data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "An object moves $8$ cm in a straight line from $A$ to $B$, turns at an angle $\\alpha$, measured in radians and chosen at random from the interval $(0,\\pi)$, and moves $5$ cm in a straight line to $C$. What is the probability that $AC < 7$? Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 6100,
"solution": "1. **Setup the coordinate system and define points**:\n - Let $B = (0, 0)$, $A = (0, -8)$.\n - The possible points of $C$ create a semi-circle of radius $5$ centered at $B$.\n\n2. **Define the circles**:\n - The circle centered at $B$ with radius $5$ is descri... |
deepscaler | [
{
"role": "user",
"content": "Consider a two-digit number in base 12 represented by $AB_{12}$, where $A$ and $B$ are duodecimal digits (0 to 11), and $A \\neq B$. When this number is reversed to $BA_{12}$, under what condition is a particular prime number a necessary factor of the difference $AB_{12} - BA_{... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 6101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $0 < α < \\frac{π}{2}$ and $\\frac{π}{2} < β < π$, with $\\cos(α + \\frac{π}{4}) = \\frac{1}{3}$ and $\\cos(\\frac{π}{4} - \\frac{β}{2}) = \\frac{\\sqrt{3}}{3}$,\n\n1. Find the value of $\\cos β$;\n2. Find the value of $\\cos(2α + β)$. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 6102,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression: $\\left(\\cos \\frac{2 \\pi}{3}+i \\sin \\frac{2 \\pi}{3}\\right)^{6} + \\left(\\cos \\frac{4 \\pi}{3}+i \\sin \\frac{4 \\pi}{3}\\right)^{6}$ using DeMoivre's Theorem. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 6103,
"solution": "We apply DeMoivre's Theorem to simplify the first expression to $\\left(\\cos 6 \\cdot \\frac{2 \\pi}{3}+\\sin 6 \\cdot \\frac{2 \\pi}{3}\\right)=(\\cos 4 \\pi+\\sin 4 \\pi)=1+0=1$. Similarly, we simplify the second expression to $\\left(\\cos 6 \\cdot \\frac{4 \\pi... |
deepscaler | [
{
"role": "user",
"content": "Find the monic quadratic polynomial, in $x,$ with real coefficients, which has $1 - i$ as a root. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^2 - 2x + 2"
} | {
"split": "train",
"index": 6104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A class has $25$ students. The teacher wants to stock $N$ candies, hold the Olympics and give away all $N$ candies for success in it (those who solve equally tasks should get equally, those who solve less get less, including, possibly, zero candies). At what smallest $N... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 6105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(a, b, c, d\\) be positive integers such that \\(a^5 =\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "757"
} | {
"split": "train",
"index": 6106,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sin 510^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 6107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A couch cost 62,500 rubles. Its price was adjusted by 20% each month, either increasing or decreasing. Over six months, the price increased three times and decreased three times (the order of these changes is unknown). Can you uniquely determine the price of the couch after six... | math | {
"style": "rule",
"ground_truth": "55296"
} | {
"split": "train",
"index": 6108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the point in the plane $3x - 4y + 5z = 30$ that is closest to the point $(1,2,3).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{11}{5}, \\frac{2}{5}, 5 \\right)"
} | {
"split": "train",
"index": 6109,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Ming has a probability of passing the English Level 4 test of $\\frac{3}{4}$. If he takes the test 3 times consecutively, what is the probability that he passes exactly one of these tests? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{64}"
} | {
"split": "train",
"index": 6110,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ of $\\triangle ABC$ respectively, and $2\\sin \\frac{7\\pi }{6}\\sin (\\frac{\\pi }{6}+C)+ \\cos C=-\\frac{1}{2}$.\n\n(1) Find $C$;\n\n(2) If $c=2\\sqrt{3}$, find the maximum area of $\\triang... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"split": "train",
"index": 6111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ordered triples of integers $(a,b,c)$ satisfy $|a+b|+c = 19$ and $ab+|c| = 97$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 6112,
