a$ to $n,$ and then $\\nu_{2} (n) > \\nu_{2} (a)$ again. Thus, Alice can keep the game going forever, and hence we are done." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2020_USAJMO_Problems_1", "eval_func": null }, "input": { "text": "Let $n \\geq 2$ be an integer. Carl has $n$ books arranged on a bookshelf.\nEach book has a height and a width. No two books have the same height, and no two\nbooks have the same width.\nInitially, the books are arranged in increasing order of height from left to right. In a\nmove, Carl picks any two adjacent books where the left book is wider and shorter than\nthe right book, and swaps their locations. Carl does this repeatedly until no further\nmoves are possible.\nProve that regardless of how Carl makes his moves, he must stop after a finite number\nof moves, and when he does stop, the books are sorted in increasing order of width\nfrom left to right.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Without loss of generality, let the books have widths and heights of 1, 2, 3,..., and n. First we will show that Carl can only make finitely many moves.\nNote that two books can never be swapped more than once with each other. This is because after a first swap, the wider book will be to the left of the other, so they can never be swapped back. Thus there can be at most ${n\\choose 2}$ moves.\nNow we will show that the books are in increasing order of width when Carl is done making moves. Assume otherwise for the sake of contradiction. Now let $f(k)$ be the width of the book in the kth spot from the left. Since the books are not in increasing order of width, there must be i such that $f(i)>f(i+1)$, for $1\\leq i\\leq n-1$. \n The ith book must be shorter than the $(i+1)$th book. \n Assume otherwise for contradiction. Then the ith book must have started to the right of the $(i+1)$th book, since the ith book is taller. However, the ith book is now to the left of the $(i+1)$th book, so they must have been swapped at some point. But since $f(i)>f(i+1)$, the ith book is also wider. But the ith book started to the right, so since it was wider it cannot have been swapped with the $(i+1)$th book, so we arrive at a contradiction.\nFrom the claim, the ith book must be shorter than the $(i+1)$th book. But since the ith book is also wider by definition (as $f(i)>f(i+1)$), Carl can swap those two books, so he still has legal moves and we arrive at a contradiction, so we are done." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2018_USAJMO_Problems_4", "eval_func": null }, "input": { "text": "Triangle $ABC$ is inscribed in a circle of radius $2$ with $\\angle ABC \\geq 90^\\circ$, and $x$ is a real number satisfying the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$, where $a=BC,b=CA,c=AB$. Find all possible values of $x$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "Notice that\n\\[x^4 + ax^3 + bx^2 + cx + 1 = \\left(x^2 + \\frac{a}{2}x\\right)^2 + \\left(\\frac{c}{2}x + 1\\right)^2 + \\left(b - \\frac{a^2}{4} - \\frac{c^2}{4}\\right)x^2.\\]\nThus, if $b > \\frac{a^2}{4} + \\frac{c^2}{4},$ then the expression above is strictly greater than $0$ for all $x,$ meaning that $x$ cannot satisfy the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0.$ It follows that $b\\le\\frac{a^2}{4} + \\frac{c^2}{4}.$\nSince $\\angle ABC\\ge 90^{\\circ},$ we have $b^2\\ge a^2 + c^2.$ From this and the above we have $4b\\le a^2 + c^2\\le b^2,$ so $4b\\le b^2.$ This is true for positive values of $b$ if and only if $b\\ge 4.$ However, since $\\triangle ABC$ is inscribed in a circle of radius $2,$ all of its side lengths must be at most the diameter of the circle, so $b\\le 4.$ It follows that $b=4.$\nWe know that $4b\\le a^2 + c^2\\le b^2.