data_source
stringclasses 1
value | prompt
listlengths 2
2
| ability
stringclasses 1
value | reward_model
dict | extra_info
dict |
|---|---|---|---|---|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "If $N$ is the number of ways to place $16$ [i]jumping [/i]rooks on an $8 \\times 8$ chessboard such that each rook attacks exactly two other rooks, find the remainder when $N$ is divided by $1000$. (A jumping rook is said to [i]attack [/i]a square if the square is in the same row or in the same column as the rook.)\n\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "840",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "73175571-0110-4580-b3c0-c86cb2cd3818"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Determine all triples $(a,b,c)$ of positive integers for which $ab-c,bc-a$, and $ca-b$ are powers of $2$. Explanation:$A$ power of $2$ is an integer of the form $2^n$, where $n$ denotes some nonnegative integer. \nHow many triples $(a, b, c)$ satisfy the conditions?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "16",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "63c67dd3-935d-4af6-850e-dd6231eedf58"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "If the four sides of a quadrilateral are $2$, $3$, $6$, and $x$, find the sum of all possible integral values for $x$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "54",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "d038a3df-4653-489c-a22f-7637db61620d"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "King William is located at $(1, 1)$ on the coordinate plane. Every day, he chooses one of the eight lattice points closest to him and moves to one of them with equal probability. When he exits the region bounded by the $x, y$ axes and $x+y = 4$, he stops moving and remains there forever. Given that after an arbitrarily large amount of time he must exit the region, the probability he ends up on $x+y = 4$ can be expressed as $\\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "65",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "5f06f137-f806-4f82-b380-960f2dd75a35"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Find the number of positive integers $n < 2018$ such that $25^n + 9^n$ is divisible by $13$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "336",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "5e095c14-a386-4c7d-bdb2-a75f8af157ab"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "设实数 $a, b$ 满足: 集合 $A=\\left\\{x \\in \\mathbf{R} \\mid x^{2}-10 x+a \\leq 0\\right\\}$ 与 $B=\\left\\{x \\in \\mathbf{R} \\mid b x \\leq b^{3}\\right\\}$的交集为 $[4,9]$, 则 $a+b$ 的值为 $\\qquad$. \n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "e4b6ed05-5520-4653-9971-fc21e4cc1777"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "It is well-known that the $n^{\\text{th}}$ triangular number can be given by the formula $\\frac{n(n+1)}{2}$. A Pythagorean triple of \\textit{square numbers} is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of \\textit{triangular numbers} (a PTTN) be an ordered triple of positive integers $(a,b,c)$ such that $a\\leq b<c$ and\n\\[\\frac{a(a+1)}{2}+\\frac{b(b+1)}{2}=\\frac{c(c+1)}{2}.\\]\nFor instance, $(3,5,6)$ is a PTTN ($6+15=21$). Here we call both $a$ and $b$ \\textit{legs} of the PTTN. Find the smallest natural number $n$ such that $n$ is a leg of \\textit{at least} six distinct PTTNs.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "54cec35a-9176-4562-845c-279da0c51122"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 32.$ Find the minimum value of\n\\[x^2 + 4xy + 4y^2 + 2z^2.\\]\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "96",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "3c476624-1d3c-4a46-bd16-4415c4d6c31e"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "全体迹为 0 的 2 阶方阵构成线性空间 M_2. 已知矩阵E_1=\\begin{bmatrix}0&1\\\\0&0\\end{bmatrix},\\quad E_2=\\begin{bmatrix}0&0\\\\ 1&0\\end{bmatrix},\\quad E_3=\\begin{bmatrix}1&0\\\\0&-1\\end{bmatrix}.定义线性变换 \\mathcal{A}_i(X)=E_iX-XE_i(i=1,2,3)，且 3 阶方阵 B=(b_{ij}) 满足 b_{ij}=\\text{tr}(\\mathcal{A}_i\\mathcal{A}_j) (1\\leq i,j\\leq 3). 则 [|\\det B|]=__________.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "128",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "5d72c9c0-e2ba-4aee-bf17-7f854f554c96"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Let $\\triangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. The altitude from $A$ intersects $BC$ at $D$.\n\nLet $\\omega_1$ and $\\omega_2$ be the incircles of $\\triangle ABD$ and $\\triangle ACD$, respectively. The common external tangent of $\\omega_1$ and $\\omega_2$ (other than $BC$) intersects $AD$ at $E$. Compute the length of $AE$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "52d5a7bd-2bcf-415b-8aa5-2840e5db3f36"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Find the minimum value of the function\n\\[f(x) = \\sqrt{-x^2 + 4x + 21} - \\sqrt{-x^2 + 3x + 10}.