"solution": "1. **Symmetry and Reduction of Cases**:\n Without loss of generality (WLOG), assume $a \\geq 0$ and $a \\geq b$. This assumption is valid because if $(a, b, c)$ is a solution, then $(-a, -b, c)$, $(b, a, c)$, and $(-b, -a, c)$ are also solutions due to the symmetr... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\frac 1n - \\frac{1}{n+1} < \\frac{1}{10}$, what is the least possible positive integer value of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 6113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $\\sqrt{1 + \\sqrt{2y-3}} = \\sqrt{6}$; find $y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 6114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Seven people arrive to dinner, but the circular table only seats six. If two seatings such that one is a rotation of the other are considered the same, then in how many different ways can we choose six people and seat them at the table? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "840"
} | {
"split": "train",
"index": 6115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "12 balls numbered 1 through 12 are placed in a bin. In how many ways can 3 balls be drawn, in order, from the bin, if each ball remains outside the bin after it is drawn? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1320"
} | {
"split": "train",
"index": 6116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\\\(a > 0\\\\), \\\\(b > 0\\\\), and \\\\(a+4b={{(ab)}^{\\\\frac{3}{2}}}\\\\).\n\n\\\\((\\\\)I\\\\()\\\\) Find the minimum value of \\\\(a^{2}+16b^{2}\\\\);\n\n\\\\((\\\\)II\\\\()\\\\) Determine whether there exist \\\\(a\\\\) and \\\\(b\\\\) such that \\\\(a+3b=6\\\\), ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 6117,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An event occurs many years ago. It occurs periodically in $x$ consecutive years, then there is a break of $y$ consecutive years. We know that the event occured in $1964$ , $1986$ , $1996$ , $2008$ and it didn't occur in $1976$ , $1993$ , $2006$ , $2013$ . What is t... | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"split": "train",
"index": 6118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways can the integers from -7 to 7 be arranged in a sequence such that the absolute value of the numbers in the sequence is nondecreasing? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 6119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two distinct natural numbers end in 5 zeros and have exactly 42 divisors. Find their sum. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "700000"
} | {
"split": "train",
"index": 6120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two people, A and B, alternately pick distinct numbers from the set \\(\\{0, 1, \\cdots, 81\\}\\). A picks first, and each person picks one number per turn. After all 82 numbers are picked, let \\(S_A\\) and \\(S_B\\) be the sums of the numbers chosen by A and B, respectively. ... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 6121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given points $A(-2,-2)$, $B(-2,6)$, $C(4,-2)$, and point $P$ moving on the circle $x^{2}+y^{2}=4$, find the maximum value of $|PA|^{2}+|PB|^{2}+|PC|^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 6122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three young married couples were captured by cannibals. Before eating the tourists, the cannibals decided to weigh them. The total weight of all six people was not an integer, but the combined weight of all the wives was exactly 171 kg. Leon weighed the same as his wife, Victor... | math | {
"style": "rule",
"ground_truth": "85.5"
} | {
"split": "train",
"index": 6123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A group consists of 5 girls and 5 boys. The members of the same gender do not know each other, and there are no two girls who have two mutual boy acquaintances. What is the maximum number of acquaintanceships within the group? How does the answer change if the group consists of... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 6124,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Steph Curry is playing the following game and he wins if he has exactly 5 points at some time. Flip a fair coin. If heads, shoot a 3-point shot which is worth 3 points. If tails, shoot a free throw which is worth 1 point. He makes \\frac{1}{2} of his 3-point shots and all of hi... | math | {
"style": "rule",
"ground_truth": "\\frac{140}{243}"
} | {
"split": "train",
"index": 6125,
"solution": "If he misses the shot, then the state of the game is the same as before he flipped the coin. Since the probability of making a free throw is \\frac{1}{2} and the probability of making a 3-point shot is \\frac{1}{4}. Therefore, given that he earns some point, the pro... |
deepscaler | [
{
"role": "user",
