data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Call a $3$-digit number geometric if it has $3$ distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "840",
"style": "rule"
} | {
"index": "8300",
"split": "train"
} |
math | [
{
"content": "What is the area of a square inscribed in a semicircle of radius 1, with one of its sides flush with the diameter of the semicircle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{5}",
"style": "rule"
} | {
"index": "8301",
"split": "train"
} |
math | [
{
"content": "What is the period of $y = \\tan \\frac{x}{2}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 \\pi",
"style": "rule"
} | {
"index": "8302",
"split": "train"
} |
math | [
{
"content": "How many of the 512 smallest positive integers written in base 8 use 5 or 6 (or both) as a digit? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "296",
"style": "rule"
} | {
"index": "8303",
"split": "train"
} |
math | [
{
"content": "Let \\( x_{1}, x_{2} \\) be the roots of the equation \\( x^{2} - x - 3 = 0 \\). Find \\(\\left(x_{1}^{5} - 20\\right) \\cdot \\left(3 x_{2}^{4} - 2 x_{2} - 35\\right)\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1063",
"style": "rule"
} | {
"index": "8304",
"split": "train"
} |
math | [
{
"content": "There exist constants $b_1,$ $b_2,$ $b_3,$ $b_4,$ $b_5,$ $b_6,$ $b_7$ such that\n\\[\\cos^7 \\theta = b_1 \\cos \\theta + b_2 \\cos 2 \\theta + b_3 \\cos 3 \\theta + b_4 \\cos 4 \\theta + b_5 \\cos 5 \\theta + b_6 \\cos 6 \\theta + b_7 \\cos 7 \\theta\\]for all angles $\\theta.$ Find $b_1^2 + b_2... | {
"ground_truth": "\\frac{1555}{4096}",
"style": "rule"
} | {
"index": "8305",
"split": "train"
} |
math | [
{
"content": "At Rachelle's school, an A counts $4$ points, a B $3$ points, a C $2$ points, and a D $1$ point. Her GPA in the four classes she is taking is computed as the total sum of points divided by 4. She is certain that she will get A's in both Mathematics and Science and at least a C in each of English a... | {
"ground_truth": "\\frac{11}{24}",
"style": "rule"
} | {
"index": "8306",
"split": "train"
} |
math | [
{
"content": "A parabola has the following optical properties: a light ray passing through the focus of the parabola and reflecting off the parabola will result in a light ray parallel to the axis of symmetry of the parabola; conversely, an incident light ray parallel to the axis of symmetry of the parabola wil... | {
"ground_truth": "\\frac{25}{4}",
"style": "rule"
} | {
"index": "8307",
"split": "train"
} |
math | [
{
"content": "In right triangle $ABC$, $\\angle C=90^{\\circ}$, $AB=9$, $\\cos B=\\frac{2}{3}$, calculate the length of $AC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3\\sqrt{5}",
"style": "rule"
} | {
"index": "8308",
"split": "train"
} |
math | [
{
"content": "How many 9-digit numbers, divisible by 2, can be formed by permuting the digits of the number 231157152? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3360",
"style": "rule"
} | {
"index": "8309",
"split": "train"
} |
math | [
{
"content": "Define $E(a,b,c) = a \\times b^c$. What positive value of $r$ is the solution to the equation $E(r,r,3) = 625$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "8310",
"split": "train"
} |
math | [
{
"content": "What is the sum of all values of $x$ for which $\\sqrt{(x+3)^{2}} = 7$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-6",
"style": "rule"
} | {
"index": "8311",
"split": "train"
} |
math | [
{
"content": "Two faces of a tetrahedron are equilateral triangles with side lengths of one unit, and the other two faces are isosceles right triangles. What is the volume of the tetrahedron? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{2}}{12}",
"style": "rule"
} | {
"index": "8312",
"split": "train"
} |
math | [
{
"content": "Given that the total amount of money originally owned by Moe, Loki, and Nick was $72, and each of Loki, Moe, and Nick gave Ott$\\, 4, determine the fractional part of the group's money that Ott now has. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"index": "8313",