$ Since $4b = b^2 = 16,$ we have $4b = a^2 + c^2 = b^2 = 16.$\nThe equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$ can be rewritten as $\\left(x^2 + \\frac{a}{2}x\\right)^2 + \\left(\\frac{c}{2}x + 1\\right)^2 = 0,$ since $b = \\frac{a^2}{4} + \\frac{c^2}{4}.$ This has a real solution if and only if the two separate terms have zeroes in common. The zeroes of $\\left(x^2 + \\frac{a}{2}x\\right)^2$ are $0$ and $-\\frac{a}{2},$ and the zero of $\\left(\\frac{c}{2}x + 1\\right)^2 = 0$ is $-\\frac{2}{c}.$ Clearly we cannot have $0=-\\frac{2}{c},$ so the only other possibility is $-\\frac{a}{2} = -\\frac{2}{c},$ which means that $ac = 4.$\nWe have a system of equations: $ac = 4$ and $a^2 + c^2 = 16.$ Solving this system gives $(a, c) = \\left(\\sqrt{6}+\\sqrt{2}, \\sqrt{6}-\\sqrt{2}\\right), \\left(\\sqrt{6}-\\sqrt{2}, \\sqrt{6}+\\sqrt{2}\\right).$ Each of these gives solutions for $x$ as $-\\frac{\\sqrt{6}+\\sqrt{2}}{2}$ and $-\\frac{\\sqrt{6}-\\sqrt{2}}{2},$ respectively. Now that we know that any valid value of $x$ must be one of these two, we will verify that both of these values of $x$ are valid.\nFirst, consider a right triangle $ABC,$ inscribed in a circle of radius $2,$ with side lengths $a = \\sqrt{6}+\\sqrt{2}, b = 4, c = \\sqrt{6}-\\sqrt{2}.$ This generates the polynomial equation\n\\[x^4 + \\left(\\sqrt{6}+\\sqrt{2}\\right)x^3 + 4x^2 + \\left(\\sqrt{6}-\\sqrt{2}\\right)x + 1 = \\left(\\frac{\\sqrt{6}-\\sqrt{2}}{2}x + 1\\right)^2\\left(\\left(\\frac{\\sqrt{6}+\\sqrt{2}}{2}x\\right)^2+1\\right) = 0.\\]\nThis is satisfied by $x=-\\frac{\\sqrt{6}+\\sqrt{2}}{2}.$\nSecond, consider a right triangle $ABC,$ inscribed in a circle of radius $2,$ with side lengths $a = \\sqrt{6}-\\sqrt{2}, b = 4, c = \\sqrt{6}+\\sqrt{2}.$ This generates the polynomial equation\n\\[x^4 + \\left(\\sqrt{6}-\\sqrt{2}\\right)x^3 + 4x^2 + \\left(\\sqrt{6}+\\sqrt{2}\\right)x + 1 = \\left(\\frac{\\sqrt{6}+\\sqrt{2}}{2}x + 1\\right)^2\\left(\\left(\\frac{\\sqrt{6}-\\sqrt{2}}{2}x\\right)^2+1\\right) = 0.\\]\nThis is satisfied by $x=-\\frac{\\sqrt{6}-\\sqrt{2}}{2}.$\nIt follows that the possible values of $x$ are $-\\frac{\\sqrt{6}+\\sqrt{2}}{2}$ and $-\\frac{\\sqrt{6}-\\sqrt{2}}{2}.$\nFun fact: these solutions correspond to a $15$-$75$-$90$ triangle." ] } } }, "output": null }, { "meta_data": { "dataset": "creative_math", "id": "2010_USAJMO_Problems_2", "eval_func": null }, "input": { "text": "Let $n > 1$ be an integer. Find, with proof, all sequences\n$x_1, x_2, \\ldots, x_{n-1}$ of positive integers with the following\nthree properties:\n(a). $x_1 < x_2 < \\cdots < x_{n-1}$;\n(b). $x_i + x_{n-i} = 2n$ for all $i=1,2,\\ldots,n-1$;\n(c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.", "file": null, "others": { "constraints": [], "instructions": { "system": "Criteria for evaluating the difference between two mathematical solutions include:\n1. If the methods used to arrive at the solutions are fundamentally different, such as algebraic manipulation versus geometric reasoning, they can be considered distinct;\n2. Even if the final results are the same, if the intermediate steps or processes involved in reaching those solutions vary significantly, the solutions can be considered different;\n3. If two solutions rely on different assumptions or conditions, they are likely to be distinct;\n4. A solution might generalize to a broader class of problems, while another solution might be specific to certain conditions. In such cases, they are considered distinct;\n5. If one solution is significantly simpler or more complex than the other, they can be regarded as essentially different, even if they lead to the same result.", "user": "Given the following