"content": "Given that $P$ is a moving point on the line $l: x-2y+4=0$, two tangents are drawn from point $P$ to the circle $C: x^{2}+y^{2}-2x=0$, with tangents intersecting at points $A$ and $B$. Find the minimum area of the circumcircle of quadrilateral $PACB$. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{4}"
} | {
"split": "train",
"index": 6126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $(1 + i)^4.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 6127,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Select two distinct integers, $m$ and $n$, randomly from the set $\\{3,4,5,6,7,8,9,10,11,12\\}$. What is the probability that $3mn - m - n$ is a multiple of $5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"split": "train",
"index": 6128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $a_{1}, a_{2}, \\ldots, a_{100}$ are positive real numbers such that $$a_{k}=\\frac{k a_{k-1}}{a_{k-1}-(k-1)}$$ for $k=2,3, \\ldots, 100$. Given that $a_{20}=a_{23}$, compute $a_{100}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "215"
} | {
"split": "train",
"index": 6129,
"solution": "If we cross multiply, we obtain $a_{n} a_{n-1}=n a_{n-1}+(n-1) a_{n}$, which we can rearrange and factor as $\\left(a_{n}-n\\right)\\left(a_{n-1}-(n-1)\\right)=n(n-1)$. Let $b_{n}=a_{n}-n$. Then, $b_{n} b_{n-1}=n(n-1)$. If we let $b_{1}=t$, then we have by induction... |
deepscaler | [
{
"role": "user",
"content": "A circle is circumscribed around $ABCD$ as follows: [asy]\npair pA, pB, pC, pD, pO;\npO = (0, 0);\npA = pO + dir(-40);\npB = pO + dir(40);\npC = pO + dir(130);\npD = pO + dir(190);\ndraw(pA--pB--pC--pA);\ndraw(pA--pD--pC--pA);\nlabel(\"$A$\", pA, SE);\nlabel(\"$B$\", pB, NE);\n... | math | {
"style": "rule",
"ground_truth": "110^\\circ"
} | {
"split": "train",
"index": 6130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives ... | math | {
"style": "rule",
"ground_truth": "471"
} | {
"split": "train",
"index": 6131,
"solution": "1. **Understanding the Problem:**\n - Ang, Ben, and Jasmin each have 5 blocks of different colors and there are 5 boxes.\n - They independently place one block in each box.\n - We need to find the probability that at least one box ends up with 3 blocks of the ... |
deepscaler | [
{
"role": "user",
"content": "Given 60 feet of fencing, what is the greatest possible number of square feet in the area of a pen, if the pen is designed as a rectangle subdivided evenly into two square areas? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "450"
} | {
"split": "train",
"index": 6132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Andy is attempting to solve the quadratic equation $$64x^2 - 96x - 48 = 0$$ by completing the square. He aims to rewrite the equation in the form $$(ax + b)^2 = c,$$ where \\(a\\), \\(b\\), and \\(c\\) are integers and \\(a > 0\\). Determine the value of \\(a + b + c\\). Let's ... | math | {
"style": "rule",
"ground_truth": "86"
} | {
"split": "train",
"index": 6133,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A quadrilateral is divided into 1000 triangles. What is the maximum number of distinct points that can be the vertices of these triangles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1002"
} | {
"split": "train",
"index": 6134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $B$ and $C$ lie on $\\overline{AD}$. The length of $\\overline{AB}$ is $4$ times the length of $\\overline{BD}$, and the length of $\\overline{AC}$ is $9$ times the length of $\\overline{CD}$. The length of $\\overline{BC}$ is what fraction of the length of $\\overline{A... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10}"
} | {
"split": "train",
"index": 6135,
"solution": "1. **Setting up the relationships:**\n - Given that $\\overline{AB} = 4\\overline{BD}$, we can express $\\overline{AB}$ and $\\overline{BD}$ in terms of a common variable, say $x$. Thus, $\\overline{BD} = x$ and $\\overline{AB} = 4x$.\n - Since $\\overline{AB} +... |
deepscaler | [
{
"role": "user",
"content": "Given that 20% of the birds are geese, 40% are swans, 10% are herons, and 20% are ducks, and the remaining are pigeons, calculate the percentage of the birds that are not herons and are ducks. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22.22\\%"
} | {
"split": "train",
"index": 6136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Victor has $3$ piles of $3$ cards each. He draws all of the cards, but cannot draw a card until all the cards above it have been drawn. (For example, for his first card, Victor must draw the top card from one of the $3$ piles.) In how many orders can Victor draw the cards... | math | {