"split": "train"
} |
math | [
{
"content": "Given three distinct points $A$, $B$, $C$ on a straight line, and $\\overrightarrow{OB}=a_{5} \\overrightarrow{OA}+a_{2012} \\overrightarrow{OC}$, find the sum of the first 2016 terms of the arithmetic sequence $\\{a_{n}\\}$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "1008",
"style": "rule"
} | {
"index": "8314",
"split": "train"
} |
math | [
{
"content": "What is the arithmetic mean of all positive two-digit multiples of 7? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "56",
"style": "rule"
} | {
"index": "8315",
"split": "train"
} |
math | [
{
"content": "Find the number of solutions to:\n\\[\\sin x = \\left(\\frac{1}{3}\\right)^x\\]\non the interval $(0, 50\\pi)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "8316",
"split": "train"
} |
math | [
{
"content": "Sofia ran $5$ laps around the $400$-meter track at her school. For each lap, she ran the first $100$ meters at an average speed of $4$ meters per second and the remaining $300$ meters at an average speed of $5$ meters per second. How much time did Sofia take running the $5$ laps? Please output the... | {
"ground_truth": "7 minutes and 5 seconds",
"style": "rule"
} | {
"index": "8317",
"split": "train"
} |
math | [
{
"content": "Let \\( z \\) be a complex number such that \\( |z| = 2 \\). Find the maximum value of\n\\[\n|(z - 2)(z + 2)^2|.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16 \\sqrt{2}",
"style": "rule"
} | {
"index": "8318",
"split": "train"
} |
math | [
{
"content": "Five socks, colored blue, brown, black, red, and purple are in a drawer. In how many different ways can we choose three socks from the drawer if the order of the socks does not matter? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "8319",
"split": "train"
} |
math | [
{
"content": "For testing a certain product, there are 6 different genuine items and 4 different defective items. The test continues until all defective items are identified. If all defective items are exactly identified by the 5th test, how many possible testing methods are there? Please output the final answe... | {
"ground_truth": "576",
"style": "rule"
} | {
"index": "8320",
"split": "train"
} |
math | [
{
"content": "Let $(x^2+1)(2x+1)^9 = a_0 + a_1(x+2) + a_2(x+2)^2 + \\ldots + a_{11}(x+2)^{11}$, then calculate the value of $a_0 + a_1 + a_2 + \\ldots + a_{11}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "8321",
"split": "train"
} |
math | [
{
"content": "Evaluate $(2-w)(2-w^2)\\cdots(2-w^{10})$ where $w=e^{2\\pi i/11}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2047",
"style": "rule"
} | {
"index": "8322",
"split": "train"
} |
math | [
{
"content": "Triangle $DEF$ is similar to triangle $ABC$. If $DE=6$, $EF=12$, and $BC=18$ units, what is the length of segment $AB$?\n\n[asy]draw((0,0)--(7,0));\ndraw((0,0)--(0,4));\ndraw((0,4)--(7,0));\nlabel(\"E\",(0,0),W);\nlabel(\"F\",(7,0),E);\nlabel(\"D\",(0,4),W);\ndraw((15,0)--(25.5,0));\ndraw((15,0)-... | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "8323",
"split": "train"
} |
math | [
{
"content": "Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $$\n\\text A. \\ \\ 21 \\qquad \\text B. \\ \\ 60 \\qquad \\text C. \\ \\ 119 \\qquad \\text D. \\ \\ 180 \\qquad \\text E. \\ \\ 231\n$$ ... | {
"ground_truth": "119",
"style": "rule"
} | {
"index": "8324",
"split": "train"
} |
math | [
{
"content": "Find the equation of the directrix of the parabola $x = -\\frac{1}{6} y^2.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "x = \\frac{3}{2}",
"style": "rule"
} | {
"index": "8325",
"split": "train"
} |
math | [
{
"content": "A neighbor bought a certain quantity of beef at two shillings a pound, and the same quantity of sausages at eighteenpence a pound. If she had divided the same money equally between beef and sausages, she would have gained two pounds in the total weight. Determine the exact amount of money she spen... | {
"ground_truth": "168",
