data_source
stringclasses 1
value | prompt
listlengths 1
1
| ability
stringclasses 1
value | reward_model
dict | extra_info
dict |
|---|---|---|---|---|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor how many values of $k$ is $12^{12}$ the least common multiple of the positive integers $6^6$ , $8^8$ , and $k$ ?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "25",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "c91159e1-d9f5-4929-a5fb-3a6148448b66"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor a given constant $b > 10,$ there are two possible triangles $ABC$ satisfying $AB = 10,$ $AC = b,$ and $\\sin B = \\frac{3}{5}.$ Find the positive difference between the lengths of side $\\overline{BC}$ in these two triangles.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "16",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "5d1cd258-c060-487a-b88a-ed7dbd87e5f3"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n对实数 x，记 \\{x\\}=x-[x]. 令S=\\int_0^{2024}\\min\\left(\\left\\{\\df{x}{8}\\right\\},\\left\\{\\df{x}{4}\\right\\}\\right)\\dd x,则 [S]=__________.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "759",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "a5284710-432a-4b08-82d8-0d199634ea33"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor how many integers $n$ between $1$ and $50$, inclusive, is \n$$\\frac{(n^2-1)!}{(n!)^n}$$ \nan integer? (Recall that $0! = 1$.)\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "34",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f58d6d54-f9ee-423a-a46c-d546e4b77d17"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the maximum possible value for the sum of the squares of the roots of \\(x^4 + ax^3 + bx^2 + cx + d\\) where \\(a\\), \\(b\\), \\(c\\), and \\(d\\) are \\(2\\), \\(0\\), \\(1\\), and \\(7\\) in some order?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "49",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "87125f5a-b1d7-47d0-a222-a85846c2f856"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $T = \\{ 1, 2, 3, \\dots, 14, 15 \\}$. A subset $S$ of $T$ is defined as \\textit{handy} if the sum of all the elements of $S$ is a multiple of $5$. For example, the empty set is handy (because its sum is $0$) and $T$ itself is handy (because its sum is $120$). Compute the number of handy subsets of $T$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "6560",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "48b61bd5-8374-4873-a137-428c63de1890"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $P_0(x) = x^3 + 313x^2 - 77x - 8$. For integers $n \\ge 1$, define $P_n(x) = P_{n - 1}(x - n)$. What is the coefficient of $x$ in $P_{20}(x)$?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "763",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "2f2d0eb8-381d-4e81-a7ee-a47919ccb8a6"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn a 14-team baseball league, each team played every other team 10 times. At the end of the season, the number of games won by each team differed from those won by the team immediately following it by the same amount. Determine the greatest number of games the last-place team could have won, assuming that no ties were allowed.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "52",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "a41236bf-947c-490f-8d31-cb846fc3921f"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the largest integer that is a divisor of \\[\n(n+1)(n+3)(n+5)(n+7)(n+9)\n\\]for all positive even integers $n$?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "15",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "e36d307f-6ea9-4355-bf38-68b941855cbd"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n方程 $\\cos \\frac{X}{4}=\\cos x$ 的通解是 $\\qquad$ ，在 $(0,24 \\pi)$ 内不相同的解有多少个？请给出不相同解的个数。\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "28918e62-4ccc-4b2a-a9aa-4769597798f2"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA cyclic quadrilateral $ABCD$ has side lengths $AB = 3$, $BC = AD = 5$, and $CD = 8$. The radius of its circumcircle can be expressed in the form $\\frac{a\\sqrt{b}}{c}$, where $a$, $b$, $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a + b + c$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "31f55c2c-9c49-435d-a64c-82ac6d4dcb0a"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLines $l$ and $m$ are perpendicular. Line $l$ partitions a convex polygon into two parts of equal area, and partitions the projection of the polygon onto $m$ into two line segments of length $a$ and $b$ respectively. Determine the maximum value of $\\left\\lfloor \\frac{1000a}{b} \\right\\rfloor$. (The floor notation $\\lfloor x \\rfloor$ denotes largest integer not exceeding $x$)\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2141",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "882397ee-9dbc-4b0b-a393-bc3a493b11d4"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the least positive four-digit solution to the following system of congruences. \\begin{align*}\n7x &\\equiv 21 \\pmod{14} \\\\\n2x+13 &\\equiv 16 \\pmod{9} \\\\\n-2x+1 &\\equiv x \\pmod{25} \\\\\n\\end{align*}\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1167",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "fee4b142-f611-42c9-af7f-12f7e8ec993e"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the largest integer $n$ such that $2^n + 65$ is equal to the square of an integer? Find the value of $n$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "c687b324-9900-4baf-9f82-20be38390bb9"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThree vertices of a cube are $P=(7,12,10)$, $Q=(8,8,1)$, and $R=(11,3,9)$. What is the side length of the cube?