"style": "rule",
"ground_truth": "1680"
} | {
"split": "train",
"index": 6137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $A^2_{}$, where $A^{}_{}$ is the sum of the absolute values of all roots of the following equation:\n$x = \\sqrt{19} + \\frac{91}{{\\sqrt{19}+\\frac{91}{{\\sqrt{19}+\\frac{91}{{\\sqrt{19}+\\frac{91}{{\\sqrt{19}+\\frac{91}{x}}}}}}}}}$ Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "383"
} | {
"split": "train",
"index": 6138,
"solution": "$x=\\sqrt{19}+\\underbrace{\\frac{91}{\\sqrt{19}+\\frac{91}{\\sqrt{19}+\\frac{91}{\\sqrt{19}+\\frac{91}{\\sqrt{19}+\\frac{91}{x}}}}}}_{x}$\n$x=\\sqrt{19}+\\frac{91}{x}$\n$x^2=x\\sqrt{19}+91$\n$x^2-x\\sqrt{19}-91 = 0$\n$\\left.\\begin{array}{l}x_1=\\frac{\\sqrt{19}+\... |
deepscaler | [
{
"role": "user",
"content": "In the table, 8 is located in the 3rd row and 2nd column, 2017 is located in the $a$th row and $b$th column. Find $a - b = \\quad$\n\\begin{tabular}{|c|c|c|}\n\\hline 1 & 4 & 5 \\\\\n\\hline 2 & 3 & 6 \\\\\n\\hline 9 & 8 & 7 \\\\\n\\hline 10 & 13 & 14 \\\\\n\\hline 11 & 12 & 15... | math | {
"style": "rule",
"ground_truth": "672"
} | {
"split": "train",
"index": 6139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the 4 digits 0, 1, 2, 3, select 3 digits to form a three-digit number without repetition. How many of these three-digit numbers are divisible by 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 6140,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the units digit of the sum of the squares of the first 2007 odd, positive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 6141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that the lines \\(l_1\\) and \\(l_2\\) are parallel, and on \\(l_1\\) and \\(l_2\\) there are 10 points \\(A_1, A_2, \\dots, A_{10}\\) and \\(B_1, B_2, \\dots, B_{10}\\), respectively. The line segments \\(A_1 B_1, A_2 B_2, \\dots, A_{10} B_{10}\\) can divide the strip-... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 6142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If 7:30 a.m. was 16 minutes ago, how many minutes will it be until 8:00 a.m.? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 6143,
"solution": "If 7:30 a.m. was 16 minutes ago, then it is currently $30+16=46$ minutes after 7:00 a.m., or 7:46 a.m. Since 8:00 a.m. is 60 minutes after 7:00 a.m., then it will be 8:00 a.m. in $60-46=14$ minutes."
} |
deepscaler | [
{
"role": "user",
"content": "Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace and the second card is a King? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{4}{663}"
} | {
"split": "train",
"index": 6144,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In equilateral triangle $A B C$, a circle \\omega is drawn such that it is tangent to all three sides of the triangle. A line is drawn from $A$ to point $D$ on segment $B C$ such that $A D$ intersects \\omega at points $E$ and $F$. If $E F=4$ and $A B=8$, determine $|A E-F D|$.... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{\\sqrt{5}} \\text{ OR } \\frac{4 \\sqrt{5}}{5}"
} | {
"split": "train",
"index": 6145,
"solution": "Without loss of generality, $A, E, F, D$ lie in that order. Let $x=A E, y=D F$. By power of a point, $x(x+4)=4^{2} \\Longrightarrow x=2 \\sqrt{5}-2$, and $y(y+4)=(x+4+y)^{2}-(4 \\sqrt{3})^{2} \\Longrightarrow y=\\frac{48-(x+4)^{2}}{2(x+2)}=\\frac{12-(1+\\sqrt{5})^{2... |
deepscaler | [
{
"role": "user",
"content": "Using the digits 1, 2, 3, 4, 5, how many even three-digit numbers less than 500 can be formed if each digit can be used more than once? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 6146,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of $12$, $15$, and $10$ minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the aver... | math | {
"style": "rule",
"ground_truth": "\\frac{88}{7}"
} | {
"split": "train",
"index": 6147,
"solution": "1. **Define the number of students in each grade:**\n Let the number of fifth graders be $f$. According to the problem, there are twice as many fourth graders as fifth graders, and twice as many third graders as fourth graders. Therefore, the number of fourth grad... |
deepscaler | [
{
"role": "user",
"content": "What is the largest positive integer that is not the sum of a positive integral multiple of $36$ and a positive composite integer that is not a multiple of $4$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "147"
} | {
"split": "train",
