diff --git "a/data/val.jsonl" "b/data/val.jsonl" deleted file mode 100644--- "a/data/val.jsonl" +++ /dev/null @@ -1,50 +0,0 @@ -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "$(2 \\times 3 \\times 4)\\left(\\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4}\\right) = $"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "26"}, "extra_info": {"index": 0, "solution": "1. **Calculate the product of the integers:** \n \\[\n 2 \\times 3 \\times 4 = 24\n \\]\n\n2. **Calculate the sum of the fractions:**\n \\[\n \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4}\n \\]\n To add these fractions, find a common denominator. The least common multiple of 2, 3, and 4 is 12. Thus, convert each fraction:\n \\[\n \\frac{1}{2} = \\frac{6}{12}, \\quad \\frac{1}{3} = \\frac{4}{12}, \\quad \\frac{1}{4} = \\frac{3}{12}\n \\]\n Adding these gives:\n \\[\n \\frac{6}{12} + \\frac{4}{12} + \\frac{3}{12} = \\frac{13}{12}\n \\]\n\n3. **Multiply the product of the integers by the sum of the fractions:**\n \\[\n 24 \\times \\frac{13}{12} = 24 \\times \\frac{13}{12} = 2 \\times 13 = 26\n \\]\n\n4. **Conclusion:**\n The calculation shows that the expression evaluates to 26. Therefore, the correct answer is $\\boxed{\\text{E}}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "650 students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "\\frac{5}{2}"}, "extra_info": {"index": 40, "solution": "1. **Identify the number of students who preferred each pasta type:** According to the problem, the number of students who preferred spaghetti is 250 and the number of students who preferred manicotti is 100.\n\n2. **Set up the ratio:** The ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti is given by:\n \\[\n \\frac{\\text{number of students who preferred spaghetti}}{\\text{number of students who preferred manicotti}} = \\frac{250}{100}\n \\]\n\n3. **Simplify the ratio:** To simplify the fraction $\\frac{250}{100}$, we divide both the numerator and the denominator by their greatest common divisor, which is 100:\n \\[\n \\frac{250}{100} = \\frac{250 \\div 100}{100 \\div 100} = \\frac{5}{2}\n \\]\n\n4. **Conclusion:** The ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti is $\\boxed{\\textbf{(E)}\\ \\dfrac{5}{2}}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "12\\sqrt{3}"}, "extra_info": {"index": 80, "solution": "1. **Draw the Diagram**: Consider a circle with center \\( O \\) and a radius \\( OA \\) of length 12. Let \\( CD \\) be the chord that is the perpendicular bisector of \\( OA \\). Since \\( CD \\) bisects \\( OA \\) perpendicularly, \\( OC \\) and \\( OD \\) are both equal to half of \\( OA \\), i.e., \\( OC = OD = 6 \\).\n\n2. **Identify the Triangle**: The line segment \\( OA \\) is bisected at point \\( M \\) by \\( CD \\), making \\( OM = 6 \\). Since \\( CD \\) is perpendicular to \\( OA \\), triangle \\( OMC \\) is a right triangle with \\( OM \\) as one leg and \\( MC \\) as the other leg.\n\n3. **Apply the Pythagorean Theorem**: In right triangle \\( OMC \\),\n \\[\n OC^2 = OM^2 + MC^2.\n \\]\n Plugging in the known values,\n \\[\n 6^2 = 6^2 + MC^2.\n \\]\n Simplifying,\n \\[\n 36 = 36 + MC^2 \\implies MC^2 = 0.\n \\]\n This is incorrect, as \\( MC \\) should not be zero. We need to correct the radius of the circle, which is \\( OC = 12 \\) (not 6, as \\( OC \\) is the hypotenuse, not the leg of the triangle).\n\n4. **Correct Calculation Using the Pythagorean Theorem**: Now, using \\( OC = 12 \\),\n \\[\n 12^2 = 6^2 + MC^2.\n \\]\n Simplifying,\n \\[\n 144 = 36 + MC^2 \\implies MC^2 = 108 \\implies MC = 6\\sqrt{3}.\n \\]\n\n5. **Calculate the Chord Length**: Since \\( MC \\) is half the length of chord \\( CD \\),\n \\[\n CD = 2 \\times MC = 2 \\times 6\\sqrt{3} = 12\\sqrt{3}.\n \\]\n\n6. **Conclusion**: The length of the chord \\( CD \\) is \\( \\boxed{12\\sqrt{3}} \\). This corresponds to choice \\(\\textbf{(D)}\\)."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "A dress originally priced at $80$ dollars was put on sale for $25\\%$ off. If $10\\%$ tax was added to the sale price, then the total selling price (in dollars) of the dress was"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "54"}, "extra_info": {"index": 120, "solution": "1. **Calculate the Sale Price After Discount:**\n The original price of the dress is $80$ dollars. The discount offered is $25\\%$. The amount of discount in dollars can be calculated as:\n \\[\n \\text{Discount} = 25\\% \\times 80 = 0.25 \\times 80 = 20 \\text{ dollars}\n \\]\n Therefore, the sale price after applying the discount is:\n \\[\n \\text{Sale Price} = \\text{Original Price} - \\text{Discount} = 80 - 20 = 60 \\text{ dollars}\n \\]\n\n2. **Calculate the Final Price Including Tax:**\n A tax of $10\\%$ is added to the sale price. The amount of tax in dollars is:\n \\[\n \\text{Tax} = 10\\% \\times 60 = 0.10 \\times 60 = 6 \\text{ dollars}\n \\]\n Thus, the total selling price of the dress, including tax, is:\n \\[\n \\text{Total Selling Price} = \\text{Sale Price} + \\text{Tax} = 60 + 6 = 66 \\text{ dollars}\n \\]\n\n3. **Conclusion:**\n The total selling price of the dress, after accounting for the discount and tax, is $66$ dollars.\n\n \\[\n \\boxed{\\text{D}}\n \\]"}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "A list of five positive integers has mean $12$ and range $18$. The mode and median are both $8$. How many different values are possible for the second largest element of the list?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "6"}, "extra_info": {"index": 160, "solution": "1. **Define Variables:**\n Let the integers in the list be $a_1, a_2, a_3, a_4, a_5$ such that $a_1 \\leq a_2 \\leq a_3 \\leq a_4 \\leq a_5$. Given that the mode and median are both $8$, we know $a_3 = 8$. Since the mode is $8$, at least one other number must be $8$. Without loss of generality, let $a_2 = 8$.\n\n2. **Use the Mean Information:**\n The mean of the numbers is $12$. Therefore, we have:\n \\[\n \\frac{a_1 + a_2 + a_3 + a_4 + a_5}{5} = 12 \\implies a_1 + a_2 + a_3 + a_4 + a_5 = 60\n \\]\n Substituting $a_2 = 8$ and $a_3 = 8$, we get:\n \\[\n a_1 + 8 + 8 + a_4 + a_5 = 60 \\implies a_1 + a_4 + a_5 = 44\n \\]\n\n3. **Use the Range Information:**\n The range of the list is $18$, which means $a_5 - a_1 = 18$. Let $a_1 = x$, then $a_5 = x + 18$.\n\n4. **Substitute and Solve for $a_4$:**\n Substituting $a_1 = x$ and $a_5 = x + 18$ into the sum equation, we have:\n \\[\n x + 8 + 8 + a_4 + (x + 18) = 60 \\implies 2x + a_4 + 34 = 60 \\implies 2x + a_4 = 26\n \\]\n Solving for $a_4$, we get:\n \\[\n a_4 = 26 - 2x\n \\]\n\n5. **Determine the Possible Values of $x$:**\n Since $a_1 \\leq a_2 = 8$, we have $x \\leq 8$. Also, $a_4 \\geq a_3 = 8$, so $26 - 2x \\geq 8 \\implies 2x \\leq 18 \\implies x \\leq 9$. Combining these, $x$ must satisfy $3 \\leq x \\leq 8$ (since $x$ must be an integer).\n\n6. **Calculate Possible Values for $a_4$:**\n For each integer value of $x$ from $3$ to $8$, calculate $a_4 = 26 - 2x$:\n - $x = 3 \\implies a_4 = 26 - 6 = 20$\n - $x = 4 \\implies a_4 = 26 - 8 = 18$\n - $x = 5 \\implies a_4 = 26 - 10 = 16$\n - $x = 6 \\implies a_4 = 26 - 12 = 14$\n - $x = 7 \\implies a_4 = 26 - 14 = 12$\n - $x = 8 \\implies a_4 = 26 - 16 = 10$\n\n7. **Count the Distinct Values for $a_4$:**\n The distinct values for $a_4$ are $10, 12, 14, 16, 18, 20$. There are $6$ distinct values.\n\n8. **Conclusion:**\n The number of different values possible for the second largest element of the list, which is $a_4$, is $\\boxed{6}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "A person starting with $64$ and making $6$ bets, wins three times and loses three times, the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss. If each wager is for half the money remaining at the time of the bet, then the final result is:"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "$37$"}, "extra_info": {"index": 200, "solution": "1. **Initial Amount and Bet Dynamics**: The person starts with $64$ dollars. Each bet is for half of the current amount. If the person wins a bet, the amount increases to $\\frac{3}{2}$ times the amount before the bet. If the person loses a bet, the amount decreases to $\\frac{1}{2}$ times the amount before the bet.\n\n2. **Effect of Winning and Losing**: \n - After a win, the new amount is $\\frac{3}{2} \\times \\text{previous amount}$.\n - After a loss, the new amount is $\\frac{1}{2} \\times \\text{previous amount}$.\n\n3. **Calculation of Final Amount**:\n - The person wins three times and loses three times. The multipliers for the wins are $\\frac{3}{2}$ each, and for the losses are $\\frac{1}{2}$ each.\n - The final amount, regardless of the order of wins and losses (due to the commutative property of multiplication), is calculated by multiplying all the multipliers together:\n \\[\n \\text{Final amount} = 64 \\times \\left(\\frac{3}{2}\\right)^3 \\times \\left(\\frac{1}{2}\\right)^3\n \\]\n\n4. **Simplifying the Expression**:\n - Simplify $\\left(\\frac{3}{2}\\right)^3 \\times \\left(\\frac{1}{2}\\right)^3$:\n \\[\n \\left(\\frac{3}{2}\\right)^3 = \\frac{27}{8} \\quad \\text{and} \\quad \\left(\\frac{1}{2}\\right)^3 = \\frac{1}{8}\n \\]\n - Multiply these results:\n \\[\n \\frac{27}{8} \\times \\frac{1}{8} = \\frac{27}{64}\n \\]\n - Therefore, the final amount is:\n \\[\n 64 \\times \\frac{27}{64} = 27\n \\]\n\n5. **Conclusion**: The person ends up with $27$ dollars. Starting with $64$ dollars and ending with $27$ dollars, the person incurs a loss of $64 - 27 = 37$ dollars.\n\nThus, the final result is a loss of $37$ dollars. The correct answer is $\\boxed{\\textbf{(C)}\\text{ a loss of }$ 37}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "A rectangular piece of paper whose length is $\\sqrt{3}$ times the width has area $A$. The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area $B$. What is the ratio $\\frac{B}{A}$?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "\\frac{4}{5}"}, "extra_info": {"index": 240, "solution": "1. **Assumptions and Setup**: Assume the width of the paper is $w$ and the length is $\\sqrt{3}w$. The area of the paper, $A$, is given by:\n \\[\n A = w \\times \\sqrt{3}w = \\sqrt{3}w^2.\n \\]\n Without loss of generality, let $w = 1$. Then the area $A = \\sqrt{3}$.\n\n2. **Dividing the Paper**: The paper is divided into three equal sections along its length, $\\sqrt{3}$. Each section has a length of $\\frac{\\sqrt{3}}{3}$.\n\n3. **Drawing the Dotted Line**: A dotted line is drawn from the first divider at $\\frac{\\sqrt{3}}{3}$ on one side to the second divider at $\\frac{2\\sqrt{3}}{3}$ on the opposite side.\n\n4. **Folding the Paper**: The paper is folded along the dotted line. The fold creates a triangular double-layered portion.\n\n5. **Geometry of the Folded Triangle**: The endpoints of the dotted line are $(\\frac{\\sqrt{3}}{3}, 0)$ and $(\\frac{2\\sqrt{3}}{3}, 1)$. The length of the dotted line, which is the base of the triangle, can be calculated using the distance formula:\n \\[\n \\text{Base length} = \\sqrt{\\left(\\frac{2\\sqrt{3}}{3} - \\frac{\\sqrt{3}}{3}\\right)^2 + (1 - 0)^2} = \\sqrt{\\left(\\frac{\\sqrt{3}}{3}\\right)^2 + 1} = \\frac{2\\sqrt{3}}{3}.\n \\]\n\n6. **Height of the Triangle**: The height of the triangle is the perpendicular distance from the midpoint of the dotted line to the opposite side (width of the paper). The midpoint of the dotted line is at $\\left(\\frac{\\sqrt{3}}{2}, \\frac{1}{2}\\right)$. The height is simply the y-coordinate of the midpoint, which is $\\frac{1}{2}$.\n\n7. **Area of the Triangle**: The area of the triangle is given by:\n \\[\n \\text{Area of triangle} = \\frac{1}{2} \\times \\text{base} \\times \\text{height} = \\frac{1}{2} \\times \\frac{2\\sqrt{3}}{3} \\times \\frac{1}{2} = \\frac{\\sqrt{3}}{6}.\n \\]\n\n8. **Area of the Folded Paper**: The folded paper excludes the area of the triangle, so:\n \\[\n B = A - \\text{Area of triangle} = \\sqrt{3} - \\frac{\\sqrt{3}}{6} = \\frac{5\\sqrt{3}}{6}.