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b3fcf40f-af44-4ea8-932b-dadafc470498"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the number of values of $x$ such that the area of the isosceles triangle with sides $x$, $65$, and $65$ is a positive integer.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "4224",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "951597a2-c3c0-484c-8a9f-4131596939ac"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the number of functions $f(x)$ from $\\{1, 2, 3, 4, 5\\}$ to $\\{1, 2, 3, 4, 5\\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\\{1, 2, 3, 4, 5\\}$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "756",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "468deb46-2313-408c-b311-71738d332927"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nJeffrey writes the numbers $1$ and $100000000 = 10^8$ on the blackboard. Every minute, if $x, y$ are on the board, Jeffery replaces them with\n\\[\\frac{x + y}{2} \\quad \\text{and} \\quad 2 \\left(\\frac{1}{x} + \\frac{1}{y}\\right)^{-1}.\\]\nAfter $2017$ minutes the two numbers are $a$ and $b$. Find $\\min(a, b)$ to the nearest integer.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "10000",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "ec572625-c31a-44fc-a9c2-2039d0b7b5b2"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n设非负实数 a_1,a_2,\\cdots,a_{100} 满足对任意 2\\leq i\\leq 99，有 \\max\\{a_{i-1}+a_i,a_i+a_{i+1}\\}\\geq i. 求 a_1+a_2+\\cdots+a_{100} 的最小值.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1683",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "47eac659-5cbd-43c4-8045-fba955d4f475"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n设正整数 $a, b, c$ 中任意两个数不互素，且 $a+b+c$ 为素数。求 $a+b+c$ 的最小值。\n\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "31",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "1d35d78e-447d-4a1a-ad40-2ba922da4037"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor some matrix $\\mathbf{P} = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}$ with nonzero integer entries,\n\\[\\begin{pmatrix} 4 & 3 \\\\ -1 & 0 \\end{pmatrix} = \\mathbf{P}^{-1} \\begin{pmatrix} 1 & 0 \\\\ 0 & 3 \\end{pmatrix} \\mathbf{P}.\\]Find the smallest possible value of $|a| + |b| + |c| + |d|.$\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "2e625936-9873-406c-a380-4760c3a2d519"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor integers $0 \\le m,n \\le 2^{2017}-1$, let $\\alpha(m,n)$ be the number of nonnegative integers $k$ for which $\\left\\lfloor \\frac{m}{2^k} \\right\\rfloor$ and $\\left\\lfloor \\frac{n}{2^k} \\right\\rfloor$ are both odd integers. Consider a $2^{2017} \\times 2^{2017}$ matrix $M$ whose $(i,j)$th entry (for $1 \\le i, j \\le 2^{2017}$) is $(-1)^{\\alpha(i-1, j-1)}$. For $1 \\le i, j \\le 2^{2017}$, let $M_{i,j}$ be the matrix with the same entries as $M$ except for the $(i,j)$th entry, denoted by $a_{i,j}$, and such that $\\det M_{i,j}=0$. Suppose that $A$ is the $2^{2017} \\times 2^{2017}$ matrix whose $(i,j)$th entry is $a_{i,j}$ for all $1 \\le i, j \\le 2^{2017}$. Compute the remainder when $\\det A$ is divided by $2017$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1382",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "062ca0e0-bda9-4b0b-ab46-b0529207261f"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDefine the sequences $a_n$ and $b_n$ as follows: $a_1 = 2017$ and $b_1 = 1$. \nFor $n > 1$, if there is a greatest integer $k > 1$ such that $a_n$ is a perfect $k$th power, then $a_{n+1} =\\sqrt[k]{a_n}$, otherwise $a_{n+1} = a_n + b_n$. If $a_{n+1} \\ge a_n$ then $b_{n+1} = b_n$, otherwise $b_{n+1} = b_n + 1$. Find $a_{2017}$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "8002",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "005b07c9-79e5-4dc5-ad5c-b6a2ec724b00"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n在正四面体 $A B C D$ 中, $E, F$ 分别在棱 $A B, A C$ 上,满足 $B E=3, E F=4$, 且 $E F$ 与面 $B C D$ 平行, 则 $\\triangle D E F$ 的面积为 $\\qquad$.原始的答案是k\\sqrt{m}+n的形式，请给出k + m + n的值。\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "35",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "2410cef2-e0a2-451f-9bcd-b53c5e109ea4"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere is a box containing $100$ balls, each of which is either orange or black. The box is equally likely to contain any number of black balls between $0$ and $100$, inclusive. A random black ball rolls out of the box. The probability that the next ball to roll out of the box is also black can be written in the form $\\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "6f7112a8-8477-4050-9e65-95def35413e4"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEach digit in the number 2001 is placed on a different card. In how many ways can three different cards be selected so that the product of the numbers on those cards is not zero?