"style": "rule"
} | {
"index": "8326",
"split": "train"
} |
math | [
{
"content": "Inside an angle of $60^{\\circ}$, there is a point located at distances $\\sqrt{7}$ and $2 \\sqrt{7}$ from the sides of the angle. Find the distance of this point from the vertex of the angle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{14 \\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "8327",
"split": "train"
} |
math | [
{
"content": "The sum of 81 consecutive integers is $9^5$. What is their median? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "729",
"style": "rule"
} | {
"index": "8328",
"split": "train"
} |
math | [
{
"content": "In a circle, a chord of length 10 cm is drawn. A tangent to the circle is drawn through one end of the chord, and a secant parallel to the tangent is drawn through the other end. The internal segment of the secant is 12 cm. Find the radius of the circle. Please output the final answer within \\box... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "8329",
"split": "train"
} |
math | [
{
"content": "Violet has one-half of the money she needs to buy her mother a necklace. After her sister gives her $\\$30$, she has three-quarters of the amount she needs. How much will Violet's father give her? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$30",
"style": "rule"
} | {
"index": "8330",
"split": "train"
} |
math | [
{
"content": "Alice, Bob, and Charlie are playing a game with 6 cards numbered 1 through 6. Each player is dealt 2 cards uniformly at random. On each player's turn, they play one of their cards, and the winner is the person who plays the median of the three cards played. Charlie goes last, so Alice and Bob deci... | {
"ground_truth": "\\frac{2}{15}",
"style": "rule"
} | {
"index": "8331",
"split": "train"
} |
math | [
{
"content": "If the inequality $x^2+ax+1\\geqslant 0$ holds for all $x\\in\\left(0, \\frac{1}{2}\\right]$, then the minimum value of $a$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{5}{2}",
"style": "rule"
} | {
"index": "8332",
"split": "train"
} |
math | [
{
"content": "There are two boxes, A and B, each containing four cards labeled with the numbers 1, 2, 3, and 4. One card is drawn from each box, and each card is equally likely to be chosen;\n(I) Find the probability that the product of the numbers on the two cards drawn is divisible by 3;\n(II) Suppose that Xi... | {
"ground_truth": "\\frac{8}{9}",
"style": "rule"
} | {
"index": "8333",
"split": "train"
} |
math | [
{
"content": "Given $f(x) = \\log_a x$, its inverse function is $g(x)$. \n(1) Solve the equation for $x$: $f(x-1) = f(a-x) - f(5-x)$; \n(2) Let $F(x) = (2m-1)g(x) + \\left( \\frac{1}{m} - \\frac{1}{2} \\right)g(-x)$. If $F(x)$ has a minimum value, find the expression for $h(m)$; \n(3) Find the maximum value ... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "8334",
"split": "train"
} |
math | [
{
"content": "A computer software retailer has 1200 copies of a new software package to sell. Given that half of them will sell right away at the original price, two-thirds of the remainder will sell later when the price is reduced by 40%, and the remaining copies will sell in a clearance sale at 75% off the or... | {
"ground_truth": "80.90",
"style": "rule"
} | {
"index": "8335",
"split": "train"
} |
math | [
{
"content": "How many sequences of 5 positive integers $(a, b, c, d, e)$ satisfy $a b c d e \\leq a+b+c+d+e \\leq 10$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "116",
"style": "rule"
} | {
"index": "8336",
"split": "train"
} |
math | [
{
"content": "Alice, Bob, and Charlie each pick a 2-digit number at random. What is the probability that all of their numbers' tens' digits are different from each others' tens' digits and all of their numbers' ones digits are different from each others' ones' digits? Please output the final answer within \\box... | {
"ground_truth": "\\frac{112}{225}",
"style": "rule"
} | {
"index": "8337",
"split": "train"
} |
math | [
{