\n \\]\n\n9. **Ratio of Areas**: The ratio of the area of the folded paper to the original paper is:\n \\[\n \\frac{B}{A} = \\frac{\\frac{5\\sqrt{3}}{6}}{\\sqrt{3}} = \\frac{5}{6}.\n \\]\n\n10. **Conclusion**: The ratio $\\frac{B}{A}$ is $\\boxed{\\textbf{(E)} \\: \\frac{5}{6}}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "\\frac {16!}{2^8}"}, "extra_info": {"index": 280, "solution": "\nTo solve this problem, we need to determine the number of ways the spider can put on its socks and shoes such that each sock is put on before its corresponding shoe on each leg.\n\n#### Step-by-step Analysis:\n\n1. **Total Items to Wear:** The spider has 8 legs, and for each leg, it has one sock and one shoe, making a total of $16$ items to wear.\n\n2. **Order Constraints:** For each leg, the sock must be put on before the shoe. This constraint applies individually to each of the 8 legs.\n\n3. **Total Permutations Without Constraints:** If there were no constraints, the spider could put on these 16 items in any order, which would give us $16!$ (16 factorial) possible permutations.\n\n4. **Applying the Constraints:** For each leg, the sock must come before the shoe. The probability that for any given leg, the sock is put on before the shoe in a random arrangement of all 16 items is $\\frac{1}{2}$. This is because, for each pair (sock and shoe), there are 2 possible orders (sock-shoe or shoe-sock), and only one of them is correct.\n\n5. **Probability for All Legs:** Since the legs are independent in terms of dressing, the probability that all legs have the sock put on before the shoe is $\\left(\\frac{1}{2}\\right)^8 = \\frac{1}{2^8}$.\n\n6. **Calculating the Number of Valid Permutations:** Given that each of the $16!$ permutations is equally likely, and the probability of a permutation being valid (socks before shoes for all legs) is $\\frac{1}{2^8}$, the number of valid permutations is:\n \\[\n \\frac{16!}{2^8}\n \\]\n\n#### Conclusion:\nThe number of different orders in which the spider can put on its socks and shoes, with the constraint that each sock must be put on before its corresponding shoe on each leg, is $\\boxed{\\frac{16!}{2^8}}$. This corresponds to choice (D) in the provided options."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "A two-digit positive integer is said to be $cuddly$ if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "1"}, "extra_info": {"index": 320, "solution": "Let's analyze the problem step by step:\n\n1. **Define the Problem:**\n A two-digit integer is represented as $10a + b$, where $a$ is the tens digit and $b$ is the units digit. The integer is called $\\emph{cuddly}$ if it satisfies the equation:\n \\[\n 10a + b = a + b^2\n \\]\n\n2. **Simplify the Equation:**\n Rearrange the equation to isolate terms involving $b$:\n \\[\n 10a + b = a + b^2 \\implies 9a = b^2 - b\n \\]\n Simplify further:\n \\[\n 9a = b(b - 1)\n \\]\n\n3. **Analyze Possible Values:**\n Since $a$ and $b$ are digits, $a$ ranges from 1 to 9 (as $a = 0$ would not make it a two-digit number), and $b$ ranges from 0 to 9. However, since the equation involves $b^2$, we focus on values of $b$ that make $b(b-1)$ a multiple of 9.\n\n4. **Check Values of $b$:**\n We compute $b(b-1)$ for $b = 0$ to $9$ and check if it is a multiple of 9:\n - $b = 0$: $0(0-1) = 0$\n - $b = 1$: $1(1-1) = 0$\n - $b = 2$: $2(2-1) = 2$\n - $b = 3$: $3(3-1) = 6$\n - $b = 4$: $4(4-1) = 12$\n - $b = 5$: $5(5-1) = 20$\n - $b = 6$: $6(6-1) = 30$\n - $b = 7$: $7(7-1) = 42$\n - $b = 8$: $8(8-1) = 56$\n - $b = 9$: $9(9-1) = 72$\n\n Only $b = 9$ results in $b(b-1) = 72$, which is a multiple of 9.\n\n5. **Determine the Corresponding $a$:**\n For $b = 9$, we have:\n \\[\n 9a = 9 \\times 8 \\implies a = 8\n \\]\n\n6. **Verify the Solution:**\n Substitute $a = 8$ and $b = 9$ back into the original number:\n \\[\n 10a + b = 10 \\times 8 + 9 = 89\n \\]\n Check if it satisfies the cuddly condition:\n \\[\n 89 = 8 + 9^2 = 8 + 81 = 89\n \\]\n\nThus, the only two-digit cuddly number is $89$.\n\n### Conclusion:\nThere is only one two-digit cuddly number, which is $\\boxed{\\textbf{(B) }1}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "Amelia has a coin that lands heads with probability $\\frac{1}{3}$, and Blaine has a coin that lands on heads with probability $\\frac{2}{5}$. Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $q-p$?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "4"}, "extra_info": {"index": 360, "solution": "To find the probability that Amelia wins, we consider the following cases:\n\n1. **Amelia wins on her first turn.**\n2. **Amelia wins on her second turn.**\n3. **Amelia wins on her third turn.**\n4. **And so on.**\n\n#### Case Analysis:\n- **Case 1:** Amelia wins on her first turn. The probability of this happening is simply the probability that Amelia's coin lands heads, which is $\\frac{1}{3}$.\n\n- **Case 2 and beyond:** For Amelia to win on her second turn or any subsequent turn, both Amelia and Blaine must first get tails, and then Amelia must get heads on her next turn. \n\n The probability that both Amelia and Blaine get tails on any given pair of turns (Amelia's turn followed by Blaine's turn) is $\\frac{2}{3} \\times \\frac{3}{5} = \\frac{2}{5}$. This is because Amelia's coin lands tails with probability $\\frac{2}{3}$ and Blaine's coin lands tails with probability $\\frac{3}{5}$.\n\n Therefore, the probability that Amelia wins on her second turn is $\\left(\\frac{2}{5}\\right) \\times \\frac{1}{3}$. For her to win on her third turn, two rounds of both getting tails must occur first, followed by Amelia getting heads. This probability is $\\left(\\frac{2}{5}\\right)^2 \\times \\frac{1}{3}$, and so on.\n\n#### Summing the Infinite Series:\nThe total probability $P$ that Amelia wins can be expressed as an infinite series:\n\\[\nP = \\frac{1}{3} + \\left(\\frac{2}{5}\\right)\\frac{1}{3} + \\left(\\frac{2}{5}\\right)^2\\frac{1}{3} + \\left(\\frac{2}{5}\\right)^3\\frac{1}{3} + \\cdots\n\\]\nFactoring out $\\frac{1}{3}$, we get:\n\\[\nP = \\frac{1}{3}\\left(1 + \\frac{2}{5} + \\left(\\frac{2}{5}\\right)^2 + \\left(\\frac{2}{5}\\right)^3 + \\cdots\\right)\n\\]\nThe expression in the parentheses is a geometric series with the first term $a = 1$ and common ratio $r = \\frac{2}{5}$. The sum $S$ of an infinite geometric series is given by $S = \\frac{a}{1 - r}$, where $|r| < 1$. Applying this formula, we find:\n\\[\nS = \\frac{1}{1 - \\frac{2}{5}} = \\frac{1}{\\frac{3}{5}} = \\frac{5}{3}\n\\]\nThus, the total probability $P$ is:\n\\[\nP = \\frac{1}{3} \\times \\frac{5}{3} = \\frac{5}{9}\n\\]\n\n#### Conclusion:\nThe probability that Amelia wins is $\\boxed{\\frac{5}{9}}$. Given that $p = 5$ and $q = 9$, the value of $q - p$ is $9 - 5 = \\boxed{\\textbf{(D)} \\ 4}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\\angle CDE = 110^\\circ$. Point $F$ lies on $\\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\\angle AFE$?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "170"}, "extra_info": {"index": 400, "solution": "1. **Identify the given information and setup**: We know that $ABCD$ is a square, and $E$ lies on the opposite half-plane determined by line $CD$ from point $A$. The angle $\\angle CDE = 110^\\circ$. Point $F$ lies on $\\overline{AD}$ such that $DE = DF$.\n\n2. **Extend $\\overline{AD}$ to point $G$**: By extending $\\overline{AD}$ to $G$ such that $\\overline{CD}$ is perpendicular to $\\overline{DG}$, we form a right angle at $D$, i.e., $\\angle CDG = 90^\\circ$.\n\n3. **Calculate $\\angle GDE$**: Since $\\angle CDE = 110^\\circ$ and $\\angle CDG = 90^\\circ$, we have:\n \\[\n \\angle GDE = \\angle CDE - \\angle CDG = 110^\\circ - 90^\\circ = 20^\\circ.\n \\]\n\n4. **Analyze triangle $DEF$**: Since $DE = DF$, triangle $DEF$ is isosceles with $DE = DF$. Therefore, the base angles $\\angle DFE$ and $\\angle DEF$ are equal. Since $\\angle GDE = 20^\\circ$ and $G, D, E, F$ are collinear, $\\angle DFE + \\angle DEF = 20^\\circ$.\n\n5. **Calculate $\\angle DFE$ and $\\angle DEF$**: Since $\\angle DFE = \\angle DEF$, we have:\n \\[\n \\angle DFE = \\angle DEF = \\frac{20^\\circ}{2} = 10^\\circ.\n \\]\n\n6. **Calculate $\\angle AFE$**: Since $F$ lies on $\\overline{AD}$ and $\\angle AFD = 180^\\circ$ (straight line), we have:\n \\[\n \\angle AFE = 180^\\circ - \\angle DFE = 180^\\circ - 10^\\circ = 170^\\circ.\n \\]\n\n7. **Conclusion**: The degree measure of $\\angle AFE$ is $\\boxed{170^\\circ}$, which corresponds to choice $\\textbf{(D)}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "\\frac{2}{5}"}, "extra_info": {"index": 440, "solution": "1. **Define Variables:**\n Let $m$ represent the total amount of money Brianna has. Let $c$ represent the total cost of all the CDs she wants to buy.\n\n2. **Establish Relationships:**\n According to the problem, Brianna uses one fifth of her money to buy one third of the CDs. This can be expressed as:\n \\[\n \\frac{1}{5}m = \\frac{1}{3}c\n \\]\n\n3. **Solve for Total Cost of CDs:**\n To find the total cost of all the CDs, we can rearrange the equation:\n \\[\n c = 3 \\left(\\frac{1}{5}m\\right) = \\frac{3}{5}m\n \\]\n This means the total cost of all the CDs is $\\frac{3}{5}m$.\n\n4. **Calculate Remaining Money:**\n After buying all the CDs, the amount of money Brianna has left is:\n \\[\n m - \\frac{3}{5}m = \\frac{5}{5}m - \\frac{3}{5}m = \\frac{2}{5}m\n \\]\n\n5. **Conclusion:**\n The fraction of her money that Brianna will have left after buying all the CDs is $\\frac{2}{5}$. Therefore, the answer is $\\boxed{\\textbf{(C) }\\frac{2}{5}}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "Consider the graphs $y=Ax^2$ and $y^2+3=x^2+4y$, where $A$ is a positive constant and $x$ and $y$ are real variables. In how many points do the two graphs intersect?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "4"}, "extra_info": {"index": 480, "solution": "#### Step-by-step Algebraic Approach:\n\n1. **Substitute $y = Ax^2$ into the second equation:**\n \\[\n y^2 + 3 = x^2 + 4y \\implies (Ax^2)^2 + 3 = x^2 + 4Ax^2\n \\]\n Simplifying, we get:\n \\[\n A^2x^4 + 3 = x^2 + 4Ax^2\n \\]\n Rearranging terms:\n \\[\n A^2x^4 - (4A+1)x^2 + 3 = 0\n \\]\n\n2. **Let $u = x^2$, then the equation becomes a quadratic in $u$:**\n \\[\n A^2u^2 - (4A+1)u + 3 = 0\n \\]\n\n3. **Apply the quadratic formula to solve for $u$:**\n \\[\n u = \\frac{4A+1 \\pm \\sqrt{(4A+1)^2 - 4 \\cdot A^2 \\cdot 3}}{2A^2}\n \\]\n Simplifying under the square root:\n \\[\n u = \\frac{4A+1 \\pm \\sqrt{16A^2 + 8A + 1 - 12A^2}}{2A^2} = \\frac{4A+1 \\pm \\sqrt{4A^2 + 8A + 1}}{2A^2}\n \\]\n\n4. **Check the discriminant for real roots:**\n \\[\n 4A^2 + 8A + 1 > 0 \\quad \\text{(always true for } A > 0\\text{)}\n \\]\n Therefore, the square root is real, and we have two distinct values for $u$.\n\n5. **Since $u = x^2$, each positive $u$ gives two values for $x$ (positive and negative):**\n \\[\n x = \\pm \\sqrt{u}\n \\]\n Thus, for each positive solution $u$, there are two corresponding $x$ values.\n\n6. **Conclusion:**\n Since there are two distinct positive solutions for $u$, and each leads to two values of $x$, there are a total of four intersection points.\n\n$\\boxed{\\text{A}}$\n\n#### Graphical Approach:\n\n1. **Rewrite the second equation in a standard form:**\n \\[\n y^2 - 4y + 3 = x^2 \\implies y^2 - 4y + 4 - 1 = x^2 \\implies (y-2)^2 - 1 = x^2\n \\]\n Rearranging:\n \\[\n (y-2)^2 - x^2 = 1\n \\]\n This is the equation of a hyperbola centered at $(0, 2)$.\n\n2. **Identify the hyperbola's branches:**\n The hyperbola opens vertically, with branches moving up and down from the center $(0, 2)$.\n\n3. **Consider the parabola $y = Ax^2$:**\n This is an upward-opening parabola with vertex at the origin.\n\n4. **Intersection points:**\n - The parabola intersects the lower branch of the hyperbola twice (once on each side of the y-axis).\n - As $x \\to \\pm \\infty$, the slope of the parabola ($2Ax$) increases without bound, ensuring it will intersect the upper branch of the hyperbola twice as well.\n\n5. **Conclusion:**\n The parabola intersects the hyperbola in four points.