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "18d000b7-e571-4dd0-98e0-b39e22ea7eca"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n一个 100\\times100 方格表的左上角小方格中有一只老鼠，右下角小方格中有一块奶酪. 老鼠希望移动到右下角小方格中吃奶酪，每次可以从一个小方格移动到相邻的小方格（两个小方格相邻指它们有公共边）. 现在在一些小方格的边上放置隔板，老鼠在移动时不能越过隔板. 称一种放置隔板的方式是\"仁慈的\"，如果放置隔板后老鼠仍能吃到奶酪. 求最小的正整数 n，使得对任意一种\"仁慈的''放置 2023 个隔板的方式，老鼠都能通过不超过 n 次移动吃到奶酪.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2220",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "e11b1e41-6ec3-46d4-90d4-c43e3815abef"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nGiven a positive integer $m,$ define the polynomial $$P_m(z) = z^4-\\frac{2m^2}{m^2+1} z^3+\\frac{3m^2-2}{m^2+1}z^2-\\frac{2m^2}{m^2+1}z+1.$$ Let $S$ be the set of roots of the polynomial $P_5(z)\\cdot P_7(z)\\cdot P_8(z) \\cdot P_{18}(z).$ Let $w$ be the point in the complex plane which minimizes $\\sum_{z \\in S} |z-w|.$ The value of $\\sum_{z \\in S} |z-w|^2$ equals $\\tfrac{a}{b}$ for relatively prime positive integers $a$ and $b.$ Compute $a+b.$\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "171",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "5ac92f1d-f1d1-4e8e-9565-cf22459a0629"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCompute the smallest base-10 positive integer greater than 5 that is a palindrome when written in both base 2 and 4.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "15",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "20699fd1-9a89-4c4d-9608-b983d9f721c4"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n设复数 \\(z, w\\) 满足: \\(|w+z|=1,\\left|w^{2}+z^{2}\\right|=4\\)，求 \\(|w z|\\) 的最小值。答案应以 \\(\\frac{k}{m}\\) 的形式给出，其中 \\(\\frac{k}{m}\\) 为最简分数。请计算 \\(k+m\\) 的值。\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "405d798f-ed8a-4208-9cdf-4a36f5599845"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n将编号为 $1,2,3, \\cdots, 9$ 的九个小球随机放置在圆周的九个等分点上，每个等分点上各放一个小球，设圆周上所有相邻两个球号码之差的绝对值之和为 $S$ ，求使 $S$ 达到最小值的放法的概率。（注：如果某种放法，经旋转或镜面反射后与另一种放法重合，则认为是相同的放法）原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "316",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "727cbb16-e827-45b2-9802-0852cd7e4046"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\mathcal{P}$ be a parallelepiped with side lengths $x$, $y$, and $z$. Suppose that the four space diagonals of $\\mathcal{P}$ have lengths $15$, $17$, $21$, and $23$. Compute $x^2 + y^2 + z^2$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "371",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f346362b-79d3-4bf2-b225-18ac089ec9fa"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nOf all the primes in the prime factorization of \\begin{align*}\\gcd(&2^23^35^57^711^{11}13^{13}17^{17}19^{19}23^{23}, \\\\ &\\quad 2^{23}3^{19}5^{17}7^{13}11^{11}13^717^519^323^2),\\end{align*}which prime appears with largest exponent?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "c44ca8fb-eef5-41c5-b6ed-9c38e94396de"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many positive integers have exactly three proper divisors (positive integral divisors excluding itself), each of which is less than 50?\n\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "109",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "c250828c-11ce-4ff3-8f62-8bffe78218a0"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA house and a store were each sold for $\\$12000$. The house was sold at a loss of $20\\%$ of its cost, and the store at a gain of $20\\%$ of its cost. Find the entire transaction's net loss or gain in dollars.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1000",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "77e46da8-59af-4554-b8d7-f17ad1a475b0"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nGiven a finite set $S \\subset \\mathbb{R}^3$, define $f(S)$ to be the minimum integer $k$ such that there exist $k$ planes that divide $\\mathbb{R}^3$ into a set of regions, where no region contains more than one point in $S$. Suppose that\n\\[M(n) = \\max\\{f(S) : |S| = n\\} \\text{ and } m(n) = \\min\\{f(S) : |S| = n\\}.\\]\nEvaluate $M(200) \\cdot m(200)$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2189",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f2ed6600-8e8c-4b1b-ab91-98894397f3f7"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA sequence \\((a_n)\\) of real numbers is defined by \\(a_0=1\\), \\(a_1=2015\\) and for all \\(n \\geq 1\\), we have:\n\\[\na_{n+1} = \\frac{n-1}{n+1}a_n - \\frac{n-2}{n^2+n}a_{n-1}.\n\\]\nCalculate the value of \n\\[\n\\frac{a_1}{a_2} - \\frac{a_2}{a_3} + \\frac{a_3}{a_4} - \\frac{a_4}{a_5} + \\ldots + \\frac{a_{2013}}{a_{2014}} - \\frac{a_{2014}}{a_{2015}}.\n\\]\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "3021",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "608c574e-7777-4aa1-b583-0eaf63128b4d"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet \\(a\\) and \\(b\\) be the roots of the equation \\(x^2-mx+2=0\\). Suppose that \\(a+\\frac{1}{b}\\) and \\(b+\\frac{1}{a}\\) are the roots of the equation \\(x^2-px+q=0\\). The original answer for the value of \\(q\\) is in the form \\(\\frac{k}{m}\\). Please find the value of \\(k + m\\).