"index": 6148,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many $n=2,3,4,\\ldots,99,100$ is the base-$n$ number $215216_n$ a multiple of $5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 6149,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 3 different balls to be placed into 5 different boxes, with at most one ball per box. How many methods are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 6150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a four-digit number $\\overline{A B C D}$ that satisfies the following properties: $\\overline{A B}$, $\\overline{B C}$, and $\\overline{C D}$ are all perfect squares (a perfect square is a number that can be expressed as the square of an integer, such as $4 = 2^2$ and $8... | math | {
"style": "rule",
"ground_truth": "13462"
} | {
"split": "train",
"index": 6151,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of two positive integers is 50 and their difference is 12. What is the value of the positive difference of the squares of the integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 6152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the quadratic polynomial $p(x)$ such that $p(-3) = 10,$ $p(0) = 1,$ and $p(2) = 5.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^2 + 1"
} | {
"split": "train",
"index": 6153,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the side length of the largest cube contained in the region $\\{(x, y, z): x^{2}+y^{2}+z^{2} \\leq 25 \\text{ and } x \\geq 0\\}$ of three-dimensional space. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5 \\sqrt{6}}{3}"
} | {
"split": "train",
"index": 6154,
"solution": "The given region is a hemisphere, so the largest cube that can fit inside it has one face centered at the origin and the four vertices of the opposite face on the spherical surface. Let the side length of this cube be $s$. Then, the radius of the circle is the hypot... |
deepscaler | [
{
"role": "user",
"content": " $A,B,C$ are points in the plane such that $\\angle ABC=90^\\circ$ . Circles with diameters $BA$ and $BC$ meet at $D$ . If $BA=20$ and $BC=21$ , then the length of segment $BD$ can be expressed in the form $\\frac{m}{n}$ where $m$ and $n$ are relatively prime ... | math | {
"style": "rule",
"ground_truth": "449"
} | {
"split": "train",
"index": 6155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diameters of two pulleys with parallel axes are 80 mm and 200 mm, respectively, and they are connected by a belt that is 1500 mm long. What is the distance between the axes of the pulleys if the belt is tight (with millimeter precision)? Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "527"
} | {
"split": "train",
"index": 6156,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest digit that is never found in the units place of an even number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 6157,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $c$ be a complex number. Suppose there exist distinct complex numbers $r$, $s$, and $t$ such that for every complex number $z$, we have\n\\[\n (z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct).\n\\]Compute the number of distinct possible values of $c$. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 6158,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $F(x, y, z) = x \\times y^z$. What positive value of $s$ is the solution to the equation $F(s, s, 2) = 1024$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8 \\cdot \\sqrt[3]{2}"
} | {
"split": "train",
"index": 6159,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of two baskets contains white and black balls such that the total number of balls in both baskets is 25. One ball is randomly drawn from each basket. It is known that the probability that both drawn balls are white is 0.54. Find the probability that both drawn balls are bl... | math | {
"style": "rule",
"ground_truth": "0.04"
} | {
"split": "train",
"index": 6160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = \\frac{x + 6}{x}.$ The sequence $(f_n)$ of functions is defined by $f_1 = f$ and\n\\[f_n = f \\circ f_{n - 1}\\]for all $n \\ge 2.$ For example,\n\\[f_2(x) = f(f(x)) = \\frac{\\frac{x + 6}{x} + 6}{\\frac{x + 6}{x}} = \\frac{7x + 6}{x + 6}\\]and\n\\[f_3(x) = f(f_2(... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 6161,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all natural numbers with the property that, when the first digit is moved to the end, the resulting number is $\\dfrac{7}{2}$ times the original one. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "153846"
} | {
"split": "train",
"index": 6162,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "1. Simplify and evaluate the expression: $\\log_{\\frac{1}{3}} \\sqrt{27} + \\lg 25 + \\lg 4 + 7^{-\\log_{7} 2} + (-0.98)^0$\n2. Given a point $P(\\sqrt{2}, -\\sqrt{6})$ on the terminal side of angle $\\alpha$, evaluate: $\\frac{\\cos \\left( \\frac{\\pi}{2} + \\alpha \\right) ... | math | {