\n\n$\\boxed{\\text{A}}$"}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "Each integer 1 through 9 is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. Which digit is most likely to be the units digit of the sum of Jack's integer and Jill's integer?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "0"}, "extra_info": {"index": 520, "solution": "1. **Understanding the Problem**: Each integer from $1$ to $9$ is placed on a slip of paper and put into a hat. Jack and Jill each draw a slip (with replacement), and we want to find the most likely units digit of the sum of the two integers they draw.\n\n2. **Setting Up the Problem**: Let $J$ be the integer Jack draws and $K$ be the integer Jill draws. Both $J$ and $K$ are uniformly distributed over the integers $1$ through $9$.\n\n3. **Calculating the Total Number of Outcomes**: Since each draw is independent and there are $9$ choices for each draw, there are $9 \\times 9 = 81$ possible outcomes for the pair $(J, K)$.\n\n4. **Summing the Integers**: The sum $S = J + K$ can range from $2$ (if both draw $1$) to $18$ (if both draw $9$).\n\n5. **Finding Units Digits**: We need to consider the units digit of $S$, which is $S \\mod 10$. We will calculate how many outcomes correspond to each possible units digit from $0$ to $9$.\n\n6. **Counting Outcomes for Each Units Digit**:\n - **Units Digit $0$**: This occurs when $S = 10, 20$. Possible pairs $(J, K)$ are $(1,9), (2,8), (3,7), (4,6), (5,5), (6,4), (7,3), (8,2), (9,1)$ for $S=10$. There are $9$ outcomes.\n - **Units Digits $1$ to $9$**: For each of these, we consider sums like $S = 1, 11, 21, \\ldots$ (only $1$ and $11$ are possible here). For each such sum, we count pairs $(J, K)$ that satisfy $J + K \\equiv n \\pmod{10}$ for $n = 1, 2, \\ldots, 9$. Each of these sums has $8$ outcomes because one of the integers $1$ through $9$ will not participate in a sum that results in each specific units digit.\n\n7. **Comparing Frequencies**:\n - The units digit $0$ occurs $9$ times.\n - Each of the units digits $1$ through $9$ occurs $8$ times.\n\n8. **Conclusion**: Since the units digit $0$ occurs more frequently than any other units digit in the sums, it is the most likely units digit of the sum of Jack's and Jill's integers.\n\nThus, the answer is $\\boxed{\\textbf{(A)}\\ 0}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "Extend the square pattern of 8 black and 17 white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "32/17"}, "extra_info": {"index": 560, "solution": "1. **Identify the original pattern dimensions and tile counts**: The original pattern consists of a square with 8 black tiles and 17 white tiles. The pattern is arranged in a way that suggests the square is 5x5 tiles in size (since $5^2 = 25$ and $8 + 17 = 25$).\n\n2. **Understand the extension**: The problem states that a border of black tiles is added around the original square. This means that the new square will be 7x7 tiles in size (since adding a border around a 5x5 square adds 1 tile to each side).\n\n3. **Calculate the number of new black tiles added**: The new border will consist of the outermost tiles in the 7x7 square. The number of tiles in a 7x7 square is $7^2 = 49$. The number of tiles in the original 5x5 square is $5^2 = 25$. Therefore, the number of new black tiles added is $49 - 25 = 24$.\n\n4. **Total number of black tiles in the extended pattern**: The original pattern had 8 black tiles, and 24 new black tiles were added in the border. Thus, the total number of black tiles is $8 + 24 = 32$.\n\n5. **Total number of white tiles in the extended pattern**: The number of white tiles remains unchanged because the extension only added black tiles. Therefore, there are still 17 white tiles.\n\n6. **Calculate the ratio of black tiles to white tiles**: The ratio of black tiles to white tiles in the extended pattern is $\\frac{32}{17}$.\n\n7. **Conclusion**: The ratio of black tiles to white tiles in the extended pattern is $\\boxed{\\textbf{(D) }32:17}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "For any positive integer $n$, define $\\boxed{n}$ to be the sum of the positive factors of $n$.\nFor example, $\\boxed{6} = 1 + 2 + 3 + 6 = 12$. Find $\\boxed{\\boxed{11}}$ ."}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "28"}, "extra_info": {"index": 600, "solution": "To solve for $\\boxed{\\boxed{11}}$, we first need to determine $\\boxed{11}$, which is the sum of the positive factors of $11$.\n\n1. **Finding $\\boxed{11}$**:\n Since $11$ is a prime number, its only positive factors are $1$ and $11$ itself. Therefore, the sum of the factors of $11$ is:\n \\[\n \\boxed{11} = 1 + 11 = 12.\n \\]\n\n2. **Finding $\\boxed{\\boxed{11}}$**:\n Now, we need to find $\\boxed{12}$, which is the sum of the positive factors of $12$.\n The positive factors of $12$ are $1, 2, 3, 4, 6,$ and $12$. Thus, the sum of these factors is:\n \\[\n \\boxed{12} = 1 + 2 + 3 + 4 + 6 + 12.\n \\]\n Calculating the sum:\n \\[\n 1 + 2 = 3, \\quad 3 + 3 = 6, \\quad 6 + 4 = 10, \\quad 10 + 6 = 16, \\quad 16 + 12 = 28.\n \\]\n Therefore, $\\boxed{12} = 28$.\n\n3. **Conclusion**:\n Since $\\boxed{\\boxed{11}} = \\boxed{12} = 28$, the final answer is:\n \\[\n \\boxed{\\textbf{(D)}\\ 28}\n \\]"}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "For the consumer, a single discount of $n\\%$ is more advantageous than any of the following discounts:\n(1) two successive $15\\%$ discounts\n(2) three successive $10\\%$ discounts\n(3) a $25\\%$ discount followed by a $5\\%$ discount\nWhat is the smallest possible positive integer value of $n$?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "29"}, "extra_info": {"index": 640, "solution": "Let the original price be $x$. We need to find the smallest integer $n$ such that a single discount of $n\\%$ is more advantageous than any of the given series of discounts.\n\n1. **Calculating the effective discount for option (1):**\n Two successive $15\\%$ discounts:\n \\[\n (1 - 0.15)(1 - 0.15)x = 0.85 \\times 0.85x = 0.7225x\n \\]\n The effective discount is:\n \\[\n x - 0.7225x = 0.2775x\n \\]\n This corresponds to a $27.75\\%$ discount.\n\n2. **Calculating the effective discount for option (2):**\n Three successive $10\\%$ discounts:\n \\[\n (1 - 0.10)(1 - 0.10)(1 - 0.10)x = 0.90 \\times 0.90 \\times 0.90x = 0.729x\n \\]\n The effective discount is:\n \\[\n x - 0.729x = 0.271x\n \\]\n This corresponds to a $27.1\\%$ discount.\n\n3. **Calculating the effective discount for option (3):**\n A $25\\%$ discount followed by a $5\\%$ discount:\n \\[\n (1 - 0.25)(1 - 0.05)x = 0.75 \\times 0.95x = 0.7125x\n \\]\n The effective discount is:\n \\[\n x - 0.7125x = 0.2875x\n \\]\n This corresponds to a $28.75\\%$ discount.\n\nTo find the smallest $n$ such that a single discount of $n\\%$ is more advantageous, $n$ must be greater than the maximum of these effective discounts:\n\\[\nn > \\max(27.75, 27.1, 28.75)\n\\]\nThus, $n > 28.75$. Since $n$ must be an integer, the smallest possible value of $n$ is $29$.\n\nTherefore, the smallest possible positive integer value of $n$ is $\\boxed{\\textbf{(C)}\\ 29}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "Given that $i^2=-1$, for how many integers $n$ is $(n+i)^4$ an integer?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "3"}, "extra_info": {"index": 680, "solution": "1. **Expand the expression**: We start by expanding $(n+i)^4$ using the binomial theorem:\n \\[\n (n+i)^4 = \\binom{4}{0}n^4i^0 + \\binom{4}{1}n^3i^1 + \\binom{4}{2}n^2i^2 + \\binom{4}{3}n^1i^3 + \\binom{4}{4}n^0i^4.\n \\]\n Simplifying each term, we get:\n \\[\n (n+i)^4 = n^4 + 4n^3i - 6n^2 - 4ni + i^4.\n \\]\n Since $i^2 = -1$, we have $i^4 = (i^2)^2 = (-1)^2 = 1$. Thus, the expression simplifies to:\n \\[\n (n+i)^4 = n^4 + 4n^3i - 6n^2 - 4ni + 1.\n \\]\n\n2. **Separate real and imaginary parts**: For $(n+i)^4$ to be an integer, the imaginary part must be zero. The imaginary part of the expression is:\n \\[\n 4n^3i - 4ni.\n \\]\n Factoring out the common terms, we get:\n \\[\n 4ni(n^2 - 1) = 0.\n \\]\n\n3. **Solve for $n$**: The equation $4ni(n^2 - 1) = 0$ holds if either $n = 0$ or $n^2 - 1 = 0$. Solving $n^2 - 1 = 0$, we find:\n \\[\n n^2 = 1 \\implies n = \\pm 1.\n \\]\n Therefore, the possible integer values of $n$ are $n = 0, 1, -1$.\n\n4. **Count the solutions**: We have found three integers ($n = 0, 1, -1$) for which $(n+i)^4$ is an integer.\n\nThus, there are $\\boxed{3}$ integers $n$ such that $(n+i)^4$ is an integer, corresponding to choice $\\text{(D)} \\ 3$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "How many numbers between $1$ and $2005$ are integer multiples of $3$ or $4$ but not $12$?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "1002"}, "extra_info": {"index": 720, "solution": "To solve this problem, we need to find the number of integers between $1$ and $2005$ that are multiples of $3$ or $4$ but not multiples of $12$. We will use the principle of inclusion-exclusion and then subtract the multiples of $12$.\n\n1. **Count multiples of $3$:**\n The multiples of $3$ from $1$ to $2005$ are given by the numbers $3, 6, 9, \\ldots, 2004$. The largest multiple of $3$ less than or equal to $2005$ is $2004$. The number of terms in this sequence can be calculated as:\n \\[\n \\left\\lfloor \\frac{2005}{3} \\right\\rfloor = 668\n \\]\n\n2. **Count multiples of $4$:**\n The multiples of $4$ from $1$ to $2005$ are given by the numbers $4, 8, 12, \\ldots, 2004$. The largest multiple of $4$ less than or equal to $2005$ is $2004$. The number of terms in this sequence can be calculated as:\n \\[\n \\left\\lfloor \\frac{2005}{4} \\right\\rfloor = 501\n \\]\n\n3. **Count multiples of $12$:**\n Since $12$ is the least common multiple of $3$ and $4$, we count the multiples of $12$ from $1$ to $2005$. The multiples of $12$ are $12, 24, 36, \\ldots, 2004$. The largest multiple of $12$ less than or equal to $2005$ is $2004$. The number of terms in this sequence can be calculated as:\n \\[\n \\left\\lfloor \\frac{2005}{12} \\right\\rfloor = 167\n \\]\n\n4. **Use the principle of inclusion-exclusion:**\n To find the numbers that are multiples of $3$ or $4$, we add the multiples of $3$ and $4$, then subtract the multiples of $12$ (since they are counted twice, once in each of the first two counts):\n \\[\n 668 + 501 - 167 = 1002\n \\]\n\n5. **Conclusion:**\n The number of integers between $1$ and $2005$ that are multiples of $3$ or $4$ but not $12$ is $1002$.\n\nThus, the correct answer is $\\boxed{\\textbf{(D) } 1002}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "How many unordered pairs of edges of a given cube determine a plane?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "42"}, "extra_info": {"index": 760, "solution": "1. **Understanding the Problem**: We need to find how many unordered pairs of edges in a cube determine a plane. Two edges determine a plane if they are either parallel or intersecting (not skew).\n\n2. **Total Number of Edges in a Cube**: A cube has 12 edges.\n\n3. **Choosing One Edge**: Choose one edge arbitrarily. There are 12 choices for this first edge.\n\n4. **Conditions for the Second Edge**:\n - **Parallel Edges**: Each edge in a cube is parallel to 3 other edges (since each face of the cube has 4 edges, and each edge is parallel to the opposite edge on the same face).\n - **Intersecting Edges**: Each edge intersects with 4 other edges that are not on the same face and not parallel.\n\n5. **Total Edges that Determine a Plane with the Chosen Edge**: Each chosen edge can pair with 3 parallel edges or 4 intersecting edges, making a total of $3 + 4 = 7$ edges that can form a plane with the chosen edge.\n\n6. **Calculating the Number of Valid Pairs**:\n - Since there are 12 edges, and each edge can form a plane-determining pair with 7 other edges, we might initially think to calculate $12 \\times 7 = 84$ pairs.\n - However, this count considers each pair twice (once for each edge as the starting edge), so we must divide by 2 to avoid double-counting: $\\frac{84}{2} = 42$.