\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "66d7c687-a852-4247-88cf-6543fe33a921"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA triangle $ABC$ with $AC = 20$ is inscribed in a circle $\\omega$. A tangent $t$ to $\\omega$ is drawn through $B$. The distance from $A$ to $t$ is $25$ and the distance from $C$ to $t$ is $16$. If $S$ denotes the area of triangle $ABC$, find the largest integer not exceeding $\\frac{S}{20}$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "24ee6a0a-5d77-4e8b-a3a5-b75d81273af4"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA zoo has a menagerie containing four pairs of different animals, one male and one female for each. The zookeeper wishes to feed the animals in a specific pattern: each time he feeds a single animal, the next one he feeds must be a different gender. If he starts by feeding the male giraffe, how many ways can he feed all the animals?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "144",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f5649567-c70a-4a6a-a100-e413eb3e68df"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is $2^{-1} + 2^{-2} + 2^{-3} + 2^{-4} + 2^{-5} + 2^{-6} \\pmod{13}$?\n\nExpress your answer as an integer from $0$ to $12$, inclusive.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "8032cf62-417b-4f4b-b8e3-1b56aaf9081a"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many noncongruent integer-sided triangles with positive area and perimeter less than $15$ are neither equilateral, isosceles, nor right triangles? Provide your answer as an integer.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "cdf9eb83-a5cb-4d0f-bfbb-61f4d1084a0c"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $P$ be the set of positive integers that are prime numbers. Find the number of subsets of $P$ that have the property that the sum of their elements is $34$, such as $\\{3, 31\\}$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "709a7396-7952-41ef-8f8a-6690ca165c03"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nKelvin the Frog was bored in math class one day, so he wrote all ordered triples $(a, b, c)$ of positive integers such that $abc=2310$ on a sheet of paper. Find the sum of all the integers he wrote down. In other words, compute $$\\sum_{\\substack{abc=2310 \\\\ a,b,c\\in \\mathbb{N}}} (a+b+c),$$ where $\\mathbb{N}$ denotes the positive integers.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "49140",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "34acdd92-1ba1-4788-8996-0b5f0b20219d"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile, the speed is constant. If the second mile is traversed in $2$ hours, find the time, in hours, needed to traverse the $n$th mile. The original answer is in the form $k(n-m)$, please give the value of $k + m$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "420ef8ac-18bd-480c-8268-9f221e45adc4"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order (each digit is greater than the previous digit)?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "34",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "59c6f560-f863-4c01-b24f-79f39d5f06ff"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n已知圆锥面 $x^{2}+y^{2}=\\frac{z^{2}}{3}$, 记沿该圆锥面从 $P(-\\sqrt{3}, 3,6)$ 到 $Q(\\sqrt{3}, 0,3)$ 的曲线长度的最小为值为 I, 则 $[10 I]=$ $\\qquad$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "60",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "9c7747d0-7289-47db-8e26-43137045fe17"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6, BC=5=DA,$ and $CD=4.$ Draw circles of radius $3$ centered at $A$ and $B,$ and circles of radius $2$ centered at $C$ and $D.$ A circle contained within the trapezoid is tangent to all four of these circles. Its radius is $\\frac{-k+m\\sqrt{n}}{p},$ where $k, m, n,$ and $p$ are positive integers, $n$ is not divisible by the square of any prime, and $k$ and $p$ are relatively prime. Find $k+m+n+p.$\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "134",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b584204a-0135-44e5-8169-dd14d8dd81c1"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIf three standard, six-faced dice are rolled, what is the probability that the sum of the three numbers rolled is 9? Express your answer as a common fraction.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "241",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "aa10fd31-97b1-4539-aef2-6effebcbc6d1"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n设 $a, b>0$ ，满足：关于 $x$ 的方程 $\\sqrt{|x|}+\\sqrt{|x+a|}=b$ 恰有三个不同的实数解 $x_{1}, x_{2}, x_{3}$, 且 $x_{1}<x_{2}<x_{3}=b$, 则 $a+b$ 的值为 $\\qquad$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "144",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "04056225-f6e2-4651-9502-f78d5ed1b6ab"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the minimum of \\( \\sum\\limits_{k=0}^{40} \\left(x+\\frac{k}{2}\\right)^2 \\) where \\( x \\) is a real number.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1435",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "8973774b-9073-4d7e-bbbc-37c14a72a7f1"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the number of ordered pairs $(x,y)$ of real numbers that satisfy\n\\[x^2 + 2y^2 + \\frac{1}{2} \\le x(2y + 1).