\\]The answer is in the form k\\sqrt{m}+n,. Please provide the value of k + m + n.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "6d272d1b-723d-47eb-baa3-57b26c4f092e"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Prinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an integer from $1$ to $50$ uniformly at random, and this number becomes their score. Dukejukem then flips a weighted coin that lands heads with probability $\\frac{3}{5}$. If he flips heads, he adds $1$ to his score. A player wins the game if their score is higher than the other player's score. The probability Dukejukem defeats Trollner to win WASS equals $\\frac{m}{n}$ where $m$ and $n$ are coprime positive integers. Compute $m+n$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "751",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "46b7065f-639b-488f-93ca-510e3191b98f"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "The sequence \\[\\log_{12}{162}, \\ \\log_{12}{x}, \\ \\log_{12}{y}, \\ \\log_{12}{z}, \\ \\log_{12}{1250}\\]is an arithmetic progression. What is $x$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "270",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "aba175f6-5674-47e9-b83e-1ef7c1df30af"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Solve in positive integers the following equation:\n\n$$ \\left[ \\sqrt{1} \\right] + \\left[ \\sqrt{2} \\right] + \\left[ \\sqrt{3} \\right] + \\ldots + \\left[ \\sqrt{x^2 - 2} \\right] + \\left[ \\sqrt{x^2 - 1} \\right] = 125, $$\n\nwhere $[a]$ denotes the integer part of the real number $a$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "1cedf39b-9910-4325-8740-a4778d5a45fb"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "$f(u)=u^{2}+a u+b-2, u=x+\\frac{1}{x}, f(u)$ 有零点, 则 $a^{2}+b^{2}$ 的最小值为 $\\qquad$.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\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "4a0305f2-2df6-4fbf-9ea2-a21a9082fc8b"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Binhao has a fair coin. He writes the number $+1$ on a blackboard. Then he flips the coin. If it comes up heads (H), he writes $+\\frac12$ , and otherwise, if he flips tails (T), he writes $-\\frac12$ . Then he flips the coin again. If it comes up heads, he writes $+\\frac14$ , and otherwise he writes $-\\frac14$ . Binhao continues to flip the coin, and on the nth flip, if he flips heads, he writes $+ \\frac{1}{2n}$ , and otherwise he writes $- \\frac{1}{2n}$ . For example, if Binhao flips HHTHTHT, he writes $1 + \\frac12 + \\frac14 - \\frac18 + \\frac{1}{16} -\\frac{1}{32} + \\frac{1}{64} -\\frac{1}{128}$ . The probability that Binhao will generate a series whose sum is greater than $\\frac17$ is $\\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + 10q$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "153",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b8a4b677-c963-4993-8a29-b3ef58b38037"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "设正整数 n 的正因数为 1=d_1<d_2<\\cdots<d_k=n，其中 k\\geq6. 求所有的 n 使得 n=d_5^2+d_6^2.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "500",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b20878f8-2276-4061-82c3-9daf33da50aa"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1462",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "758923b5-846f-4980-97b8-45c72136c684"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "What is the smallest positive integer $n$ such that $\\frac{n}{2010}$ is a terminating decimal?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "201",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "3c60a368-c009-4eed-af06-eb01b1a52b3a"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure, we have $3$ rows of small congruent equilateral triangles, with $5$ small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $2003$ small equilateral triangles? Provide your answer as an integer.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1507509",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "d63f9c3c-aacc-45e8-98d9-5d03d8415a90"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Let $a$ be a positive real number such that \\(4a^2+\\frac{1}{a^2}=117.\\) Find \\(8a^3+\\frac{1}{a^3}.\\)\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1265",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "995eabca-c7f6-4a3f-84c2-80f82042e751"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "The graph of the equation $x^{y}=y^{x}$ in the first quadrant (i.e., the region where $x>0$ and $y>0$) consists of a straight line and a curve. Find the sum of the coordinates of the intersection points of a straight line and a curve, rounded down.