\n\n7. **Conclusion**: The number of unordered pairs of edges that determine a plane in a cube is $\\boxed{42}$, corresponding to choice $\\textbf{(D)}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "If $\\frac{2+3+4}{3}=\\frac{1990+1991+1992}{N}$, then $N=$"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "1991"}, "extra_info": {"index": 800, "solution": "1. **Calculate the left-hand side of the equation:**\n \\[\n \\frac{2+3+4}{3} = \\frac{9}{3} = 3\n \\]\n\n2. **Set up the equation:**\n Given that\n \\[\n \\frac{2+3+4}{3} = \\frac{1990+1991+1992}{N}\n \\]\n substituting the value from step 1, we have:\n \\[\n 3 = \\frac{1990+1991+1992}{N}\n \\]\n\n3. **Calculate the sum in the numerator on the right-hand side:**\n \\[\n 1990 + 1991 + 1992 = 5973\n \\]\n\n4. **Substitute the sum into the equation:**\n \\[\n 3 = \\frac{5973}{N}\n \\]\n\n5. **Solve for \\(N\\):**\n \\[\n N = \\frac{5973}{3} = 1991\n \\]\n\nThus, the value of \\(N\\) that satisfies the equation is \\(1991\\).\n\n\\[\n\\boxed{\\text{(D)}\\ 1991}\n\\]"}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "If $f(x) = 5x^2 - 2x - 1$, then $f(x + h) - f(x)$ equals:"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "h(10x+5h-2)"}, "extra_info": {"index": 840, "solution": "To find $f(x+h) - f(x)$ for the function $f(x) = 5x^2 - 2x - 1$, we start by substituting $x+h$ into the function and then subtracting $f(x)$.\n\n1. **Substitute $x+h$ into $f$:**\n \\[\n f(x+h) = 5(x+h)^2 - 2(x+h) - 1\n \\]\n Expand $(x+h)^2$:\n \\[\n (x+h)^2 = x^2 + 2xh + h^2\n \\]\n Substitute back into $f(x+h)$:\n \\[\n f(x+h) = 5(x^2 + 2xh + h^2) - 2(x + h) - 1\n \\]\n Distribute and simplify:\n \\[\n f(x+h) = 5x^2 + 10xh + 5h^2 - 2x - 2h - 1\n \\]\n\n2. **Subtract $f(x)$ from $f(x+h)$:**\n \\[\n f(x+h) - f(x) = (5x^2 + 10xh + 5h^2 - 2x - 2h - 1) - (5x^2 - 2x - 1)\n \\]\n Simplify by canceling terms:\n \\[\n f(x+h) - f(x) = 5x^2 + 10xh + 5h^2 - 2x - 2h - 1 - 5x^2 + 2x + 1\n \\]\n \\[\n f(x+h) - f(x) = 10xh + 5h^2 - 2h\n \\]\n\n3. **Factor out $h$ from the expression:**\n \\[\n f(x+h) - f(x) = h(10x + 5h - 2)\n \\]\n\nThus, the expression for $f(x+h) - f(x)$ is $h(10x + 5h - 2)$, which corresponds to choice $\\textbf{(D)}$.\n\n$\\boxed{(\\textbf{D})}$"}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "If $y = 2x$ and $z = 2y$, then $x + y + z$ equals"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "$7x$"}, "extra_info": {"index": 880, "solution": "Given the equations:\n1. \\( y = 2x \\)\n2. \\( z = 2y \\)\n\nWe need to find the expression for \\( x + y + z \\).\n\n**Step 1:** Substitute the value of \\( y \\) from equation 1 into equation 2:\n\\[ z = 2y = 2(2x) = 4x \\]\n\n**Step 2:** Substitute the values of \\( y \\) and \\( z \\) into the expression \\( x + y + z \\):\n\\[ x + y + z = x + 2x + 4x \\]\n\n**Step 3:** Combine like terms:\n\\[ x + 2x + 4x = 7x \\]\n\nThus, \\( x + y + z = 7x \\).\n\nTherefore, the correct answer is:\n\\[ \\boxed{\\text{(D)}\\ 7x} \\]"}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "If the sum of the first $3n$ positive integers is $150$ more than the sum of the first $n$ positive integers, then the sum of the first $4n$ positive integers is"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "300"}, "extra_info": {"index": 920, "solution": "1. **Set up the equation based on the problem statement:**\n The sum of the first $3n$ positive integers is given by the formula $\\frac{3n(3n+1)}{2}$, and the sum of the first $n$ positive integers is given by $\\frac{n(n+1)}{2}$. According to the problem, the sum of the first $3n$ integers is $150$ more than the sum of the first $n$ integers. Therefore, we can write the equation:\n \\[\n \\frac{3n(3n+1)}{2} = \\frac{n(n+1)}{2} + 150\n \\]\n\n2. **Simplify the equation:**\n Multiply through by $2$ to clear the fraction:\n \\[\n 3n(3n+1) = n(n+1) + 300\n \\]\n Expanding both sides:\n \\[\n 9n^2 + 3n = n^2 + n + 300\n \\]\n Rearrange to form a quadratic equation:\n \\[\n 9n^2 + 3n - n^2 - n - 300 = 0 \\Rightarrow 8n^2 + 2n - 300 = 0\n \\]\n Simplify the quadratic equation:\n \\[\n 4n^2 + n - 150 = 0\n \\]\n\n3. **Solve the quadratic equation:**\n Use the quadratic formula where $a = 4$, $b = 1$, and $c = -150$:\n \\[\n n = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a} = \\frac{-1 \\pm \\sqrt{1 + 2400}}{8} = \\frac{-1 \\pm \\sqrt{2401}}{8}\n \\]\n Simplifying further:\n \\[\n n = \\frac{-1 \\pm 49}{8} \\Rightarrow n = \\frac{48}{8} = 6 \\quad \\text{or} \\quad n = \\frac{-50}{8} = -\\frac{25}{4}\n \\]\n Since $n$ must be a positive integer, we choose $n = 6$.\n\n4. **Calculate the sum of the first $4n$ positive integers:**\n Now, $4n = 4 \\times 6 = 24$. The sum of the first $24$ positive integers is:\n \\[\n \\frac{24 \\times 25}{2} = 300\n \\]\n\n5. **Conclude with the final answer:**\n The sum of the first $4n$ positive integers is $\\boxed{300}$. This corresponds to choice $\\text{(A)}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "In a mathematics contest with ten problems, a student gains 5 points for a correct answer and loses 2 points for an incorrect answer. If Olivia answered every problem and her score was 29, how many correct answers did she have?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "7"}, "extra_info": {"index": 960, "solution": "Let $c$ be the number of correct answers Olivia had, and $w$ be the number of incorrect answers. Since there are 10 problems in total, we have:\n\\[ c + w = 10 \\]\n\nFor each correct answer, Olivia gains 5 points, and for each incorrect answer, she loses 2 points. Therefore, her total score can be expressed as:\n\\[ 5c - 2w = 29 \\]\n\nWe can solve these equations simultaneously. First, express $w$ in terms of $c$ from the first equation:\n\\[ w = 10 - c \\]\n\nSubstitute this expression for $w$ into the second equation:\n\\[ 5c - 2(10 - c) = 29 \\]\n\\[ 5c - 20 + 2c = 29 \\]\n\\[ 7c - 20 = 29 \\]\n\\[ 7c = 49 \\]\n\\[ c = 7 \\]\n\nThus, Olivia had 7 correct answers. To verify, calculate the score with 7 correct answers:\n\\[ 5 \\times 7 - 2 \\times (10 - 7) = 35 - 6 = 29 \\]\n\nThis confirms that the calculations are correct. Therefore, the number of correct answers Olivia had is $\\boxed{\\text{(C)}\\ 7}$. $\\blacksquare$"}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "In the adjoining figure triangle $ABC$ is such that $AB = 4$ and $AC = 8$. IF $M$ is the midpoint of $BC$ and $AM = 3$, what is the length of $BC$?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "2\\sqrt{31}"}, "extra_info": {"index": 1000, "solution": "1. **Identify the Given Information:**\n - Triangle $ABC$ with $AB = 4$, $AC = 8$.\n - $M$ is the midpoint of $BC$.\n - $AM = 3$.\n\n2. **Set Up the Variables:**\n - Let $BM = MC = x$. Therefore, $BC = 2x$.\n\n3. **Apply Stewart's Theorem:**\n - Stewart's Theorem states that for a triangle $ABC$ with a cevian $AD$ to side $BC$, where $D$ is between $B$ and $C$, and $BD = m$, $DC = n$, $AD = d$, $AB = c$, $AC = b$, then:\n \\[\n b^2 \\cdot m + c^2 \\cdot n = a(d^2 + mn)\n \\]\n - In our case, $AD = AM = 3$, $BD = BM = x$, $DC = MC = x$, $AB = 4$, $AC = 8$, and $BC = 2x$. Plugging in these values, we get:\n \\[\n 8^2 \\cdot x + 4^2 \\cdot x = 2x(3^2 + x^2)\n \\]\n \\[\n 64x + 16x = 2x(9 + x^2)\n \\]\n \\[\n 80x = 18x + 2x^3\n \\]\n \\[\n 2x^3 - 62x = 0\n \\]\n \\[\n 2x(x^2 - 31) = 0\n \\]\n\n4. **Solve for $x$:**\n - From $2x(x^2 - 31) = 0$, we have either $x = 0$ or $x^2 - 31 = 0$.\n - $x = 0$ is not possible as it would imply $BC = 0$, which is not feasible.\n - Therefore, $x^2 = 31 \\implies x = \\sqrt{31}$.\n\n5. **Find $BC$:**\n - Since $BC = 2x$, substituting $x = \\sqrt{31}$, we get $BC = 2\\sqrt{31}$.\n\n6. **Conclusion:**\n - The length of $BC$ is $\\boxed{2\\sqrt{31}}$, which corresponds to choice $\\textbf{(B)}\\ 2\\sqrt{31}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "In this diagram the center of the circle is $O$, the radius is $a$ inches, chord $EF$ is parallel to chord $CD$. $O$,$G$,$H$,$J$ are collinear, and $G$ is the midpoint of $CD$. Let $K$ (sq. in.) represent the area of trapezoid $CDFE$ and let $R$ (sq. in.) represent the area of rectangle $ELMF.$ Then, as $CD$ and $EF$ are translated upward so that $OG$ increases toward the value $a$, while $JH$ always equals $HG$, the ratio $K:R$ becomes arbitrarily close to:"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "\\frac{1}{\\sqrt{2}}+\\frac{1}{2}"}, "extra_info": {"index": 1040, "solution": "1. **Identify the given information and setup the problem:**\n - The center of the circle is $O$, and the radius is $a$ inches.\n - Chords $CD$ and $EF$ are parallel, and $O$, $G$, $H$, $J$ are collinear with $G$ as the midpoint of $CD$.\n - $OG = a - 2h$, where $h = JH = HG$.\n - We need to find the ratio $K:R$ as $OG$ approaches $a$ (or equivalently as $h$ approaches $0$).\n\n2. **Calculate the lengths of segments using the Pythagorean Theorem:**\n - Since $OC$ and $OE$ are radii, their lengths are $a$.\n - Length of $EH = \\sqrt{a^2 - (OG + h)^2} = \\sqrt{a^2 - (a - h)^2}$.\n - Length of $CG = \\sqrt{a^2 - OG^2} = \\sqrt{a^2 - (a - 2h)^2}$.\n\n3. **Calculate the areas of rectangle $EHGL$ and trapezoid $EHGC$:**\n - Area of rectangle $EHGL = EH \\cdot HG = h \\sqrt{a^2 - (a - h)^2}$.\n - Area of trapezoid $EHGC = \\frac{HG}{2}(EH + CG) = \\frac{h}{2}(\\sqrt{a^2 - (a - h)^2} + \\sqrt{a^2 - (a - 2h)^2})$.\n\n4. **Simplify the expressions for the areas:**\n - Substitute $OG = a - 2h$ into the expressions:\n - $EH = \\sqrt{2ah - h^2}$,\n - $CG = \\sqrt{4ah - 4h^2}$.\n - Thus, area of rectangle $EHGL = h\\sqrt{2ah - h^2}$.\n - Area of trapezoid $EHGC = \\frac{h}{2}(\\sqrt{2ah - h^2} + \\sqrt{4ah - 4h^2})$.\n\n5. **Find the limit of the ratio $K:R$ as $h \\rightarrow 0$:**\n - Simplify the ratio $\\frac{K}{R} = \\frac{\\frac{h}{2}(\\sqrt{2ah - h^2} + \\sqrt{4ah - 4h^2})}{h\\sqrt{2ah - h^2}}$.\n - Cancel $h$ and simplify the expression under the limit:\n \\[\n \\lim_{h\\rightarrow 0}\\frac{\\frac{1}{2}(\\sqrt{2ah - h^2} + \\sqrt{4ah - 4h^2})}{\\sqrt{2ah - h^2}} = \\lim_{h\\rightarrow 0}\\frac{1}{2}\\left(1 + \\frac{\\sqrt{4ah - 4h^2}}{\\sqrt{2ah - h^2}}\\right).\n \\]\n - As $h \\rightarrow 0$, $\\sqrt{4ah - 4h^2} \\approx 2\\sqrt{ah}$ and $\\sqrt{2ah - h^2} \\approx \\sqrt{2ah}$, so the ratio simplifies to:\n \\[\n \\frac{1}{2}\\left(1 + \\frac{2}{\\sqrt{2}}\\right) = \\frac{1}{2} + \\frac{1}{\\sqrt{2}}.\n \\]\n\n6. **Conclude with the final answer:**\n - The ratio $K:R$ becomes arbitrarily close to $\\boxed{\\textbf{(D) }\\frac{1}{2}+\\frac{1}{\\sqrt{2}}}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "Josh writes the numbers $1,2,3,\\dots,99,100$. He marks out $1$, skips the next number $(2)$, marks out $3$, and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number $(2)$, skips the next number $(4)$, marks out $6$, skips $8$, marks out $10$, and so on to the end. Josh continues in this manner until only one number remains. What is that number?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "64"}, "extra_info": {"index": 1080, "solution": "Josh's method of marking out numbers can be understood as a process of elimination where in each round, every second number from the remaining list is marked out. Let's analyze the pattern step by step:\n\n1. **First Pass:**\n - Josh marks out every second number starting from $1$. This means all odd numbers are marked out: $1, 3, 5, \\dots, 99$.\n - Remaining numbers after the first pass: $2, 4, 6, \\dots, 100$ (all even numbers).\n\n2. **Second Pass:**\n - Starting with the first number in the new list ($2$), Josh marks out every second number: $2, 6, 10, \\dots, 98$.\n - Remaining numbers after the second pass: $4, 8, 12, \\dots, 100$ (every fourth number).\n\n3. **Third Pass:**\n - Starting with the first number in the new list ($4$), Josh marks out every second number: $4, 12, 20, \\dots, 100$.\n - Remaining numbers after the third pass: $8, 16, 24, \\dots, 96$ (every eighth number).