\\]\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "47ede0e4-3356-479b-b2a8-e9766486c256"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n方程 x^3-9x^2+8x+2=0 有 3 个实根 p,q,r，则 \\df{1}{p^2}+\\df{1}{q^2}+\\df{1}{r^2}=__________.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "25",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "bfe7ce93-5585-4363-aabf-d6de22b6e9d4"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $S = \\{1, \\cdots, 6\\}$ and $\\mathcal{P}$ be the set of all nonempty subsets of $S$. Let $N$ equal the number of functions $f:\\mathcal{P} \\to S$ such that if $A, B \\in \\mathcal{P}$ are disjoint, then $f(A) \\neq f(B)$. Determine the number of positive integer divisors of $N$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "9792",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "5fe15d3c-f590-4d18-a29a-7705df55e832"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIf $n$ and $k$ are positive integers such that $5<\\frac nk<6$, then what is the smallest possible value of $\\frac{\\mathop{\\text{lcm}}[n,k]}{\\gcd(n,k)}$?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "22",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "5916be78-314c-4913-b012-7b5075e744d0"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCompute the sum of all the roots of \n\\((2x+3)(x-4)+(2x+3)(x-6)=0\\). \nExpress your answer as a fraction in simplest form \\(\\frac{k}{m}\\), and provide the value of \\(k + m\\).\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "35f0edbd-c5c1-4afd-afd2-918812c1ae4a"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n设函数 f(x) 的定义域为 \\bR，且 f(x-1) 为偶函数，f(x-2) 为奇函数. 当 x\\in[1,2] 时，f(x)=ax^2+bx. 若 f(0)+f(3)=-1，则 f(\\df{9}{2})=__________.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "1d95e47e-c84f-4cdc-b424-3b28fb77ff48"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nGiven that $13^{-1} \\equiv 29 \\pmod{47}$, find $34^{-1} \\pmod{47}$, as a residue modulo 47. (Give a number between 0 and 46, inclusive.)\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "18",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "d37260c0-84f5-4fe3-9a59-acd95f63b84e"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTwo congruent equilateral triangles $\\triangle ABC$ and $\\triangle DEF$ lie on the same side of line $BC$ so that $B$, $C$, $E$, and $F$ are collinear as shown. A line intersects $\\overline{AB}$, $\\overline{AC}$, $\\overline{DE}$, and $\\overline{EF}$ at $W$, $X$, $Y$, and $Z$, respectively, such that $\\frac{AW}{BW} = \\frac{2}{9}$, $\\frac{AX}{CX} = \\frac{5}{6}$, and $\\frac{DY}{EY} = \\frac{9}{2}$. The ratio $\\frac{EZ}{FZ}$ can then be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "33",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "6d0b145b-d3bc-4855-b513-58047b718c69"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n一个六面体的各个面和一个正八面体的各个面都是边长为 $a$ 的正三角形, 这样两个多面体的内切球的半径之比是一个既约分数 $\\frac{m}{n}$ 。那么, 积 $m \\cdot n$ 是\n$\\qquad$ $\\cdot$\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f387fa35-ca89-409a-88ab-2f9a9657ff39"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine the largest integer $n$ such that $7^{2048} - 1$ is divisible by $2^n$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b30444ad-ed14-47e0-8a5a-148e0017f3ac"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n13 LHS Students attend the LHS Math Team tryouts. The students are numbered $1, 2, \\ldots, 13$. Their scores are $s_1, s_2, \\ldots, s_{13}$, respectively. There are 5 problems on the tryout, each of which is given a weight, labeled $w_1, w_2, \\ldots, w_5$. Each score $s_i$ is equal to the sum of the weights of all problems solved by student $i$. On the other hand, each weight $w_j$ is assigned to be $\\frac{1}{\\sum_{s_i}}$, where the sum is over all the scores of students who solved problem $j$. (If nobody solved a problem, the score doesn't matter). If the largest possible average score of the students can be expressed in the form $\\frac{\\sqrt{a}}{b}$, where $a$ is square-free, find $a+b$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "78",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "3812d1fb-65f1-4ba8-8241-ece6766ae23f"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThree concentric circles have radii $3,$ $4,$ and $5.$ An equilateral triangle with one vertex on each circle has side length $s.$ The largest possible area of the triangle can be written as $a + \\tfrac{b}{c} \\sqrt{d},$ where $a,$ $b,$ $c,$ and $d$ are positive integers, $b$ and $c$ are relatively prime, and $d$ is not divisible by the square of any prime. Find $a+b+c+d.$\n\n\n\n\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "41",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "432d5a78-88b0-4716-adf1-0b8d6e188e99"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe diagram below shows a large square with each of its sides divided into four equal segments. The shaded square, whose sides are diagonals drawn to these division points, has an area of $13$. Find the area of the large square.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "104",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "07e856a6-1edb-44f2-afec-ba56295cc2e1"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere are $100$ people standing in a line from left to right. Half of them are randomly chosen to face right (with all $\\binom{100}{50}$ possible choices being equally likely), and the others face left. Then, while there is a pair of people who are facing each other and have no one between them, the leftmost such pair leaves the line. Compute the expected number of people remaining once this process terminates.