"content": "What is the coefficient of $x^3$ in the expansion of $$(x+2\\sqrt3)^7?$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5040",
"style": "rule"
} | {
"index": "8338",
"split": "train"
} |
math | [
{
"content": "Zach has twelve identical-looking chocolate eggs. Exactly three of the eggs contain a special prize inside. Zach randomly gives three of the twelve eggs to each of Vince, Wendy, Xin, and Yolanda. What is the probability that only one child will receive an egg that contains a special prize (that is... | {
"ground_truth": "1/55",
"style": "rule"
} | {
"index": "8339",
"split": "train"
} |
math | [
{
"content": "The numbers \\(1, 2, 3, \\ldots, 10\\) are written in some order around a circle. Peter computed the sums of all 10 triples of neighboring numbers and wrote the smallest of these sums on the board. What is the maximum possible number that could have been written on the board? Please output the fin... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "8340",
"split": "train"
} |
math | [
{
"content": "Calculate: \\(\\left(2 \\frac{2}{3} \\times\\left(\\frac{1}{3}-\\frac{1}{11}\\right) \\div\\left(\\frac{1}{11}+\\frac{1}{5}\\right)\\right) \\div \\frac{8}{27} = 7 \\underline{1}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7 \\frac{1}{2}",
"style": "rule"
} | {
"index": "8341",
"split": "train"
} |
math | [
{
"content": "Given that \\( p \\) is a prime number and \\( r \\) is the remainder when \\( p \\) is divided by 210, if \\( r \\) is a composite number that can be expressed as the sum of two perfect squares, find \\( r \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "169",
"style": "rule"
} | {
"index": "8342",
"split": "train"
} |
math | [
{
"content": "The teacher wrote a two-digit number on the board. Each of the three boys made two statements.\n\n- Andrey: \"This number ends in the digit 6\" and \"This number is divisible by 7.\"\n- Borya: \"This number is greater than 26\" and \"This number ends in the digit 8.\"\n- Sasha: \"This number is di... | {
"ground_truth": "91",
"style": "rule"
} | {
"index": "8343",
"split": "train"
} |
math | [
{
"content": "Let $P$ be the product of the first $100$ positive odd integers. Find the largest integer $k$ such that $P$ is divisible by $3^k .$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "8344",
"split": "train"
} |
math | [
{
"content": "The distance traveled by the center \\( P \\) of a circle with radius 1 as it rolls inside a triangle with side lengths 6, 8, and 10, returning to its initial position. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "8345",
"split": "train"
} |
math | [
{
"content": "Calvin has a bag containing 50 red balls, 50 blue balls, and 30 yellow balls. Given that after pulling out 65 balls at random (without replacement), he has pulled out 5 more red balls than blue balls, what is the probability that the next ball he pulls out is red? Please output the final answer wi... | {
"ground_truth": "\\frac{9}{26}",
"style": "rule"
} | {
"index": "8346",
"split": "train"
} |
math | [
{
"content": "Let $x$ and $y$ be two positive real numbers such that $x + y = 35.$ Enter the ordered pair $(x,y)$ for which $x^5 y^2$ is maximized. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(25,10)",
"style": "rule"
} | {
"index": "8347",
"split": "train"
} |
math | [
{
"content": "Find the largest integer $n$ satisfying the following conditions:\n(i) $n^2$ can be expressed as the difference of two consecutive cubes;\n(ii) $2n + 79$ is a perfect square. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "181",
"style": "rule"
} | {
"index": "8348",
"split": "train"
} |
math | [
{
"content": "On May 19, 2022, Jinyan Airport in Dazhou was officially opened, covering a total area of 2940 acres, with an estimated investment of approximately 26.62 billion yuan. Express this amount in scientific notation. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2.662 \\times 10^{9}",
"style": "rule"
} | {
"index": "8349",
"split": "train"
} |
math | [
{