\n\n4. **Continuing the Pattern:**\n - Each pass effectively doubles the gap between remaining numbers. After the fourth pass, the remaining numbers are every sixteenth number, and so on.\n\n5. **Final Passes:**\n - The pattern continues until the gap between the numbers exceeds the largest number in the list. The last number remaining is the largest number that is a power of $2$ within the range, as each pass doubles the gap (or equivalently, marks out every second number in the progressively halved list).\n\n - The powers of $2$ within the range $1$ to $100$ are $1, 2, 4, 8, 16, 32, 64$. The process will eliminate all but the largest power of $2$ that can still be doubled within the range. The next power, $128$, is outside the range, so $64$ is the last number remaining.\n\nThus, the last number remaining after all passes is $\\boxed{\\textbf{(D)}\\ 64}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "Let $ABCD$ be a unit square. Let $Q_1$ be the midpoint of $\\overline{CD}$. For $i=1,2,\\dots,$ let $P_i$ be the intersection of $\\overline{AQ_i}$ and $\\overline{BD}$, and let $Q_{i+1}$ be the foot of the perpendicular from $P_i$ to $\\overline{CD}$. What is \n\\[\\sum_{i=1}^{\\infty} \\text{Area of } \\triangle DQ_i P_i \\, ?\\]"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "\\frac{1}{4}"}, "extra_info": {"index": 1120, "solution": "\n#### Step 1: Understanding the Problem\nWe are given a unit square $ABCD$ and a sequence of points $Q_i$ and $P_i$ defined recursively. We need to find the sum of the areas of triangles $\\triangle DQ_iP_i$ for all $i \\geq 1$.\n\n#### Step 2: Analyzing the First Triangle $\\triangle DQ_1P_1$\n- $Q_1$ is the midpoint of $\\overline{CD}$, so $DQ_1 = \\frac{1}{2}$.\n- $\\triangle DQ_1P_1$ is similar to $\\triangle ABP_1$ with a ratio of 1:2.\n- The intersection $P_1$ of $\\overline{AQ_1}$ and $\\overline{BD}$ can be found using the similarity of triangles. Since $\\overline{BD}$ is a diagonal of the square, $P_1$ divides $\\overline{BD}$ in the ratio of the distances from $A$ and $B$ to $Q_1$.\n- The height $P_1Q_2$ from $P_1$ to $\\overline{CD}$ is $\\frac{1}{3}$ due to the similarity ratio.\n\nThus, the area of $\\triangle DQ_1P_1$ is:\n\\[\n\\text{Area} = \\frac{1}{2} \\times \\text{base} \\times \\text{height} = \\frac{1}{2} \\times \\frac{1}{2} \\times \\frac{1}{3} = \\frac{1}{12}\n\\]\n\n#### Step 3: Generalizing for $\\triangle DQ_iP_i$\n- Each subsequent triangle $\\triangle DQ_iP_i$ follows a similar pattern, with $DQ_i = \\frac{1}{2^{i}}$ and $P_iQ_{i+1} = \\frac{1}{3 \\cdot 2^{i-1}}$.\n- The area of each triangle is then:\n\\[\n\\text{Area of } \\triangle DQ_iP_i = \\frac{1}{2} \\times DQ_i \\times P_iQ_{i+1} = \\frac{1}{2} \\times \\frac{1}{2^i} \\times \\frac{1}{3 \\cdot 2^{i-1}} = \\frac{1}{12 \\cdot 2^{2i-1}}\n\\]\n\n#### Step 4: Summing the Series\nThe sum of the areas of all such triangles is:\n\\[\n\\sum_{i=1}^{\\infty} \\text{Area of } \\triangle DQ_iP_i = \\sum_{i=1}^{\\infty} \\frac{1}{12 \\cdot 2^{2i-1}} = \\frac{1}{12} \\sum_{i=1}^{\\infty} \\frac{1}{2^{2i-1}}\n\\]\nThis is a geometric series with a common ratio of $\\frac{1}{4}$, starting from $\\frac{1}{2}$. The sum of this series is:\n\\[\n\\frac{1}{12} \\times \\frac{\\frac{1}{2}}{1 - \\frac{1}{4}} = \\frac{1}{12} \\times \\frac{\\frac{1}{2}}{\\frac{3}{4}} = \\frac{1}{12} \\times \\frac{2}{3} = \\frac{1}{18}\n\\]\nThis calculation seems incorrect based on the options provided. Let's recheck using the telescoping series approach from the original solution.\n\n#### Step 5: Using the Telescoping Series\nFrom the original solution, we have:\n\\[\n\\frac{1}{2} \\sum_{n=2}^\\infty \\left(\\frac{1}{n} - \\frac{1}{n+1}\\right) = \\frac{1}{2} \\left(1 - \\frac{1}{2}\\right) = \\frac{1}{4}\n\\]\nThis telescoping series correctly simplifies to $\\frac{1}{4}$, confirming the sum of the areas.\n\n#### Conclusion\nThe sum of the areas of the triangles $\\triangle DQ_iP_i$ for all $i \\geq 1$ is $\\boxed{\\textbf{(B)}\\ \\frac{1}{4}}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "Let $\\{a_k\\}_{k=1}^{2011}$ be the sequence of real numbers defined by $a_1=0.201,$ $a_2=(0.2011)^{a_1},$ $a_3=(0.20101)^{a_2},$ $a_4=(0.201011)^{a_3}$, and in general,\n\\[a_k=\\begin{cases}(0.\\underbrace{20101\\cdots 0101}_{k+2\\text{ digits}})^{a_{k-1}}\\qquad\\text{if }k\\text{ is odd,}\\\\(0.\\underbrace{20101\\cdots 01011}_{k+2\\text{ digits}})^{a_{k-1}}\\qquad\\text{if }k\\text{ is even.}\\end{cases}\\]Rearranging the numbers in the sequence $\\{a_k\\}_{k=1}^{2011}$ in decreasing order produces a new sequence $\\{b_k\\}_{k=1}^{2011}$. What is the sum of all integers $k$, $1\\le k \\le 2011$, such that $a_k=b_k?$"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "1341"}, "extra_info": {"index": 1160, "solution": "To solve this problem, we need to understand the behavior of the sequence $\\{a_k\\}_{k=1}^{2011}$ and determine when $a_k = b_k$, where $\\{b_k\\}_{k=1}^{2011}$ is the sequence $\\{a_k\\}_{k=1}^{2011}$ rearranged in decreasing order.\n\n#### Step 1: Analyze the sequence $\\{a_k\\}_{k=1}^{2011}$\nThe sequence is defined recursively, with each term being a power of a decimal less than 1 raised to the previous term. The base of the power alternates between having an additional '01' or '011' at the end as $k$ increases.\n\n#### Step 2: Establish the behavior of the sequence\nWe start by comparing the first few terms:\n- **Comparing $a_1$ and $a_2$:**\n \\[\n a_1 = 0.201, \\quad a_2 = (0.2011)^{a_1}\n \\]\n Since $0.2011 > 0.201$ and $0 < a_1 < 1$, we have:\n \\[\n a_2 = (0.2011)^{0.201} > (0.201)^{0.201} = a_1\n \\]\n Thus, $a_1 < a_2$.\n\n- **Comparing $a_2$ and $a_3$:**\n \\[\n a_3 = (0.20101)^{a_2}\n \\]\n Since $0.20101 < 0.2011$ and $0 < a_2 < 1$, we have:\n \\[\n a_3 = (0.20101)^{a_2} < (0.2011)^{a_2} = a_2\n \\]\n Thus, $a_3 < a_2$.\n\n- **Comparing $a_3$ and $a_4$:**\n \\[\n a_4 = (0.201011)^{a_3}\n \\]\n Since $0.201011 > 0.20101$ and $0 < a_3 < 1$, we have:\n \\[\n a_4 = (0.201011)^{a_3} > (0.20101)^{a_3} = a_3\n \\]\n Thus, $a_3 < a_4$.\n\n#### Step 3: General pattern and induction hypothesis\nFrom the comparisons, we observe that:\n- When $k$ is odd, $a_k < a_{k+1}$.\n- When $k$ is even, $a_k > a_{k+1}$.\n\nThis suggests a zigzag pattern where each odd-indexed term is less than the next term, and each even-indexed term is greater than the next term.\n\n#### Step 4: Determine when $a_k = b_k$\nGiven the zigzag pattern, the sequence $\\{a_k\\}$ is rearranged in decreasing order as $\\{b_k\\}$ such that:\n\\[\nb_1 = a_2 > b_2 = a_4 > \\ldots > b_{1005} = a_{2010} > b_{1006} = a_{2011} > \\ldots > b_{2010} = a_3 > b_{2011} = a_1\n\\]\nThus, $a_k = b_k$ only when $k$ is an even number from 2 to 2010 inclusive.\n\n#### Step 5: Calculate the sum of all such $k$\nThe sum of an arithmetic sequence where $k$ starts at 2, ends at 2010, and increments by 2 is:\n\\[\n\\text{Sum} = \\frac{\\text{Number of terms}}{2} \\times (\\text{First term} + \\text{Last term})\n\\]\nThe number of terms is $\\frac{2010 - 2}{2} + 1 = 1005$, and the sum is:\n\\[\n\\text{Sum} = \\frac{1005}{2} \\times (2 + 2010) = 1005 \\times 1006 = 1011030\n\\]\nHowever, this is not one of the options. We need to find the sum of indices $k$ such that $a_k = b_k$. Since $a_k = b_k$ only for $k = 2011$, the sum is:\n\\[\n\\boxed{\\textbf{(C)}\\ 1341}\n\\]"}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "Let $t_n = \\frac{n(n+1)}{2}$ be the $n$th triangular number. Find\n\\[\\frac{1}{t_1} + \\frac{1}{t_2} + \\frac{1}{t_3} + ... + \\frac{1}{t_{2002}}\\]"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "\\frac {4004}{2003}"}, "extra_info": {"index": 1200, "solution": "The problem statement and the solution provided seem unrelated. The problem statement asks for the sum of the reciprocals of the first 2002 triangular numbers, while the solution discusses logarithms and bases, which is irrelevant to the problem. Let's solve the original problem regarding triangular numbers.\n\n1. **Understanding Triangular Numbers**: The $n$-th triangular number, $t_n$, is given by the formula:\n \\[\n t_n = \\frac{n(n+1)}{2}\n \\]\n This represents the sum of the first $n$ natural numbers.\n\n2. **Sum of Reciprocals of Triangular Numbers**: We need to find:\n \\[\n \\sum_{n=1}^{2002} \\frac{1}{t_n}\n \\]\n where $t_n = \\frac{n(n+1)}{2}$.\n\n3. **Simplifying the Expression**: Substitute the expression for $t_n$:\n \\[\n \\sum_{n=1}^{2002} \\frac{1}{\\frac{n(n+1)}{2}} = \\sum_{n=1}^{2002} \\frac{2}{n(n+1)}\n \\]\n We can simplify $\\frac{2}{n(n+1)}$ using partial fractions:\n \\[\n \\frac{2}{n(n+1)} = \\frac{2}{n} - \\frac{2}{n+1}\n \\]\n\n4. **Telescoping Series**: The series now becomes a telescoping series:\n \\[\n \\sum_{n=1}^{2002} \\left(\\frac{2}{n} - \\frac{2}{n+1}\\right)\n \\]\n Most terms cancel out, leaving:\n \\[\n \\left(\\frac{2}{1} - \\frac{2}{2}\\right) + \\left(\\frac{2}{2} - \\frac{2}{3}\\right) + \\cdots + \\left(\\frac{2}{2002} - \\frac{2}{2003}\\right)\n \\]\n Simplifying this, we get:\n \\[\n 2 - \\frac{2}{2003}\n \\]\n\n5. **Final Calculation**: Simplify the expression:\n \\[\n 2 - \\frac{2}{2003} = \\frac{4006}{2003} - \\frac{2}{2003} = \\frac{4004}{2003}\n \\]\n\nThus, the sum of the reciprocals of the first 2002 triangular numbers is $\\boxed{\\frac{4004}{2003}}$. This corresponds to choice $\\text{(C)}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$. Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "12"}, "extra_info": {"index": 1240, "solution": "1. **Define the problem in terms of algebra**: Let the two-digit number Mary thought of be $x$. According to the problem, she performs the following operations on $x$:\n - Multiplies by $3$: $3x$\n - Adds $11$: $3x + 11$\n - Switches the digits of the result.\n\n2. **Analyze the range of the final number**: The switched digits result in a number between $71$ and $75$, inclusive.\n\n3. **Consider the properties of digit switching**: Let's denote the result of $3x + 11$ as a two-digit number $10a + b$ (where $a$ and $b$ are the tens and units digits respectively). Switching the digits gives $10b + a$. We know $10b + a$ is between $71$ and $75$.\n\n4. **Set up the equation and solve for possible values of $x$**:\n - Since $10b + a$ must be between $71$ and $75$, we consider each case:\n - If $10b + a = 71$, then reversing gives $10a + b = 17$. Solving $3x + 11 = 17$ gives $3x = 6$, so $x = 2$. This is not a two-digit number, so it's invalid.\n - If $10b + a = 72$, then reversing gives $10a + b = 27$. Solving $3x + 11 = 27$ gives $3x = 16$, which is not possible since $x$ must be an integer.\n - If $10b + a = 73$, then reversing gives $10a + b = 37$. Solving $3x + 11 = 37$ gives $3x = 26$, which is not possible since $x$ must be an integer.\n - If $10b + a = 74$, then reversing gives $10a + b = 47$. Solving $3x + 11 = 47$ gives $3x = 36$, so $x = 12$. This is a valid two-digit number.\n - If $10b + a = 75$, then reversing gives $10a + b = 57$. Solving $3x + 11 = 57$ gives $3x = 46$, which is not possible since $x$ must be an integer.\n\n5. **Verify the solution**: The only valid solution from the above analysis is when $x = 12$. We can verify this:\n - Multiply $12$ by $3$ to get $36$.\n - Add $11$ to get $47$.\n - Switching the digits of $47$ gives $74$, which is within the range $71$ to $75$.\n\n6. **Conclude with the final answer**: Therefore, the original number Mary thought of is $\\boxed{\\textbf{(B)}\\ 12}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "On a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. The speed of sound is 1088 feet per second and one mile is 5280 feet. Estimate, to the nearest half-mile, how far Snoopy was from the flash of lightning."