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "50",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "fb913174-070a-4d16-8872-1db79e15825a"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the integer closest in value to\n\\[\\frac{1}{\\sqrt[4]{5^4+1}-\\sqrt[4]{5^4-1}}.\\]\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "250",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "237e82fb-5fff-4bc9-b4de-7d3dd4f6aad3"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEach face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube? Provide your answer as a simplified fraction in the form \\frac{k}{m}, and find the value of k + m.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "19",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "826f3e43-af7a-463c-965c-3cb416ba01b3"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nMy clock chimes two times 15 minutes after the hour, four times 30 minutes after the hour and six times 45 minutes after the hour. The clock also chimes eight times on each hour in addition to chiming the number of times equal to the hour. (So at 2:00 p.m., the clock chimes $8 + 2 = 10$ times.) Starting at 12:05 a.m., how many times does the clock chime in a 24-hour period?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "636",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "88eb27ad-2152-4d16-b7f0-67f018a940eb"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $LMT$ represent a 3-digit positive integer where $L$ and $M$ are nonzero digits. Suppose that the 2-digit number $MT$ divides $LMT$. Compute the difference between the maximum and minimum possible values of $LMT$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "880",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "8fbf5877-b66b-43e5-aa77-d629df4c6022"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nRobert tiles a $420 \\times 420$ square grid completely with $1 \\times 2$ blocks. He notices that the two diagonals of the grid pass through a total of $n$ blocks. Find the sum of all possible values of $n$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2517",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "d4f3093f-9129-47db-bf7e-749b4e5a3432"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA natural number $n$ is considered \\textit{good} if it can be expressed as the sum of $r$ consecutive positive integers, where $r \\geq 2$. Determine the number of good numbers within the set $\\{1, 2, \\dots, 100\\}$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "93",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "208c62ec-e721-443d-a2c3-a117bab13aa0"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA rectangle has an area of $16$ and a perimeter of $18$. Determine the length of the diagonal of the rectangle.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "9192f0de-491f-430e-9037-3ed4de3821a9"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the greatest prime number $p$ such that $p^3$ divides $$\\frac{122!}{121}+ 123!:$$\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "61",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "3e21052c-f5e8-4bfb-976a-af89f3198f16"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere are circles $\\omega_1$ and $\\omega_2$. They intersect in two points, one of which is the point $A$. Point $B$ lies on $\\omega_1$ such that $AB$ is tangent to $\\omega_2$. The tangent to $\\omega_1$ at $B$ intersects $\\omega_2$ at points $C$ and $D$, where $D$ is closer to $B$. The line $AD$ intersects $\\omega_1$ again at $E$. Given that $BD = 3$ and $CD = 13$, find the ratio $\\frac{EB}{ED}$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "6f2e71b6-ee6e-4bcd-8da5-d8c1da23886d"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn triangle $ABC$, where $AB = 28$, $AC = 36$, and $BC = 32$, let $D$ be the point on segment $BC$ such that $\\angle BAD = \\angle DAC$. Let $E$ be the unique point such that $DE \\parallel AB$ and line $AE$ is tangent to the circumcircle of $\\triangle ABC$. Find the length of segment $AE$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "18",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "15f84155-8516-4eba-8279-67c76d09415d"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nPatrick started walking at a constant rate along a straight road from school to the park. One hour after Patrick left, Tanya started running along the same road from school to the park. One hour after Tanya left, Jose started bicycling along the same road from school to the park. Tanya ran at a constant rate of $2$ miles per hour faster than Patrick walked, Jose bicycled at a constant rate of $7$ miles per hour faster than Tanya ran, and all three arrived at the park at the same time. The distance from the school to the park is $\\frac{m}{n}$ miles, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "277",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "a380db75-a4cc-484c-b235-fcce050b7516"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere are $2022$ grids in a row. Two people, A and B, play a game with these grids. Initially, each odd-numbered grid from the left is marked with A's name, and each even-numbered grid from the left is marked with B's name. The players take turns performing the following action, starting with player A:\n\n- One should select two grids marked with their own name, such that these two grids are not adjacent, and all grids between them are marked with the opponent's name. Then, change the name in all the grids between those two grids to one's own name.