mathematical problem:\n{question}\nAnd some typical solutions:\n{solutions}\nPlease output a novel solution distinct from the given ones for this math problem." }, "references": { "solutions": [ "The sequence is $2, 4, 6, \\ldots, 2n-2$.Let $n > 1$ be an integer. Find, with proof, all sequences\n$x_1, x_2, \\ldots, x_{n-1}$ of positive integers with the following\nthree properties:\n(a). $x_1 < x_2 < \\cdots < x_{n-1}$;\n(b). $x_i + x_{n-i} = 2n$ for all $i=1,2,\\ldots,n-1$;\n(c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.The sequence is $2, 4, 6, \\ldots, 2n-2$.\nProof 1:\nWe will prove that any sequence $x_1, \\ldots, x_{n-1}$, that satisfies the given conditions, is an arithmetic progression with $x_1$ as both the first term and the increment. Once this is proved, condition (b) implies that $x_1 + x_{n-1} = x_1 + (n-1)x_1 = nx_1 = 2n$. Therefore $x_1 = 2$, and the sequence is just the even numbers from $2$ to $2n-2$. The sequence of successive even numbers clearly satisfies all three conditions, and we are done.\nFirst a degenerate case. If $n = 2$, there is only one element $x_1$, and condition (b) gives $x_1 + x_1 = 4$ or $x_1 = 2$. Conditions (a) and (c) are vacuously true.\nOtherwise, for $n > 2$, we will prove by induction on $m$ that the difference $x_{n-m} - x_{n-1-m} = x_1$ for all $m \\in [1, n-2]$, which makes all the differences $x_{n-1} - x_{n-2} = \\ldots = x_2 - x_1 = x_1$, i.e. the sequence is an arithmetic progression with $x_1$ as the first term and increment as promised.\nSo first the $m=1$ case. With $n > 2$, $x_{n-2}$ exists and is less than $x_{n-1}$ by condition (a). Now since by condition (b) $x_1 + x_{n-1} = 2n$, we conclude that $x_1 + x_{n-2} < 2n$, and therefore by condition (c) $x_1 + x_{n-2} = x_k$ for some $k$. Now, since $x_1 > 0$, $x_k > x_{n-2}$ and can only be $x_{n-1}$. So $x_1 + x_{n-2} = x_{n-1}$.\nNow for the induction step on all values of $m$. Suppose we have shown that for all $i \\le m$, $x_1 + x_{n-1-i} = x_{n-i}$. If $m = n-2$ we are done, otherwise $m < n-2$, and by condition (c) $x_1 + x_{n-2-m} = x_k$ for some $k$. This $x_k$ is larger than $x_{n-2-m}$, but smaller than $x_1 + x_{n-1-m} = x_{n-m}$ by the inductive hypothesis. It then follows that $x_1 + x_{n-2-m} = x_{n-1-m}$, the only element of the sequence between $x_{n-2-m}$ and $x_{n-m}$. This establishes the result for $i=m+1$.\nSo, by induction $x_1 + x_{n-1-m} = x_{n-m}$ for all $m \\in [1, n-2]$, which completes the proof.\nProof 2:\nLet $S=\\{x_1,x_2,...,x_{n-1}\\}$. Notice that \\[x_1 < x_1+x_1 < x_1+x_2 < \\dots < x_1+x_{n-2} < 2n.\\]\nThen by condition (c), we must have $x_1,x_1+x_1,...,x_1+x_{n-2}\\in S$. This implies that $x_1=x_1,x_1+x_1=x_2,...,x_1+x_{n-2}=x_{n-1}$, or that $x_k=kx_1$. Then we have $x_1+x_{n-1}=n(x_1)=2n \\rightarrow x_1=2$, and the rest is trivial.", "The claim is that in this sequence, if there are $2$ elements $a,b$ where $a,b0$. Adding $x_1$ to both sides, $2x_1>x_1$. Therefore, since the $n-2$ expressions are all at least $2x_1$, they have to be equal to $x_2, x_3, \\dots, x_{n-1}$.\nSince there are $n-2$ distinct expressions $x_1+x_i$ equal to $n-2$ distinct expressions $x_j$, we have that each $x_1+x_i$ is equal to one $x_j$. Using the orders of the $x_1+x_i$ and the $x_j$, we find that\n\\begin{align*} x_1+x_1 &= x_2 \\\\ x_1+x_2 &= x_3 \\\\ ...& \\\\ x_1+x_{n-2} &= x_{n-1}. \\end{align*}\nThis gives us\n\\begin{align*} x_1 &= x_1 \\\\ x_2 &= 2x_1 \\\\ x_3 &= 3x_1 \\\\ &\\text{ }...\\\\ x_{n-1} &= (n-1)x_1\\\\ \\end{align*}\nNow, we can use property (b) with $i=1$, so $x_1 + x_{n-1} = 2n$, which means $x_1 = 2$. This gives the sequence \\[\\boxed{2, 4, 6, \\dots, 2(n-1)}\\]" ] } } }, "output": null } ]