}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "2"}, "extra_info": {"index": 1280, "solution": "1. **Calculate the distance the sound traveled**: The problem states that the speed of sound is 1088 feet per second and it took 10 seconds for the sound of thunder to reach Snoopy after the flash of lightning. Therefore, the distance the sound traveled can be calculated by multiplying the speed of sound by the time it took for the sound to reach:\n \\[\n \\text{Distance} = \\text{Speed} \\times \\text{Time} = 1088 \\, \\text{feet/second} \\times 10 \\, \\text{seconds} = 10880 \\, \\text{feet}\n \\]\n\n2. **Convert the distance from feet to miles**: Given that one mile is 5280 feet, we convert the distance from feet to miles:\n \\[\n \\text{Distance in miles} = \\frac{\\text{Distance in feet}}{\\text{Feet per mile}} = \\frac{10880 \\, \\text{feet}}{5280 \\, \\text{feet/mile}} \\approx 2.06 \\, \\text{miles}\n \\]\n\n3. **Round to the nearest half-mile**: The distance in miles (2.06 miles) needs to be rounded to the nearest half-mile. Since 2.06 is closer to 2 than to 2.5, we round it down to 2 miles.\n\n4. **Select the correct answer**: The problem asks us to estimate the distance to the nearest half-mile, and our calculation shows that Snoopy was approximately 2 miles away from the flash of lightning. Therefore, the correct answer is:\n \\[\n \\boxed{\\text{C}}\n \\]"}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "Points $A = (3,9)$, $B = (1,1)$, $C = (5,3)$, and $D=(a,b)$ lie in the first quadrant and are the vertices of quadrilateral $ABCD$. The quadrilateral formed by joining the midpoints of $\\overline{AB}$, $\\overline{BC}$, $\\overline{CD}$, and $\\overline{DA}$ is a square. What is the sum of the coordinates of point $D$?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "10"}, "extra_info": {"index": 1320, "solution": "1. **Identify the midpoints of sides of the quadrilateral**: \n - The midpoint of $\\overline{AB}$ is calculated as:\n \\[\n \\left(\\frac{3+1}{2}, \\frac{9+1}{2}\\right) = (2, 5)\n \\]\n - The midpoint of $\\overline{BC}$ is calculated as:\n \\[\n \\left(\\frac{1+5}{2}, \\frac{1+3}{2}\\right) = (3, 2)\n \\]\n\n2. **Determine the possible coordinates of the other two midpoints**:\n - Since the quadrilateral formed by the midpoints is a square, the sides of the square are either horizontal or vertical, and the diagonals are equal in length. The coordinates of the other two midpoints must be such that they form a square with $(2,5)$ and $(3,2)$.\n - The vector from $(2,5)$ to $(3,2)$ is $(1, -3)$. To form a square, the next vector should be perpendicular to $(1, -3)$ and of the same magnitude. The perpendicular vectors to $(1, -3)$ are $(3, 1)$ and $(-3, -1)$.\n - Adding these vectors to $(3,2)$, we get:\n \\[\n (3,2) + (3,1) = (6,3) \\quad \\text{and} \\quad (3,2) + (-3,-1) = (0,1)\n \\]\n - Since $(0,1)$ would imply the next point is $(0,1)$, which does not form a square with the given points in the first quadrant, we discard this option.\n\n3. **Calculate the coordinates of point $D$**:\n - Since $(6,3)$ is the midpoint of $\\overline{CD}$, we can find $D$ by using the midpoint formula in reverse:\n \\[\n D = (2 \\times 6 - 5, 2 \\times 3 - 3) = (7, 3)\n \\]\n\n4. **Sum the coordinates of point $D$**:\n - The sum of the coordinates of point $D$ is:\n \\[\n 7 + 3 = \\boxed{10}\n \\]"}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "Rectangle $ABCD$ has $AB = 6$ and $BC = 3$. Point $M$ is chosen on side $AB$ so that $\\angle AMD = \\angle CMD$. What is the degree measure of $\\angle AMD$?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "45"}, "extra_info": {"index": 1360, "solution": "1. **Identify the given information and the goal:** We are given a rectangle $ABCD$ with $AB = 6$ and $BC = 3$. We need to find the degree measure of $\\angle AMD$ given that $\\angle AMD = \\angle CMD$.\n\n2. **Assign variables and use the Pythagorean Theorem:** Let $AM = x$. Then, $MB = 6 - x$. Since $D$ and $C$ are on the same horizontal line and $M$ is on line $AB$, we can use the Pythagorean Theorem to find $MD$ and $MC$:\n \\[ MD = \\sqrt{x^2 + 9} \\quad \\text{(since $AD = 3$)} \\]\n \\[ MC = \\sqrt{(6-x)^2 + 9} \\quad \\text{(since $BC = 3$)} \\]\n\n3. **Apply the Law of Cosines in $\\triangle MDC$:** Since $\\angle AMD = \\angle CMD = \\theta$, we have:\n \\[ MD^2 + MC^2 - 2 \\cdot MD \\cdot MC \\cdot \\cos(\\theta) = DC^2 \\]\n \\[ (\\sqrt{x^2 + 9})^2 + (\\sqrt{(6-x)^2 + 9})^2 - 2 \\cdot \\sqrt{x^2 + 9} \\cdot \\sqrt{(6-x)^2 + 9} \\cdot \\cos(\\theta) = 36 \\]\n Simplifying, we get:\n \\[ x^2 + 9 + (6-x)^2 + 9 - 2 \\cdot \\sqrt{x^2 + 9} \\cdot \\sqrt{(6-x)^2 + 9} \\cdot \\cos(\\theta) = 36 \\]\n \\[ 2x^2 - 12x + 54 - 2 \\cdot \\sqrt{(x^2 + 9)((6-x)^2 + 9)} \\cdot \\cos(\\theta) = 36 \\]\n\n4. **Simplify and solve for $x$:** Rearranging and simplifying the equation:\n \\[ 2x^2 - 12x + 18 = 2 \\cdot \\sqrt{(x^2 + 9)((6-x)^2 + 9)} \\cdot \\cos(\\theta) \\]\n \\[ x^2 - 6x + 9 = \\sqrt{(x^2 + 9)((6-x)^2 + 9)} \\cdot \\cos(\\theta) \\]\n Since $\\cos(\\theta)$ must be a real number between -1 and 1, and the equation must hold true, we test possible values of $x$ that simplify the equation. By symmetry and the condition $\\angle AMD = \\angle CMD$, $x = 3$ is a plausible choice, simplifying calculations.\n\n5. **Verify $x = 3$ and calculate $\\theta$:** If $x = 3$, then $AM = MB = 3$, $MD = MC = \\sqrt{3^2 + 9} = \\sqrt{18} = 3\\sqrt{2}$, and $\\triangle AMD$ and $\\triangle CMD$ are congruent. Thus, $\\theta = 45^\\circ$.\n\n6. **Conclusion:** The degree measure of $\\angle AMD$ is $\\boxed{45^\\circ}$. $\\blacksquare$"}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "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"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "B"}, "extra_info": {"index": 1400, "solution": "\nTo solve this problem, we need to understand how the squares are hinged together and how they fold to form a cube. We will consider two different configurations to determine which color is opposite the white face.\n\n#### Configuration 1:\n1. Assume that the green (G) square is the base of the cube.\n2. Since the squares are hinged, the adjacent squares to G can be folded up to form the sides of the cube.\n3. If we place blue (B) at the back and white (W) at the front, the remaining colors will occupy the other faces.\n4. In this configuration, the face opposite to white (W) is blue (B).\n\n#### Configuration 2:\n1. Let yellow (Y) be the top face of the cube.\n2. Fold green (G), orange (O), and white (W) down from the top face.\n3. In this setup, blue (B) will naturally fold to become the back face of the cube.\n4. Thus, the face opposite to white (W) is again blue (B).\n\nIn both configurations, we find that the face opposite the white face is blue. Therefore, the correct answer is:\n\n$\\boxed{\\text{(A)}\\ B}$"}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "Teams $A$ and $B$ are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team $A$ has won $\\frac{2}{3}$ of its games and team $B$ has won $\\frac{5}{8}$ of its games. Also, team $B$ has won $7$ more games and lost $7$ more games than team $A.$ How many games has team $A$ played?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "42"}, "extra_info": {"index": 1440, "solution": "1. **Understanding the problem**: \n - Team $A$ has won $\\frac{2}{3}$ of its games, so it has lost $\\frac{1}{3}$ of its games.\n - Team $B$ has won $\\frac{5}{8}$ of its games, so it has lost $\\frac{3}{8}$ of its games.\n - Team $B$ has won $7$ more games and lost $7$ more games than team $A$.\n\n2. **Setting up equations**:\n - Let $a$ be the total number of games played by team $A$.\n - Then, team $A$ has won $\\frac{2}{3}a$ games and lost $\\frac{1}{3}a$ games.\n - Team $B$ has played $a + 14$ games (since they have played $7$ more wins and $7$ more losses than team $A$).\n - Team $B$ has won $\\frac{5}{8}(a + 14)$ games and lost $\\frac{3}{8}(a + 14)$ games.\n\n3. **Using the information about the number of games won**:\n - From the problem, team $B$ has won $7$ more games than team $A$:\n \\[\n \\frac{5}{8}(a + 14) = \\frac{2}{3}a + 7\n \\]\n - Solving for $a$:\n \\[\n \\frac{5}{8}a + \\frac{5}{8} \\times 14 = \\frac{2}{3}a + 7\n \\]\n \\[\n \\frac{5}{8}a + \\frac{70}{8} = \\frac{2}{3}a + 7\n \\]\n \\[\n \\frac{5}{8}a - \\frac{2}{3}a = 7 - \\frac{70}{8}\n \\]\n \\[\n \\left(\\frac{5}{8} - \\frac{16}{24}\\right)a = 7 - \\frac{35}{4}\n \\]\n \\[\n \\left(\\frac{15}{24} - \\frac{16}{24}\\right)a = \\frac{28 - 35}{4}\n \\]\n \\[\n -\\frac{1}{24}a = -\\frac{7}{4}\n \\]\n \\[\n a = 42\n \\]\n\n4. **Verification**:\n - Team $A$ plays $42$ games, wins $\\frac{2}{3} \\times 42 = 28$ games, and loses $\\frac{1}{3} \\times 42 = 14$ games.\n - Team $B$ plays $42 + 14 = 56$ games, wins $\\frac{5}{8} \\times 56 = 35$ games, and loses $\\frac{3}{8} \\times 56 = 21$ games.\n - The difference in wins and losses between teams $A$ and $B$ is indeed $7$ games, confirming our solution.\n\nThus, the number of games team $A$ has played is $\\boxed{42}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "The area of a rectangle remains unchanged when it is made $2 \\frac{1}{2}$ inches longer and $\\frac{2}{3}$ inch narrower, or when it is made $2 \\frac{1}{2}$ inches shorter and $\\frac{4}{3}$ inch wider. Its area, in square inches, is:"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "20"}, "extra_info": {"index": 1480, "solution": "1. **Set up the equations based on the problem statement:**\n We are given that the area of the rectangle remains unchanged under two transformations:\n - When the rectangle is made $2 \\frac{1}{2}$ inches longer and $\\frac{2}{3}$ inch narrower.\n - When the rectangle is made $2 \\frac{1}{2}$ inches shorter and $\\frac{4}{3}$ inch wider.\n\n Let the original dimensions of the rectangle be $x$ inches (length) and $y$ inches (width). The area of the rectangle is $xy$. According to the problem, we have:\n \\[\n xy = \\left(x + \\frac{5}{2}\\right)\\left(y - \\frac{2}{3}\\right) = \\left(x - \\frac{5}{2}\\right)\\left(y + \\frac{4}{3}\\right)\n \\]\n\n2. **Expand and simplify both equations:**\n - For the first transformation:\n \\[\n \\left(x + \\frac{5}{2}\\right)\\left(y - \\frac{2}{3}\\right) = xy + \\frac{5}{2}y - \\frac{2}{3}x - \\frac{5}{3}\n \\]\n - For the second transformation:\n \\[\n \\left(x - \\frac{5}{2}\\right)\\left(y + \\frac{4}{3}\\right) = xy + \\frac{4}{3}x - \\frac{5}{2}y - \\frac{10}{3}\n \\]\n\n3. **Set the expressions for unchanged area equal to each other:**\n \\[\n xy + \\frac{5}{2}y - \\frac{2}{3}x - \\frac{5}{3} = xy + \\frac{4}{3}x - \\frac{5}{2}y - \\frac{10}{3}\n \\]\n Simplifying this by subtracting $xy$ from both sides and rearranging terms, we get:\n \\[\n \\frac{5}{2}y - \\frac{2}{3}x - \\frac{5}{3} = \\frac{4}{3}x - \\frac{5}{2}y - \\frac{10}{3}\n \\]\n \\[\n 5y - \\frac{4}{3}x + \\frac{5}{2}y = \\frac{2}{3}x + \\frac{10}{3} + \\frac{5}{3}\n \\]\n \\[\n 5y + \\frac{5}{2}y = \\frac{4}{3}x + \\frac{2}{3}x + \\frac{15}{3}\n \\]\n \\[\n \\frac{15}{2}y = 2x + 5\n \\]\n \\[\n 2x = \\frac{15}{2}y - 5\n \\]\n\n4. **Solve for $y$ using the derived equation:**\n \\[\n 5y + \\frac{5}{3} = \\frac{15}{2}y - 5\n \\]\n \\[\n \\frac{15}{2}y - 5y = 5 + \\frac{5}{3}\n \\]\n \\[\n \\frac{5}{2}y = \\frac{20}{3}\n \\]\n \\[\n y = \\frac{20}{3} \\cdot \\frac{2}{5} = \\frac{8}{3}\n \\]\n\n5. **Find $x$ using the value of $y$:**\n \\[\n 2x = \\frac{15}{2} \\cdot \\frac{8}{3} - 5\n \\]\n \\[\n 2x = \\frac{120}{6} - 5 = 20 - 5 = 15\n \\]\n \\[\n x = \\frac{15}{2}\n \\]\n\n6. **Calculate the area of the rectangle:**\n \\[\n xy = \\left(\\frac{15}{2}\\right)\\left(\\frac{8}{3}\\right) = \\frac{120}{6} = 20\n \\]\n\nThus, the area of the rectangle is $\\boxed{20}$, corresponding to choice $\\textbf{(E)}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "The domain of the function $f(x)=\\log_{\\frac{1}{2}}(\\log_4(\\log_{\\frac{1}{4}}(\\log_{16}(\\log_{\\frac{1}{16}}x))))$ is an interval of length $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "271"}, "extra_info": {"index": 1520, "solution": "To find the domain of the function \n\\[ f(x) = \\log_{\\frac{1}{2}}\\left(\\log_4\\left(\\log_{\\frac{1}{4}}\\left(\\log_{16}\\left(\\log_{\\frac{1}{16}}x\\right)\\right)\\right)\\right), \\]\nwe need to ensure that each logarithm in the nested expression is defined and positive, as the logarithm function is only defined for positive arguments.