\n\nThe game continues until one player can no longer make a move. Find the largest positive integer $m$ such that:\n\n- No matter how player B acts, player A can always ensure that there are at least $m$ grids marked with A's name.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1011",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "d8ad1943-d93f-4d64-b717-57f63b554304"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n已知函数 f:\\bR\\rightarrow\\bR，若对任意实数 x,y,z 都有\\df{1}{3}f(xy)+\\df{1}{3}f(xz)-f(x)f(yz)\\geq\\df{1}{9},求 \\ds{i=1}{100}[if(i)]，其中 [x] 代表不超过 x 的最大整数.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1650",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "20247aa6-e432-4080-b98b-96013b760783"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA 5x5x5 cube is formed by assembling 125 unit cubes. Nine unit squares are painted on each of the six faces of the cube according to the pattern shown. How many of the 125 unit cubes have no paint on them? [asy]\nfill((2,0)--(3,0)--(3,5)--(2,5)--cycle,gray(.7));\nfill((0,2)--(0,3)--(5,3)--(5,2)--cycle,gray(.7));\npath p=(0,0)--(5,0);\n\ndraw(p,linewidth(1));\ndraw(shift(0,1)*p,linewidth(1));\ndraw(shift(0,2)*p,linewidth(1));\ndraw(shift(0,3)*p,linewidth(1));\ndraw(shift(0,4)*p,linewidth(1));\ndraw(shift(0,5)*p,linewidth(1));\n\ndraw((0,0)--(0,5),linewidth(1));\ndraw((1,0)--(1,5),linewidth(1));\ndraw((2,0)--(2,5),linewidth(1));\ndraw((3,0)--(3,5),linewidth(1));\ndraw((4,0)--(4,5),linewidth(1));\ndraw((5,0)--(5,5),linewidth(1));\n\n[/asy]\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "83",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b550cdfd-a2dd-4a0b-9aa1-026e12a063f0"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSuppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed—a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle? The original answer is in \\(\\frac{k}{m}\\) format, please give the value of k + m.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "bae9b795-e39f-4d24-8443-f2e48054bf1c"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many functions $f : \\{1,2,3,4,5,6\\} \\to \\{1,2,3,4,5,6\\}$ have the property that $f(f(x)) + f(x) + x$ is divisible by $3$ for all $x \\in \\{1,2,3,4,5,6\\}$?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "64",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "4ad3677e-50b8-4933-9114-c503e793aee6"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n13. Let $\\triangle T B D$ be a triangle with $T B=6, B D=8$, and $D T=7$. Let $I$ be the incenter of $\\triangle T B D$, and let $T I$ intersect the circumcircle of $\\triangle T B D$ at $M \\neq T$. Let lines $T B$ and $M D$ intersect at $Y$, and let lines $T D$ and $M B$ intersect at $X$. Let the circumcircles of $\\triangle Y B M$ and $\\triangle X D M$ intersect at $Z \\neq M$. If the area of $\\triangle Y B Z$ is $x$ and the area of $\\triangle X D Z$ is $y$, then the ratio $\\frac{x}{y}$ can be expressed as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "97",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f01b9de6-b1a4-431d-b065-d08c251369fe"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n求一切实数 $p$ ，使得三次方程 $5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p$ 的三个根均为正整数。\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "76",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "bc6cffab-388d-49cb-9aa0-40eee922cf17"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTriangle $ABC^{}_{}$ has $AB=9^{}_{}$ and $BC: AC=40: 41^{}_{}$. What's the largest area that this triangle can have?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "820",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "d5b84bf3-659e-4e79-aa49-c58cdd3ca6e8"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n将一枚均匀的硬币连续抛掷 $n$ 次，以 $p_{n}$ 表示未出现连续 3 次正面的概率. 求数列 $\\left\\{p_{n}\\right\\}$ 的其极限.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "ecc6b85c-7e9f-43d0-b44a-626ba980b826"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n设多项式 \\(f(x)\\) 的各项系数都是非负实数，且 \\(f(1)=f^{\\prime}(1)=f^{\\prime \\prime}(1)=f^{\\prime \\prime \\prime}(1)=1\\)。求 \\(f(x)\\) 的常数项的最小值。答案的形式为 \\(\\frac{k}{m}\\)，请给出 \\(k + m\\) 的值。\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f1c3e405-9b68-4d87-b387-2320daaf607d"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n给定正整数 $k$. 若整数集 $\\mathbf{Z}$ 的子集序列 $I_{1} \\supseteq I_{2} \\supseteq \\cdots \\supseteq I_{k}$ 满足对 $i=1,2, \\cdots, k, 168$ $\\in I_{i}$ ，且对于任意的 $x 、 y \\in I_{i}$ ，均有 $x-y \\in I_{i}$ $\\left(x 、 y\\right.$ 可以相等），则称 $\\left(I_{1}, I_{2}, \\cdots, I_{k}\\right)$ 为 $\\mathbf{Z}$ 上的一条 $k$ —链。问： $\\mathbf{Z}$ 上共有多少条 $k$ —链？请说明理由。请计算当k=5时的答案。请给出计算出的整数值。