"content": "How many 9 step paths are there from $E$ to $G$ which pass through $F$?[asy]size(4cm,4cm);int w=6;int h=5;int i;pen p=fontsize(9);for (i=0; i<h; ++i){draw((0,i) -- (w-1,i));}for (i=0; i<w; ++i){draw((i, 0)--(i,h-1));}label(\"G\", (w-1,0), SE, p);label(\"E\", (0,h-1), NW, p);label(\"F\", (3,3), NE,... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "8350",
"split": "train"
} |
math | [
{
"content": "If\n\\[(1 + \\tan 1^\\circ)(1 + \\tan 2^\\circ)(1 + \\tan 3^\\circ) \\dotsm (1 + \\tan 45^\\circ) = 2^n,\\]then find $n.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "8351",
"split": "train"
} |
math | [
{
"content": "Consider the paths from \\((0,0)\\) to \\((6,3)\\) that only take steps of unit length up and right. Compute the sum of the areas bounded by the path, the \\(x\\)-axis, and the line \\(x=6\\) over all such paths. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "756",
"style": "rule"
} | {
"index": "8352",
"split": "train"
} |
math | [
{
"content": "The number of children in the families $A$, $B$, $C$, $D$, and $E$ are shown in the table below:\n\n| | $A$ | $B$ | $C$ | $D$ | $E$ |\n|---------|-----|-----|-----|-----|-----|\n| Boys | $0$ | $1$ | $0$ | $1$ | $1$ |\n| Girls | $0$ | $0$ | $1$ | $1$ | $2$ |\n\n$(1)$ If a girl is rando... | {
"ground_truth": "\\frac{6}{5}",
"style": "rule"
} | {
"index": "8353",
"split": "train"
} |
math | [
{
"content": "Let $V=\\{1, \\ldots, 8\\}$. How many permutations $\\sigma: V \\rightarrow V$ are automorphisms of some tree? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30212",
"style": "rule"
} | {
"index": "8354",
"split": "train"
} |
math | [
{
"content": "What is the area, in square units, of a triangle whose vertices are at $(4, -1)$, $(10, 3)$ and $(4, 5)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "8355",
"split": "train"
} |
math | [
{
"content": "Given sets \\( A = \\{1, 2, 3\\} \\) and \\( B = \\{4, 5, 6, 7\\} \\), with a mapping \\( f: A \\rightarrow B \\), if for \\( x \\in A \\), the expression \\( x + f(x) + x \\cdot f(x) \\) is odd, then the number of such mappings \\( f \\) is: Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "8356",
"split": "train"
} |
math | [
{
"content": "Let the solution set of the inequality about $x$, $|x-2| < a$ ($a \\in \\mathbb{R}$), be $A$, and $\\frac{3}{2} \\in A$, $-\\frac{1}{2} \\notin A$. \n(1) For any $x \\in \\mathbb{R}$, the inequality $|x-1| + |x-3| \\geq a^2 + a$ always holds true, and $a \\in \\mathbb{N}$. Find the value of $a$. ... | {
"ground_truth": "\\frac{1 + 2\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "8357",
"split": "train"
} |
math | [
{
"content": "Given that \\( O \\) is the circumcenter of \\(\\triangle ABC\\), where \\(|AB|=2\\), \\(|AC|=1\\), and \\(\\angle BAC = \\frac{2}{3} \\pi\\). Let \\(\\overrightarrow{AB} = \\mathbf{a}\\) and \\(\\overrightarrow{AC} = \\mathbf{b}\\). If \\(\\overrightarrow{AO} = \\lambda_1 \\mathbf{a} + \\lambda_2... | {
"ground_truth": "\\frac{13}{6}",
"style": "rule"
} | {
"index": "8358",
"split": "train"
} |
math | [
{
"content": "We have 10 springs, each originally $0.5 \\mathrm{~m}$ long with a spring constant of $200 \\mathrm{~N}/\\mathrm{m}$. A mass of $2 \\mathrm{~kg}$ is hung on each spring, and the springs, along with the masses, are hung in a series. What is the length of the resulting chain? (Neglect the mass of th... | {
"ground_truth": "10.39",
"style": "rule"
} | {
"index": "8359",
"split": "train"
} |
math | [
{
"content": "Consider the integer \\[M = 8 + 88 + 888 + 8888 + \\cdots + \\underbrace{88\\ldots 88}_\\text{150 digits}.\\] Find the sum of the digits of $M$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "8360",
"split": "train"
} |
math | [
{
"content": "Vertex E of equilateral triangle ∆ABE is inside square ABCD. F is the intersection point of diagonal BD and line segment AE. If AB has length √(1+√3), calculate the area of ∆ABF. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "8361",
"split": "train"
} |
math | [
{