\n\n1. **Innermost Logarithm**: $\\log_{\\frac{1}{16}}x$\n - This is defined for $x > 0$.\n - To be positive, $\\log_{\\frac{1}{16}}x > 0 \\Rightarrow x < 1$ (since the base $\\frac{1}{16} < 1$).\n\n2. **Second Logarithm**: $\\log_{16}\\left(\\log_{\\frac{1}{16}}x\\right)$\n - This requires $\\log_{\\frac{1}{16}}x > 0 \\Rightarrow x < 1$.\n - To be positive, $\\log_{16}\\left(\\log_{\\frac{1}{16}}x\\right) > 0 \\Rightarrow \\log_{\\frac{1}{16}}x > 1 \\Rightarrow x < \\left(\\frac{1}{16}\\right)^1 = \\frac{1}{16}$.\n\n3. **Third Logarithm**: $\\log_{\\frac{1}{4}}\\left(\\log_{16}\\left(\\log_{\\frac{1}{16}}x\\right)\\right)$\n - This requires $\\log_{16}\\left(\\log_{\\frac{1}{16}}x\\right) > 0 \\Rightarrow x < \\frac{1}{16}$.\n - To be positive, $\\log_{\\frac{1}{4}}\\left(\\log_{16}\\left(\\log_{\\frac{1}{16}}x\\right)\\right) > 0 \\Rightarrow \\log_{16}\\left(\\log_{\\frac{1}{16}}x\\right) < 1 \\Rightarrow \\log_{\\frac{1}{16}}x < 16 \\Rightarrow x > \\left(\\frac{1}{16}\\right)^{16} = \\frac{1}{256}$.\n\n4. **Fourth Logarithm**: $\\log_4\\left(\\log_{\\frac{1}{4}}\\left(\\log_{16}\\left(\\log_{\\frac{1}{16}}x\\right)\\right)\\right)$\n - This requires $\\log_{\\frac{1}{4}}\\left(\\log_{16}\\left(\\log_{\\frac{1}{16}}x\\right)\\right) > 0 \\Rightarrow \\frac{1}{256} < x < \\frac{1}{16}$.\n - To be positive, $\\log_4\\left(\\log_{\\frac{1}{4}}\\left(\\log_{16}\\left(\\log_{\\frac{1}{16}}x\\right)\\right)\\right) > 0 \\Rightarrow \\log_{\\frac{1}{4}}\\left(\\log_{16}\\left(\\log_{\\frac{1}{16}}x\\right)\\right) < 1$.\n\n5. **Outermost Logarithm**: $\\log_{\\frac{1}{2}}\\left(\\log_4\\left(\\log_{\\frac{1}{4}}\\left(\\log_{16}\\left(\\log_{\\frac{1}{16}}x\\right)\\right)\\right)\\right)$\n - This requires $\\log_4\\left(\\log_{\\frac{1}{4}}\\left(\\log_{16}\\left(\\log_{\\frac{1}{16}}x\\right)\\right)\\right) > 0 \\Rightarrow \\frac{1}{256} < x < \\frac{1}{16}$.\n\nThus, the domain of $f(x)$ is the interval $\\left(\\frac{1}{256}, \\frac{1}{16}\\right)$. The length of this interval is $\\frac{1}{16} - \\frac{1}{256} = \\frac{16}{256} - \\frac{1}{256} = \\frac{15}{256}$.\n\nThe fraction $\\frac{15}{256}$ is already in simplest form (since 15 and 256 are relatively prime). Therefore, $m = 15$ and $n = 256$, and $m+n = 15 + 256 = 271$.\n\nThus, the final answer is $\\boxed{\\textbf{(C) } 271}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "6"}, "extra_info": {"index": 1560, "solution": "1. **Interpreting the Graph**: The problem states that Suzanna rides her bike at a constant rate, and the graph shows that every 5 minutes, her distance increases by 1 mile.\n\n2. **Total Riding Time**: Suzanna rides for a total of half an hour. Half an hour is equivalent to 30 minutes.\n\n3. **Calculating Distance per 5 Minutes**: Since every 5 minutes corresponds to 1 mile, we need to determine how many 5-minute intervals are in 30 minutes. We calculate this as:\n \\[\n \\frac{30 \\text{ minutes}}{5 \\text{ minutes per mile}} = 6 \\text{ intervals}\n \\]\n\n4. **Total Distance**: Each interval corresponds to 1 mile, so in 6 intervals, Suzanna would have ridden:\n \\[\n 6 \\text{ intervals} \\times 1 \\text{ mile per interval} = 6 \\text{ miles}\n \\]\n\n5. **Conclusion**: Therefore, Suzanna would have ridden a total of 6 miles in half an hour.\n\n \\[\n \\boxed{\\textbf{(C)}\\ 6}\n \\]"}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "The number $2013$ is expressed in the form $2013 = \\frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$,where $a_1 \\ge a_2 \\ge \\cdots \\ge a_m$ and $b_1 \\ge b_2 \\ge \\cdots \\ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "2"}, "extra_info": {"index": 1600, "solution": "1. **Prime Factorization of 2013**: \n The prime factorization of $2013$ is $2013 = 3 \\times 11 \\times 61$.\n\n2. **Form of the Expression**:\n We need to express $2013$ in the form $\\frac{a_1!a_2!\\cdots a_m!}{b_1!b_2!\\cdots b_n!}$, where $a_1 \\ge a_2 \\ge \\cdots \\ge a_m$ and $b_1 \\ge b_2 \\ge \\cdots \\ge b_n$ are positive integers.\n\n3. **Minimizing $a_1 + b_1$**:\n To minimize $a_1 + b_1$, we need to minimize $a_1$ while ensuring that $61$ (the largest prime factor of $2013$) is included in the numerator. Thus, $a_1$ must be at least $61$ to have $61$ as a factor in $a_1!$. Therefore, we set $a_1 = 61$.\n\n4. **Determining $b_1$**:\n Since $b_1$ must be less than $61$ (to minimize $a_1 + b_1$) and we need to cancel out factors in the denominator, we consider the largest prime less than $61$, which is $59$. To ensure that $59$ does not introduce an extra factor in the numerator, we set $b_1 = 59$.\n\n5. **Calculation of $|a_1 - b_1|$**:\n We have $a_1 = 61$ and $b_1 = 59$. Therefore, the absolute difference is $|61 - 59| = 2$.\n\n6. **Verification of the Expression**:\n We can express $2013$ as $\\frac{61! \\cdot 19! \\cdot 11!}{59! \\cdot 20! \\cdot 10!}$. Here, the factorials are chosen such that:\n - $61!$ provides the factor of $61$.\n - $19!$ and $11!$ in the numerator and $20!$ and $10!$ in the denominator are chosen to balance out the factors of $3$ and $11$ in $2013$.\n\n7. **Conclusion**:\n The smallest possible value of $|a_1 - b_1|$ under the given conditions is $\\boxed{\\textbf{(B)}\\ 2}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\\in P$ with integer coordinates is it true that $|4x+3y| \\leq 1000$?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "40"}, "extra_info": {"index": 1640, "solution": "1. **Identify the axis of symmetry**: Given the focus of the parabola $P$ at $(0,0)$ and points $(4,3)$ and $(-4,-3)$ on $P$, we observe that the line connecting these points has a slope of $\\frac{3 - (-3)}{4 - (-4)} = \\frac{6}{8} = \\frac{3}{4}$. This suggests that the axis of symmetry of the parabola makes an angle $\\theta$ with the x-axis such that $\\tan\\theta = \\frac{3}{4}$.\n\n2. **Rotate the coordinate system**: To simplify the equation of the parabola, we rotate the coordinate system by $\\theta$ where $\\cos\\theta = \\frac{4}{5}$ and $\\sin\\theta = \\frac{3}{5}$. The transformation equations between the original coordinates $(x, y)$ and the rotated coordinates $(\\widetilde{x}, \\widetilde{y})$ are:\n \\[\n \\begin{align*}\n x &= \\frac{4}{5} \\widetilde{x} - \\frac{3}{5} \\widetilde{y}, \\\\\n y &= \\frac{3}{5} \\widetilde{x} + \\frac{4}{5} \\widetilde{y}.\n \\end{align*}\n \\]\n\n3. **Equation of the parabola in the rotated system**: In the rotated system, the parabola still has its focus at $(0,0)$ and passes through $(5,0)$ and $(-5,0)$. The directrix is thus $\\widetilde{y} = -5$. The equation of the parabola, using the definition that any point $(\\widetilde{x}, \\widetilde{y})$ on it is equidistant from the focus and the directrix, is:\n \\[\n \\widetilde{x}^2 + \\widetilde{y}^2 = (\\widetilde{y} + 5)^2 \\implies \\widetilde{y} = \\frac{1}{10}(\\widetilde{x}^2 - 25).\n \\]\n\n4. **Transform back and find integer points**: We need to find integer points $(x, y)$ such that $|4x + 3y| \\leq 1000$. From the transformation, we have $|4x + 3y| = 5|\\widetilde{x}|$. Thus, $|\\widetilde{x}| \\leq 200$. Since $\\widetilde{x}$ must be a multiple of 5 (from the transformation equations), let $\\widetilde{x} = 5a$. Then $a$ ranges from $-40$ to $40$, but we exclude $0$, giving $81$ possible values for $a$.\n\n5. **Counting valid $\\widetilde{x}$ values**: We need to count the odd multiples of 5 within $[-195, 195]$. These are given by $\\pm 5, \\pm 15, \\ldots, \\pm 195$, which are $20$ positive and $20$ negative values, totaling $40$ values.\n\nThus, there are $\\boxed{40}$ points $(x, y)$ on the parabola $P$ with integer coordinates such that $|4x + 3y| \\leq 1000$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "The real value of $x$ such that $64^{x-1}$ divided by $4^{x-1}$ equals $256^{2x}$ is:"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "-\\frac{1}{3}"}, "extra_info": {"index": 1680, "solution": "To solve the equation $\\frac{64^{x-1}}{4^{x-1}} = 256^{2x}$, we start by expressing all terms with base 2:\n\n1. **Expressing each term with base 2:**\n - $64 = 2^6$, so $64^{x-1} = (2^6)^{x-1} = 2^{6(x-1)}$\n - $4 = 2^2$, so $4^{x-1} = (2^2)^{x-1} = 2^{2(x-1)}$\n - $256 = 2^8$, so $256^{2x} = (2^8)^{2x} = 2^{16x}$\n\n2. **Rewriting the equation:**\n \\[\n \\frac{2^{6(x-1)}}{2^{2(x-1)}} = 2^{16x}\n \\]\n Simplifying the left-hand side using the laws of exponents:\n \\[\n 2^{6(x-1) - 2(x-1)} = 2^{16x}\n \\]\n \\[\n 2^{4x - 4} = 2^{16x}\n \\]\n\n3. **Equating the exponents:**\n \\[\n 4x - 4 = 16x\n \\]\n Solving for $x$:\n \\[\n 4x - 16x = 4\n \\]\n \\[\n -12x = 4\n \\]\n \\[\n x = -\\frac{4}{12} = -\\frac{1}{3}\n \\]\n\nThus, the value of $x$ that satisfies the original equation is $-\\frac{1}{3}$.\n\n$\\boxed{\\text{B}}$"}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "The sum of the first eighty positive odd integers subtracted from the sum of the first eighty positive even integers is"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "80"}, "extra_info": {"index": 1720, "solution": "1. **Identify the pattern in the sequence of odd and even integers:**\n - The $n$-th odd integer can be represented as $2n - 1$.\n - The $n$-th even integer can be represented as $2n$.\n\n2. **Calculate the difference between the $n$-th even and $n$-th odd integer:**\n - The difference between the $n$-th even and $n$-th odd integer is:\n \\[\n 2n - (2n - 1) = 2n - 2n + 1 = 1.\n \\]\n\n3. **Sum the differences over the first eighty integers:**\n - Since each pair of the $n$-th even and odd integers contributes a difference of $1$, and there are $80$ such pairs, the total difference is:\n \\[\n 80 \\times 1 = 80.\n \\]\n\n4. **Conclusion:**\n - The sum of the first eighty positive odd integers subtracted from the sum of the first eighty positive even integers is $\\boxed{\\textbf{(E)}\\ 80}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "There are $10$ people standing equally spaced around a circle. Each person knows exactly $3$ of the other $9$ people: the $2$ people standing next to her or him, as well as the person directly across the circle. How many ways are there for the $10$ people to split up into $5$ pairs so that the members of each pair know each other?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "12"}, "extra_info": {"index": 1760, "solution": "To solve this problem, we need to consider the relationships each person has with others in the circle. Each person knows exactly three others: the two adjacent people and the one directly across the circle. We need to find how many ways we can pair up these 10 people such that each pair consists of people who know each other.\n\n1. **Label the People**: Let's label the people standing in the circle from $1$ to $10$. Person $1$ knows persons $2$, $10$, and $6$ (directly across in a circle of $10$). Similarly, person $2$ knows persons $1$, $3$, and $7$ (directly across), and so on.\n\n2. **Identify Known Pairs**: From the arrangement, each person knows exactly three others:\n - Each person knows the two adjacent persons.\n - Each person knows one person directly across the circle.\n \n This gives us the following pairs: $(1,2)$, $(2,3)$, $(3,4)$, $(4,5)$, $(5,6)$, $(6,7)$, $(7,8)$, $(8,9)$, $(9,10)$, $(10,1)$ for adjacent pairs, and $(1,6)$, $(2,7)$, $(3,8)$, $(4,9)$, $(5,10)$ for across-the-circle pairs.\n\n3. **Count Valid Pairings**: We need to pair up all $10$ people using these known pairs. We can observe that:\n - Each person is involved in exactly three pairs.\n - We need to ensure that no person is used in more than one pair in our final selection.\n\n To find the number of valid pairings, we can use a combinatorial approach or a case-by-case analysis. However, due to the symmetry and the specific relationships, we can simplify the counting:\n - **Adjacent Pairings**: If we choose to pair people only with their adjacent neighbors, we can start by pairing $(1,2)$, then we must pair $(3,4)$, and so on, leading to $(1,2)$, $(3,4)$, $(5,6)$, $(7,8)$, $(9,10)$.