\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2016",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "942ed614-8671-4c08-9e4e-b61bb0e43585"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSuppose $A \\subseteq \\{0,1,\\dots,29\\}$. It satisfies that for any integer $k$ and any two members $a,b \\in A$ (where $a$ and $b$ are allowed to be the same), $a + b + 30k$ is always not the product of two consecutive integers. Please find $A$ with the largest possible cardinality.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "4aee130e-3ad9-4b51-ad95-bbdb3ee09045"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet triangle $ABC$ have $\\angle A = 70^\\circ$, $\\angle B = 60^\\circ$, and $\\angle C = 50^\\circ$. Extend altitude $BH$ past $H$ to point $D$ so that $BD = BC$. Find $\\angle BDA$ in degrees.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "50",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "1bebb179-426d-4dc6-998d-4e7015ffe72f"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $m$ be a positive integer, and suppose that $9$ is its own inverse $\\pmod m$, but $3$ is $\\textbf{not}$ its own inverse $\\pmod m$.\n\nHow many possible values for $m$ are there?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "0dcd19fb-cbfb-4504-a43f-95d96511d722"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n已知 $a$ 为实数, 且对任意 $k \\in[-1,1]$, 当 $x \\in(0,6]$ 时, $6 \\ln x+x^{2}-8 x+a \\leqslant k x$ 恒成立, 则 $a$的最大值是 $\\qquad$.原始答案形式为$a-b \\ln c$，求a+b+c的值。\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "18",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "5ff8dd60-7af3-42b3-b87c-b695d50c05fb"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n设整数 n>4，(x+2\\sqrt{y}-1)^n 的展开式中 x^{n-4} 与 xy 两项的系数相等，则 n 的值为________.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "51",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "fc991e76-0963-4120-9b44-c27d8925bffb"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n一个球与正四面体的六条棱都相切。若正四面体的棱长为 $a$，则这个球的体积的原始答案是\\frac{\\sqrt{a}\\pi a^b}{c}的格式，请给出a+b+c的值。\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "29",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "735f4fae-a20e-439a-bf35-673147771643"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n定义在 \\(\\mathbf{R}\\) 上的函数 \\(f(x)\\) 满足 \\(f(0)=0\\), \\(f(x)+f(1-x)=1\\), \\(f\\left(\\frac{x}{5}\\right)=\\frac{1}{2} f(x)\\)，且当 \\(0 \\leqslant x_{1}<x_{2} \\leqslant 1\\) 时, \\(f\\left(x_{1}\\right) \\leqslant f\\left(x_{2}\\right)\\)。求 \\(f\\left(\\frac{1}{2007}\\right)\\) 的值。答案应为\\(\\frac{k}{m}\\)的形式，请给出k+m的值。\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "33",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "1765ca2d-2324-4e9d-b1b8-251de2864ae9"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n设复数 $z=(\\omega+2)^{3}(\\omega-3)^{2}$ 。求当 $\\omega$取遍所有模长为 1 的复数时, $z$ 的模长的最大值。原始答案为$m^{\\frac{p}{q}}$的形式，其中$p,q$互质，请给出$m+p+q$的值。\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "e851ab41-50f9-440f-b5b1-802f43603b15"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\triangle ABC$ be an isosceles triangle such that $BC = 30$ and $AB = AC.$ We have that $I$ is the incenter of $\\triangle ABC,$ and $IC = 18.$ What is the length of the inradius of the triangle?The answer is in the form k\\sqrt{m}+n,. Please provide the value of k + m + n.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f094441d-2831-435c-b995-fb677f16b323"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $a, b, c, d, e,$ and $f$ be real numbers. Define the polynomials: \n\\[ P(x) = 2x^4 - 26x^3 + ax^2 + bx + c \\] \nand \n\\[ Q(x) = 5x^4 - 80x^3 + dx^2 + ex + f. \\] \nLet $S$ be the set of all complex numbers which are a root of either $P$ or $Q$ (or both). Given that $S = \\{1,2,3,4,5\\}$, compute $P(6) \\cdot Q(6).$\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2400",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "54f4b5dd-619d-43d4-bab1-85cd48c23d87"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEuler discovered that the polynomial $p(n) = n^2 - n + 41$ yields prime numbers for many small positive integer values of $n$. What is the smallest positive integer $n$ for which $p(n)$ and $p(n+1)$ share a common factor greater than $1$?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "41",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "6e1c746c-6afa-4ac5-a97e-b00f005f6c3c"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEquilateral triangle $ABC$ has area $1$. $A'$, $B'$, and $C'$ are the midpoints of $BC$, $CA$, and $AB$, respectively. $A''$, $B''$, $C''$ are the midpoints of $B'C'$, $C'A'$, and $A'B'$, respectively. The area of trapezoid $BB''C''C$ can be written as $\\frac{m}{n}$ for relative prime positive integers $m$ and $n$. Find $m+n$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "4436d9d8-0f11-4098-b3c6-2f45eaa02540"
}
|
math_dapo
|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the sum of all positive integers $m$ such that $2^m$ can be expressed as a sum of four factorials (of positive integers).\n\nNote: The factorials do not have to be distinct. For example, $2^4=16$ counts, because it equals $3!+3!+2!+2!$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "3f0387a7-58b3-41a7-8613-c1442abe1d48"
}
|
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