"content": "The base of a pyramid is an isosceles triangle with a base of 6 and a height of 9. Each lateral edge is 13. Find the volume of the pyramid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "8362",
"split": "train"
} |
math | [
{
"content": "By how many zeros does the number 2012! end? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "501",
"style": "rule"
} | {
"index": "8363",
"split": "train"
} |
math | [
{
"content": "Suppose that $p$ is prime and $1007_p+306_p+113_p+125_p+6_p=142_p+271_p+360_p$. How many possible values of $p$ are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "8364",
"split": "train"
} |
math | [
{
"content": "In the book \"Nine Chapters on the Mathematical Art,\" a tetrahedron with all four faces being right-angled triangles is called a \"biēnào.\" Given that tetrahedron $ABCD$ is a \"biēnào,\" $AB\\bot $ plane $BCD$, $BC\\bot CD$, and $AB=\\frac{1}{2}BC=\\frac{1}{3}CD$. If the volume of this tetrahedr... | {
"ground_truth": "14\\pi",
"style": "rule"
} | {
"index": "8365",
"split": "train"
} |
math | [
{
"content": "There are 7 parking spaces in a row in a parking lot, and now 4 cars need to be parked. If 3 empty spaces need to be together, calculate the number of different parking methods. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "8366",
"split": "train"
} |
math | [
{
"content": "Thirty teams play a tournament where every team plays every other team exactly once. Each game results either in a win or a loss with a $50\\%$ chance for either outcome. Calculate the probability that all teams win a unique number of games. Express your answer as $\\frac{m}{n}$ where $m$ and $n$ ... | {
"ground_truth": "409",
"style": "rule"
} | {
"index": "8367",
"split": "train"
} |
math | [
{
"content": "The quartic (4th-degree) polynomial P(x) satisfies $P(1)=0$ and attains its maximum value of $3$ at both $x=2$ and $x=3$ . Compute $P(5)$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-24",
"style": "rule"
} | {
"index": "8368",
"split": "train"
} |
math | [
{
"content": "Find the number of ways to partition a set of $10$ elements, $S = \\{1, 2, 3, . . . , 10\\}$ into two parts; that is, the number of unordered pairs $\\{P, Q\\}$ such that $P \\cup Q = S$ and $P \\cap Q = \\emptyset$ . Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "511",
"style": "rule"
} | {
"index": "8369",
"split": "train"
} |
math | [
{
"content": "In a kindergarten's junior group, there are two identical small Christmas trees and five children. The teachers want to divide the children into two circles around each tree, with at least one child in each circle. The teachers distinguish the children but do not distinguish the trees: two such di... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "8370",
"split": "train"
} |
math | [
{
"content": "A slant asymptote of the rational expression $y = \\frac{2x^2 + 3x - 7}{x-3}$ is the line approached by the equation as $x$ approaches $\\infty$ or $-\\infty$. If this line is of the form $y = mx + b$, find $m+b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "8371",
"split": "train"
} |
math | [
{
"content": "Let \\( a, b \\in \\{2, 3, \\cdots, 8\\} \\). Find the maximum value of \\(\\frac{a}{10b + a} + \\frac{b}{10a + b}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{89}{287}",
"style": "rule"
} | {
"index": "8372",
"split": "train"
} |
math | [
{
"content": "A tank with a mass of $m_{1}=2$ kg rests on a cart with a mass of $m_{2}=10$ kg, which is accelerated with an acceleration of $a=5 \\, \\text{m/s}^2$. The coefficient of friction between the tank and the cart is $\\mu=0.6$. Determine the frictional force acting on the tank from the cart. Please ou... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "8373",
"split": "train"
} |
math | [
{
"content": "Calculate $1010101_2 + 111000_2$ and write your answer in base $10$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "141",
"style": "rule"
} | {
"index": "8374",
"split": "train"
} |
math | [
{