\n - **Across-the-Circle Pairings**: If we choose to pair people across the circle, we can start by pairing $(1,6)$, then we must pair $(2,7)$, and so on, leading to $(1,6)$, $(2,7)$, $(3,8)$, $(4,9)$, $(5,10)$.\n\n These are two distinct ways to pair everyone up. However, there are more combinations possible by mixing adjacent and across-the-circle pairings. The key is to ensure that each person is paired exactly once without overlap.\n\n4. **Calculate Total Pairings**: The total number of ways to achieve this, considering the constraints and symmetry of the circle, is $12$. This includes all combinations of adjacent and across-the-circle pairings that satisfy the condition that each person is paired exactly once.\n\nThus, the total number of ways to pair up the $10$ people into $5$ pairs such that the members of each pair know each other is $\\boxed{\\textbf{(B)}\\ 12}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ in such a way that the sum of the three interior angles at $A$ is $360^{\\circ}$. Thus the three polygons form a new polygon with $A$ as an interior point. What is the largest possible perimeter that this polygon can have?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "21"}, "extra_info": {"index": 1800, "solution": "1. **Identify the Problem Requirements:**\n We need to find three non-overlapping regular polygons, at least two of which are congruent, that meet at a point $A$ such that the sum of their interior angles at $A$ is $360^\\circ$. The sides of each polygon are of length $1$. We aim to maximize the perimeter of the new polygon formed by these three polygons.\n\n2. **Understanding the Interior Angles:**\n The interior angle of a regular polygon with $n$ sides is given by:\n \\[\n \\text{Interior angle} = 180^\\circ \\left(\\frac{n-2}{n}\\right)\n \\]\n For three polygons with sides $a$, $b$, and $c$, the sum of their interior angles at $A$ must be $360^\\circ$:\n \\[\n 180^\\circ \\left(\\frac{a-2}{a}\\right) + 180^\\circ \\left(\\frac{b-2}{b}\\right) + 180^\\circ \\left(\\frac{c-2}{c}\\right) = 360^\\circ\n \\]\n\n3. **Simplifying the Equation:**\n Simplify the equation by dividing through by $180^\\circ$:\n \\[\n \\frac{a-2}{a} + \\frac{b-2}{b} + \\frac{c-2}{c} = 2\n \\]\n Assuming $a = c$ (since at least two polygons are congruent), the equation becomes:\n \\[\n 2\\cdot\\frac{a-2}{a} + \\frac{b-2}{b} = 2\n \\]\n\n4. **Transforming into a Diophantine Equation:**\n Multiply through by $ab$ to clear the denominators:\n \\[\n 2b(a-2) + a(b-2) = 2ab\n \\]\n Simplify and rearrange:\n \\[\n 2ab - 4b + ab - 2a = 2ab \\implies ab - 4b - 2a = 0\n \\]\n Add $8$ to both sides:\n \\[\n (a-4)(b-2) = 8\n \\]\n\n5. **Finding Possible Values for $a$ and $b$:**\n The possible factorizations of $8$ considering $a, b \\geq 3$ are:\n - $(a-4, b-2) = (1, 8)$, yielding $(a, b) = (5, 10)$\n - $(a-4, b-2) = (2, 4)$, yielding $(a, b) = (6, 6)$\n - $(a-4, b-2) = (4, 2)$, yielding $(a, b) = (8, 4)$\n - $(a-4, b-2) = (8, 1)$, yielding $(a, b) = (12, 3)$\n\n6. **Calculating the Perimeters:**\n The perimeter of the resulting polygon is $(a-2) + (b-2) + (c-2) = (a+b+c)-6$. Substituting the values:\n - For $(5, 10, 5)$: Perimeter = $(5+10+5)-6 = 14$\n - For $(6, 6, 6)$: Perimeter = $(6+6+6)-6 = 12$\n - For $(8, 4, 8)$: Perimeter = $(8+4+8)-6 = 14$\n - For $(12, 3, 12)$: Perimeter = $(12+3+12)-6 = 21$\n\n7. **Conclusion:**\n The largest possible perimeter is obtained with the configuration $(12, 3, 12)$, which gives a perimeter of $\\boxed{21}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "Two boys start moving from the same point A on a circular track but in opposite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first meet at the point A again, then the number of times they meet, excluding the start and finish, is"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "13"}, "extra_info": {"index": 1840, "solution": "1. **Define the problem setup**: Two boys start at the same point A on a circular track and move in opposite directions with speeds of 5 ft/s and 9 ft/s respectively. We need to find the number of times they meet excluding the start and finish.\n\n2. **Calculate relative speed**: Since they are moving in opposite directions, their relative speed is the sum of their individual speeds, which is $5 + 9 = 14$ ft/s.\n\n3. **Determine the time to meet at point A again**: Let $d$ be the circumference of the track. The time it takes for them to meet at point A again is the time it takes for one of them to complete a full lap relative to the other. This time is given by $\\frac{d}{14}$ seconds.\n\n4. **Determine the number of meetings in one lap**: To find the number of times they meet (excluding the start and finish), we need to consider how often their paths cross in one complete lap around the track. Since they are moving towards each other, they will meet whenever the distance they collectively cover is a multiple of the circumference $d$. The number of times they meet in one lap is given by the least common multiple (LCM) of their laps per unit time. The boy with speed 5 ft/s covers $\\frac{5}{d}$ laps per second, and the boy with speed 9 ft/s covers $\\frac{9}{d}$ laps per second. The LCM of $\\frac{5}{d}$ and $\\frac{9}{d}$ in terms of laps per second is $\\frac{45}{d}$ (since LCM of 5 and 9 is 45).\n\n5. **Calculate the number of meetings per second**: They meet $\\frac{45}{d}$ times per second. To find the number of meetings in one complete lap around the track, multiply this rate by the time for one complete lap, which is $\\frac{d}{14}$ seconds. Thus, the number of meetings per lap is $\\frac{45}{d} \\times \\frac{d}{14} = \\frac{45}{14}$.\n\n6. **Simplify and find the integer number of meetings**: Simplifying $\\frac{45}{14}$ gives approximately 3.214. However, since they can only meet an integer number of times, we consider the nearest integers. We need to verify if this calculation correctly reflects the number of meetings excluding the start and finish.\n\n7. **Verification and adjustment**: The calculation above gives an approximation. We need to verify by considering the actual number of laps completed by each boy in $\\frac{d}{14}$ seconds. The first boy completes $\\frac{5}{14}d$ laps, and the second boy completes $\\frac{9}{14}d$ laps. The number of times they meet (excluding the start and finish) is the number of times their paths cross, which is indeed 13 times as calculated earlier.\n\n8. **Conclusion**: The boys meet each other 13 times excluding the start and finish.\n\nThus, the correct answer is $\\boxed{\\textbf{(A)}\\ 13}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "What expression describes the set of values of $a$ for which the curves $x^2+y^2=a^2$ and $y=x^2-a$ in the real $xy$-plane intersect at exactly $3$ points?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "a>\\frac{1}{2}"}, "extra_info": {"index": 1880, "solution": "To solve this problem, we need to analyze the conditions under which the circle $x^2 + y^2 = a^2$ and the parabola $y = x^2 - a$ intersect at exactly three points.\n\n1. **Equation Setup**: Substitute $y = x^2 - a$ into the circle equation:\n \\[\n x^2 + (x^2 - a)^2 = a^2\n \\]\n Simplifying, we get:\n \\[\n x^2 + x^4 - 2ax^2 + a^2 = a^2\n \\]\n \\[\n x^4 - (2a - 1)x^2 = 0\n \\]\n Factoring out $x^2$, we have:\n \\[\n x^2(x^2 - (2a - 1)) = 0\n \\]\n\n2. **Root Analysis**: This equation has roots at $x = 0$ and $x = \\pm \\sqrt{2a - 1}$. The root $x = 0$ corresponds to a single intersection point on the parabola and circle. The roots $x = \\pm \\sqrt{2a - 1}$ will provide additional intersection points if $2a - 1 > 0$, i.e., $a > \\frac{1}{2}$.\n\n3. **Intersection Points**:\n - At $x = 0$, substituting into $y = x^2 - a$, we get $y = -a$. The point of intersection is $(0, -a)$.\n - At $x = \\pm \\sqrt{2a - 1}$, substituting into $y = x^2 - a$, we get $y = 2a - 1 - a = a - 1$. The points of intersection are $(\\pm \\sqrt{2a - 1}, a - 1)$.\n\n4. **Counting Points**: \n - If $a \\leq \\frac{1}{2}$, then $2a - 1 \\leq 0$, and there are no real solutions for $x = \\pm \\sqrt{2a - 1}$, resulting in only the point $(0, -a)$.\n - If $a > \\frac{1}{2}$, then $2a - 1 > 0$, and there are real solutions for $x = \\pm \\sqrt{2a - 1}$, resulting in three points: $(0, -a)$ and $(\\pm \\sqrt{2a - 1}, a - 1)$.\n\n5. **Conclusion**: The curves intersect at exactly three points when $a > \\frac{1}{2}$. Therefore, the correct answer is:\n \\[\n \\boxed{\\textbf{(E) }a>\\frac12}\n \\]\n$\\blacksquare$"}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "3127"}, "extra_info": {"index": 1920, "solution": "1. **Identify the smallest primes greater than 50**: The problem specifies that the number must not have any prime factors less than 50. The smallest primes greater than 50 are 53 and 59.\n\n2. **Calculate the product of the smallest primes**: To find the smallest number that meets the criteria, we calculate the product of 53 and 59:\n \\[\n 53 \\times 59 = 3127\n \\]\n\n3. **Check if 3127 is a perfect square**: We need to ensure that 3127 is not a perfect square. The square root of 3127 is approximately 55.92, which is not an integer. Thus, 3127 is not a perfect square.\n\n4. **Verify that 3127 is not prime**: Since 3127 is the product of two primes (53 and 59), it is not a prime number itself.\n\n5. **Check the conditions for other answer choices**:\n - **3133**: This is not the product of the two smallest primes greater than 50.\n - **3137**: Same as above.\n - **3139**: Same as above.\n - **3149**: Same as above.\n\n6. **Conclusion**: Since 3127 is the product of the two smallest primes greater than 50, is not a perfect square, and is not a prime itself, it meets all the conditions set by the problem. None of the other choices meet these criteria as directly and simply as 3127.\n\nThus, the smallest positive integer that is neither prime nor square and has no prime factor less than 50 is $\\boxed{\\textbf{(A)}\\ 3127}$."}} -{"data_source": "DeepScaleR", "prompt": [{"role": "user", "content": "When $15$ is appended to a list of integers, the mean is increased by $2$. When $1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $1$. How many integers were in the original list?"}], "ability": "math", "reward_model": {"style": "rule", "ground_truth": "4"}, "extra_info": {"index": 1960, "solution": "Let $n$ be the number of integers in the original list and $m$ be the mean of these integers. The sum of the integers in the original list is then $mn$.\n\n1. **Appending 15 to the list:**\n When 15 is added to the list, the new list has $n+1$ integers, and the new mean is $m+2$. The sum of the integers in the new list is $mn + 15$. Therefore, the equation for the sum with the new mean is:\n \\[\n (m+2)(n+1) = mn + 15\n \\]\n Expanding and simplifying:\n \\[\n mn + 2n + m + 2 = mn + 15\n \\]\n \\[\n 2n + m + 2 = 15\n \\]\n \\[\n m + 2n = 13 \\quad \\text{(Equation 1)}\n \\]\n\n2. **Appending 1 to the enlarged list:**\n After appending 1 to the list that already includes the 15, the list now has $n+2$ integers, and the new mean is $m+1$. The sum of the integers in this further enlarged list is $mn + 16$. Therefore, the equation for the sum with the new mean is:\n \\[\n (m+1)(n+2) = mn + 16\n \\]\n Expanding and simplifying:\n \\[\n mn + 2m + n + 2 = mn + 16\n \\]\n \\[\n 2m + n + 2 = 16\n \\]\n \\[\n 2m + n = 14 \\quad \\text{(Equation 2)}\n \\]\n\n3. **Solving the system of equations:**\n We have two equations:\n \\[\n \\begin{cases}\n m + 2n = 13 \\\\\n 2m + n = 14\n \\end{cases}\n \\]\n Multiply the first equation by 2:\n \\[\n 2m + 4n = 26\n \\]\n Subtract the second equation from this result:\n \\[\n 2m + 4n - (2m + n) = 26 - 14\n \\]\n \\[\n 3n = 12\n \\]\n \\[\n n = 4\n \\]\n Substitute $n = 4$ back into the first equation:\n \\[\n m + 2 \\times 4 = 13\n \\]\n \\[\n m + 8 = 13\n \\]\n \\[\n m = 5\n \\]\n\nThus, the number of integers in the original list is $\\boxed{\\textbf{(A)}\\ 4}$."}}