"style": "rule",
"ground_truth": "\\frac{-\\sqrt{3} - 1}{3}"
} | {
"split": "train",
"index": 6163,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the arithmetic mean of 14, 22 and 36? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 6164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive three-digit integers have a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by 11? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 6165,
"solution": "1. **Formulate the congruences:** We are given that a number $n$ leaves a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by 11. We can express these conditions using congruences:\n \\[\n n \\equiv 2 \\pmod... |
deepscaler | [
{
"role": "user",
"content": "Calculate \\(7 \\cdot 9\\frac{2}{5}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65\\frac{4}{5}"
} | {
"split": "train",
"index": 6166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two spinners are divided into fifths and sixths, respectively. If each of these spinners is spun once, what is the probability that the product of the results of the two spins will be an even number? The numbers on the first spinner are 3, 5, 6, 7, and 9. The numbers on the sec... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{15}"
} | {
"split": "train",
"index": 6167,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $21^{-1} \\equiv 17 \\pmod{53}$, find $32^{-1} \\pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 6168,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $1$ to $15$ in clockwise order. Committee rules state that a M... | math | {
"style": "rule",
"ground_truth": "346"
} | {
"split": "train",
"index": 6169,
"solution": "The arrangements must follow the pattern MVEMVE....... where each MVE consists of some Martians followed by some Venusians followed by some Earthlings, for $1, 2, 3, 4,$ or $5$ MVE's. If there are $k$ MVE's, then by stars and bars, there are ${4 \\choose k-1}$ choic... |
deepscaler | [
{
"role": "user",
"content": "Given the wins of five baseball teams as displayed in the graph, identify the number of games won by the Patriots. The clues are as follows:\n1. The Tigers won more than the Eagles.\n2. The Patriots won more than the Cubs, but fewer than the Mounties.\n3. The Cubs won more than... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 6170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The formula expressing the relationship between $x$ and $y$ in the table is:\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline x & 2 & 3 & 4 & 5 & 6\\\n\\hline y & 0 & 2 & 6 & 12 & 20\\\n\\hline\n\\end{tabular} Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "y = x^{2}-3x+2"
} | {
"split": "train",
"index": 6171,
"solution": "To find the correct formula that expresses the relationship between $x$ and $y$ in the given table, we will substitute each pair $(x, y)$ into the formulas provided in the options and check for consistency.\n\nThe table provided is:\n\\[\\begin{tabular}{|c|c|c|c|c|c... |
deepscaler | [
{
"role": "user",
"content": "If the system of equations $$\\begin{aligned} & |x+y|=99 \\\\ & |x-y|=c \\end{aligned}$$ has exactly two real solutions $(x, y)$, find the value of $c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 6172,
"solution": "If $c<0$, there are no solutions. If $c>0$ then we have four possible systems of linear equations given by $x+y= \\pm 99, x-y= \\pm c$, giving four solutions $(x, y)$. So we must have $c=0$, and then we do get two solutions ( $x=y$, so they must both equal $\\pm 99 ... |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(\\frac{1}{2},\\frac{1}{2}\\sin x+\\frac{\\sqrt{3}}{2}\\cos x)$, $\\overrightarrow{b}=(1,y)$, if $\\overrightarrow{a}\\parallel\\overrightarrow{b}$, let the function be $y=f(x)$.\n\n$(1)$ Find the smallest positive period of the function $y=f(... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 6173,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x)$ is a real valued function of the real variable $x$, and $f(x)$ is not identically zero, and for all $a$ and $b$, $f(a+b)+f(a-b)=2f(a)+2f(b)$, then for all $x$ and $y$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(-x)=f(x)"
} | {
"split": "train",
"index": 6174,
"solution": "1. **Finding $f(0)$:**\n Let $a = b = 0$ in the given functional equation:\n \\[\n f(a+b) + f(a-b) = 2f(a) + 2f(b)\n \\]\n Substituting $a = b = 0$:\n \\[\n f(0+0) + f(0-0) = 2f(0) + 2f(0)\n \\]\n Simplifying, we get:\n \\[\n 2f(0) = 4f(0)\n ... |
deepscaler | [
{