"content": "Given the lines $x-y-1=0$ and $x-y-5=0$ both intersect circle $C$ creating chords of length 10, find the area of circle $C$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27\\pi",
"style": "rule"
} | {
"index": "8375",
"split": "train"
} |
math | [
{
"content": "Xiaoming's home is 30 minutes away from school by subway and 50 minutes by bus. One day, due to some reasons, Xiaoming first took the subway and then transferred to the bus, taking 40 minutes to reach the school. The transfer process took 6 minutes. Calculate the time Xiaoming spent on the bus tha... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "8376",
"split": "train"
} |
math | [
{
"content": "What is the units digit of $31^3+13^3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "8377",
"split": "train"
} |
math | [
{
"content": "Given $tan({θ+\\frac{π}{{12}}})=2$, find $sin({\\frac{π}{3}-2θ})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{5}",
"style": "rule"
} | {
"index": "8378",
"split": "train"
} |
math | [
{
"content": "Suppose that $a$ is a multiple of 4 and $b$ is a multiple of 8. Which of the following statements are true?\n\nA. $a+b$ must be even.\nB. $a+b$ must be a multiple of 4.\nC. $a+b$ must be a multiple of 8.\nD. $a+b$ cannot be a multiple of 8.\n\nAnswer by listing your choices in alphabetical order, ... | {
"ground_truth": "\\text{A,B}",
"style": "rule"
} | {
"index": "8379",
"split": "train"
} |
math | [
{
"content": "Given that $\\alpha$ is an angle in the third quadrant, the function $f(\\alpha)$ is defined as:\n$$f(\\alpha) = \\frac {\\sin(\\alpha - \\frac {\\pi}{2}) \\cdot \\cos( \\frac {3\\pi}{2} + \\alpha) \\cdot \\tan(\\pi - \\alpha)}{\\tan(-\\alpha - \\pi) \\cdot \\sin(-\\alpha - \\pi)}.$$ \n1. Simplif... | {
"ground_truth": "\\frac{6\\sqrt{2} - 1}{10}",
"style": "rule"
} | {
"index": "8380",
"split": "train"
} |
math | [
{
"content": "There are 12 sprinters in an international track event final, including 5 Americans. Gold, silver, and bronze medals are awarded to the first, second, and third place finishers respectively. Determine the number of ways the medals can be awarded if no more than two Americans receive medals. Please... | {
"ground_truth": "1260",
"style": "rule"
} | {
"index": "8381",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=- \\sqrt {2}\\sin (2x+ \\frac {\\pi}{4})+6\\sin x\\cos x-2\\cos ^{2}x+1$, where $x\\in\\mathbb{R}$. \n(I) Find the smallest positive period of $f(x)$. \n(II) Find the maximum and minimum values of $f(x)$ in the interval $\\left[0, \\frac {\\pi}{2}\\right]$. Please output... | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "8382",
"split": "train"
} |
math | [
{
"content": "Let $(x, y)$ be a point in the cartesian plane, $x, y>0$. Find a formula in terms of $x$ and $y$ for the minimal area of a right triangle with hypotenuse passing through $(x, y)$ and legs contained in the $x$ and $y$ axes. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 x y",
"style": "rule"
} | {
"index": "8383",
"split": "train"
} |
math | [
{
"content": "In an acute triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. Given $a=4$, $b=5$, and the area of $\\triangle ABC$ is $5\\sqrt{3}$, find the values of $c$ and $\\sin A$. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "\\frac{2\\sqrt{7}}{7}",
"style": "rule"
} | {
"index": "8384",
"split": "train"
} |
math | [
{
"content": "My friend and I decided to dine at a new Mediterranean restaurant. I am a vegan and also avoid soy. The restaurant features six dishes that are vegan-friendly. These vegan dishes constitute one-fourth of the entire menu. However, five of these vegan dishes contain soy. What fraction of the menu ca... | {
"ground_truth": "\\frac{1}{24}",
"style": "rule"
} | {
"index": "8385",
"split": "train"
} |
math | [
{
"content": "Positive integers are put into the following table.\n\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}\n\\hline\n1 & 3 & 6 & 10 & 15 & 21 & 28 & 36 & & \\\\ \\hline\n2 & 5 & 9 & 14 & 20 & 27 & 35 & 44 & & \\\\ \\hline\n4 & 8 & 13 & 19 & 26 & 34 & 43 & 53 & & \\\\ \\hline\n7 & 12 & 18 & ... | {