"role": "user",
"content": "What is the maximum number of possible points of intersection of a circle and a triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 6175,
"solution": "To determine the maximum number of possible points of intersection between a circle and a triangle, we need to consider the interaction between the circle and each side of the triangle. \n\n1. **Intersection of a Circle and a Line Segment**: \n A circle can inters... |
deepscaler | [
{
"role": "user",
"content": "A function \\( g(x) \\) is defined for all real numbers \\( x \\). For all non-zero values \\( x \\), we have\n\\[ 3g(x) + g\\left(\\frac{1}{x}\\right) = 7x + 6. \\]\nLet \\( T \\) denote the sum of all of the values of \\( x \\) for which \\( g(x) = 2005 \\). Compute the integ... | math | {
"style": "rule",
"ground_truth": "763"
} | {
"split": "train",
"index": 6176,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the complex plane, $z,$ $z^2,$ $z^3$ represent, in some order, three vertices of a non-degenerate equilateral triangle. Determine all possible perimeters of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"split": "train",
"index": 6177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the points A(1, -2) and B(5, 6) are equidistant from the line $l: ax + y + 1 = 0$, determine the value(s) of the real number $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 6178,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many non-empty subsets $S$ of $\\{1, 2, 3, \\ldots, 10\\}$ satisfy the following two conditions?\n1. No two consecutive integers belong to $S$.\n2. If $S$ contains $k$ elements, then $S$ contains no number less than $k$. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 6179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum and minimum values of the function $f(x)=1+x-x^{2}$ in the interval $[-2,4]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-11"
} | {
"split": "train",
"index": 6180,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\cos 135^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 6181,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence is defined by $a_{0}=1$ and $a_{n}=2^{a_{n-1}}$ for $n \\geq 1$. What is the last digit (in base 10) of $a_{15}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 6182,
"solution": "6. Certainly $a_{13} \\geq 2$, so $a_{14}$ is divisible by $2^{2}=4$. Writing $a_{14}=4 k$, we have $a_{15}=2^{4 k}=16^{k}$. But every power of 16 ends in 6, so this is the answer."
} |
deepscaler | [
{
"role": "user",
"content": "Jane Doe invested some amount of money into a savings account and mutual funds. The total amount she invested was \\$320,000. If she invested 6 times as much in mutual funds as she did in the savings account, what was her total investment in mutual funds? Let's think step by st... | math | {
"style": "rule",
"ground_truth": "274,285.74"
} | {
"split": "train",
"index": 6183,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the circle \\\\({{x}^{2}}+{{y}^{2}}-6x-2y+3=0\\\\) and the distance from its center to the line \\\\(x+ay-1=0\\\\) is \\\\(1,\\\\) then the real number \\\\(a=\\\\)_______ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\dfrac{3}{4}"
} | {
"split": "train",
"index": 6184,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a tetrahedron \\( A B C D \\) with side lengths \\( A B = 41 \\), \\( A C = 7 \\), \\( A D = 18 \\), \\( B C = 36 \\), \\( B D = 27 \\), and \\( C D = 13 \\), let \\( d \\) be the distance between the midpoints of edges \\( A B \\) and \\( C D \\). Find the value of \\( d... | math | {
"style": "rule",
"ground_truth": "137"
} | {
"split": "train",
"index": 6185,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of the function $f(x)=3x^{6}-2x^{5}+x^{3}+1$ at $x=2$ using the Horner's method (also known as the Qin Jiushao algorithm) to determine the value of $v_{4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 6186,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate \\begin{align*}\n\\left(c^c-c(c-1)^c\\right)^c\n\\end{align*} when $c=3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 6187,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "20 players are playing in a Super Smash Bros. Melee tournament. They are ranked $1-20$, and player $n$ will always beat player $m$ if $n<m$. Out of all possible tournaments where each player plays 18 distinct other players exactly once, one is chosen uniformly at random. Find t... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 6188,
"solution": "Consider instead the complement of the tournament: The 10 possible matches that are not played. In order for each player to play 18 games in the tournament, each must appear once in these 10 unplayed matches. Players $n$ and $n+1$ will win the same number of games i... |
deepscaler | [
{
"role": "user",
"content": "For each vertex of the triangle \\(ABC\\), the angle between the altitude and the angle bisector drawn from that vertex was determined. It turned out that these angles at vertices \\(A\\) and \\(B\\) are equal to each other and are less than the angle at vertex \\(C\\). What is... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 6189,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the base ten equivalent of $12345_{6}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1865"
} | {
"split": "train",
"index": 6190,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line joining $(2,3)$ and $(5,1)$ divides the square shown into two parts. What fraction of the area of the square is above this line? The square has vertices at $(2,1)$, $(5,1)$, $(5,4)$, and $(2,4)$. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 6191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given: Circle $C$ passes through point $D(0,1)$, $E(-2,1)$, $F(-1,\\sqrt{2})$, $P$ is any point on the line $l_{1}: y=x-2$, and the line $l_{2}: y=x+1$ intersects circle $C$ at points $A$ and $B$. <br/>$(Ⅰ)$ Find the equation of circle $C$;<br/>$(Ⅱ)$ Find the minimum value of $... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 6192,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular $2015$ -simplex $\\mathcal P$ has $2016$ vertices in $2015$ -dimensional space such that the distances between every pair of vertices are equal. Let $S$ be the set of points contained inside $\\mathcal P$ that are closer to its center than any of its vertice... | math | {
"style": "rule",
"ground_truth": "520"
} | {
"split": "train",
"index": 6193,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What percent of square $EFGH$ is shaded? All angles in the diagram are right angles. [asy]\nimport graph;\ndefaultpen(linewidth(0.7));\n\nxaxis(0,8,Ticks(1.0,NoZero));\nyaxis(0,8,Ticks(1.0,NoZero));\n\nfill((0,0)--(2,0)--(2,2)--(0,2)--cycle);\nfill((3,0)--(5,0)--(5,5)--(0,5)--(... | math | {
"style": "rule",
"ground_truth": "67\\%"
} | {
"split": "train",
"index": 6194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It can be shown that for any positive integer $n,$\n\\[\\begin{pmatrix} 1 & 1 \\\\ 1 & 0 \\end{pmatrix}^n = \\begin{pmatrix} F_{n + 1} & F_n \\\\ F_n & F_{n - 1} \\end{pmatrix},\\]where $F_n$ denotes the $n$th Fibonacci number.\n\nCompute $F_{784} F_{786} - F_{785}^2.$ Let's th... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 6195,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $f(x)$ is a rational function such that $3f\\left(\\frac{1}{x}\\right) + \\frac{2f(x)}{x} = x^{2}$ for $x \\neq 0$. Find $f(-2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{67}{20}"
} | {
"split": "train",
"index": 6196,
"solution": "Let $x = \\frac{-1}{2}$. Then $$\\begin{align*} 3f(-2) + \\frac{2f\\left(\\frac{-1}{2}\\right)}{\\frac{-1}{2}} = & \\frac{1}{4} \\\\ \\Rightarrow 3f(-2) - 4f\\left(\\frac{-1}{2}\\right) & = \\frac{1}{4} \\tag{1} \\end{align*}$$ Let $x = -2$. Then $$\\begin{align*} &... |
deepscaler | [
{
"role": "user",
"content": "For $a$ a positive real number, let $x_{1}, x_{2}, x_{3}$ be the roots of the equation $x^{3}-a x^{2}+a x-a=0$. Determine the smallest possible value of $x_{1}^{3}+x_{2}^{3}+x_{3}^{3}-3 x_{1} x_{2} x_{3}$. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 6197,
"solution": "Note that $x_{1}+x_{2}+x_{3}=x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{1}=a$. Then $$\\begin{aligned} & x_{1}^{3}+x_{2}^{3}+x_{3}^{3}-3 x_{1} x_{2} x_{3}=\\left(x_{1}+x_{2}+x_{3}\\right)\\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-\\left(x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{1}\\right)\\... |
deepscaler | [
{
"role": "user",
"content": "The list $p, q, r, s$ consists of four consecutive integers listed in increasing order. If $p + s = 109$, what is the value of $q + r$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "109"
} | {
"split": "train",
"index": 6198,
"solution": "Since $p, q, r, s$ is a list of consecutive integers in increasing order, then $q$ is 1 more than $p$ and $r$ is 1 less than $s$. This means that $q + r = (p + 1) + (s - 1) = p + s = 109$. Therefore, $q + r = 109$."
} |
deepscaler | [
{
"role": "user",
"content": "The value of x that satisfies the equation \\( x^{x^x} = 2 \\) is calculated. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 6199,
"solution": ""
} |
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