"ground_truth": "(62, 2)",
"style": "rule"
} | {
"index": "8386",
"split": "train"
} |
math | [
{
"content": "How many $3$-digit positive integers have digits whose product equals $36$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "8387",
"split": "train"
} |
math | [
{
"content": "Let the integer part and decimal part of $2+\\sqrt{6}$ be $x$ and $y$ respectively. Find the values of $x$, $y$, and the square root of $x-1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "8388",
"split": "train"
} |
math | [
{
"content": "Given that 10 spots for the finals of the 2009 National High School Mathematics Competition are to be distributed to four different schools in a certain district, with the requirement that one school gets 1 spot, another gets 2 spots, a third gets 3 spots, and the last one gets 4 spots, calculate ... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "8389",
"split": "train"
} |
math | [
{
"content": "Supposed that $x$ and $y$ are nonzero real numbers such that $\\frac{3x+y}{x-3y}=-2$. What is the value of $\\frac{x+3y}{3x-y}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "8390",
"split": "train"
} |
math | [
{
"content": "For what is the smallest $n$ such that there exist $n$ numbers within the interval $(-1, 1)$ whose sum is 0 and the sum of their squares is 42? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "44",
"style": "rule"
} | {
"index": "8391",
"split": "train"
} |
math | [
{
"content": "Let $F_1 = (0,1)$ and $F_ 2= (4,1).$ Then the set of points $P$ such that\n\\[PF_1 + PF_2 = 6\\]form an ellipse. The equation of this ellipse can be written as\n\\[\\frac{(x - h)^2}{a^2} + \\frac{(y - k)^2}{b^2} = 1.\\]Find $h + k + a + b.$ Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "6 + \\sqrt{5}",
"style": "rule"
} | {
"index": "8392",
"split": "train"
} |
math | [
{
"content": "In a triangle $ABC$, points $M$ and $N$ are on sides $AB$ and $AC$, respectively, such that $MB = BC = CN$. Let $R$ and $r$ denote the circumradius and the inradius of the triangle $ABC$, respectively. Express the ratio $MN/BC$ in terms of $R$ and $r$. Please output the final answer within \\boxed... | {
"ground_truth": "\\sqrt{1 - \\frac{2r}{R}}",
"style": "rule"
} | {
"index": "8393",
"split": "train"
} |
math | [
{
"content": "Triangle \\( ABC \\) has a right angle at \\( B \\). Point \\( D \\) lies on side \\( BC \\) such that \\( 3 \\angle BAD = \\angle BAC \\). Given \\( AC = 2 \\) and \\( CD = 1 \\), compute \\( BD \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{8}",
"style": "rule"
} | {
"index": "8394",
"split": "train"
} |
math | [
{
"content": "How many positive integers, not exceeding 100, are multiples of 2 or 3 but not 4? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "8395",
"split": "train"
} |
math | [
{
"content": "Let $f:X\\rightarrow X$, where $X=\\{1,2,\\ldots ,100\\}$, be a function satisfying:\n1) $f(x)\\neq x$ for all $x=1,2,\\ldots,100$;\n2) for any subset $A$ of $X$ such that $|A|=40$, we have $A\\cap f(A)\\neq\\emptyset$.\nFind the minimum $k$ such that for any such function $f$, there exist a subse... | {
"ground_truth": "69",
"style": "rule"
} | {
"index": "8396",
"split": "train"
} |
math | [
{
"content": "Find all prime numbers $ p,q,r$, such that $ \\frac{p}{q}\\minus{}\\frac{4}{r\\plus{}1}\\equal{}1$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(7, 3, 2), (5, 3, 5), (3, 2, 7)",
"style": "rule"
} | {
"index": "8397",
"split": "train"
} |
math | [
{
"content": "What is the smallest four-digit number that is divisible by $45$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1008",
"style": "rule"
} | {
"index": "8398",
"split": "train"
} |
math | [
{
"content": "Calculate \\[\\left|\\left(2 + 2i\\right)^6 + 3\\right|\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "515",
"style": "rule"
} | {
"index": "8399",
"split": "train"
} |
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