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "dc1a0907-aba5-4d07-a25a-0c5bdcd7869e"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Azar and Carl play a game of tic-tac-toe. Azar places an $X$ in one of the boxes in a $3$-by-$3$ array of boxes, then Carl places an $O$ in one of the remaining boxes. After that, Azar places an $X$ in one of the remaining boxes, and so on until all boxes are filled or one of the players has three of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third $O$. How many ways can the board look after the game is over?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "148",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b70fc505-0800-49f0-99de-1710e7004315"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Point $B$ is on $\\overline{AC}$ with $AB = 9$ and $BC = 21.$ Point $D$ is not on $\\overline{AC}$ so that $AD = CD,$ and $AD$ and $BD$ are integers. Let $s$ be the sum of all possible perimeters of $\\triangle ACD$. Find $s.$\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "380",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "d26601bc-c3f7-4af8-adbc-e9acb89e6312"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "In triangle $ABC$, the angle $C$ is obtuse, $m(\\angle A) = 2m(\\angle B)$, and the side lengths are integers. Find the smallest possible perimeter of this triangle.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "77",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b8249ac2-cac3-4a9c-b110-67eba298fc25"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Point $A$ lies on the circumference of a circle $\\Omega$ with radius $78$. Point $B$ is placed such that $AB$ is tangent to the circle and $AB=65$, while point $C$ is located on $\\Omega$ such that $BC=25$. Compute the length of $\\overline{AC}$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "60",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "a5f4eabc-c4de-42a0-b5a2-d4a3e6497c6d"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "The least positive angle $\\alpha$ for which $$\\left(\\frac34-\\sin^2(\\alpha)\\right)\\left(\\frac34-\\sin^2(3\\alpha)\\right)\\left(\\frac34-\\sin^2(3^2\\alpha)\\right)\\left(\\frac34-\\sin^2(3^3\\alpha)\\right)=\\frac1{256}$$ has a degree measure of $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "131",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "1ad3f665-1845-48b5-a0c9-54388ba803a4"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "求六个元素的置换群 $S_{6}$ 中 6 阶元素的个数。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "240",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b95efc1d-7bc2-494b-bc1a-84479dd57776"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Let \\(ABCD\\) be an isosceles trapezoid with parallel bases \\(\\overline{AB}\\) and \\(\\overline{CD}\\), where \\(AB > CD\\). Line segments from a point inside \\(ABCD\\) to the vertices divide the trapezoid into four triangles with areas \\(2\\), \\(3\\), \\(4\\), and \\(5\\), starting with the triangle with base \\(\\overline{CD}\\) and moving clockwise. The original answer for the ratio \\(\\frac{AB}{CD}\\) is in the form \\(k + \\sqrt{m}\\). Please calculate the value of \\(k + m\\).\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "6168d66e-2133-4704-97cc-64bac471b73f"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Recall that if $b$ is a residue $\\pmod{m}$, then the modular inverse of $b$ is the residue $c$ for which $bc \\equiv 1\\pmod{m}$. The table below shows the inverses of the first 9 positive residues modulo 47. \\begin{tabular}{c|ccccccccc}\n$b$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\\ \\hline\ninverse of $\\,b$ & 1 & 24 & 16 & 12 & 19 & 8 & 27 & 6 & 21\n\\end{tabular}Find the modular inverse of $35\\pmod{47}$.\n\nExpress your answer as an integer from $0$ to $46$, inclusive.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "43",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "fd145fd9-9788-4a94-baab-89905e68d21d"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "如图， $A B$ 是单位圆的直径，在 $A B$ 上任取一点 $D$ ，作 $D C \\perp A B$, 交圆周于 $C$ 。若点 $D$ 的坐标为 $(x, 0)$,则当 $x \\in$ $\\qquad$时，线段 $A D 、 B D 、 C D$ 可以构成锐角三角形。请提供区间的两个端点之和。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "76efbe6b-3575-4909-815c-013657b47d8c"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Line segment $\\overline{AB}$ is a diameter of a circle with $AB = 24$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\\triangle ABC$ traces out a closed curve missing two points. To the nearest positive integer, find the area of the region bounded by this curve.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "50",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "26c6c9f0-ba6d-41f3-a5e6-756309beb50f"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Let $s(m)$ denote the sum of the digits of the positive integer $m$. Find the largest positive integer that has no digits equal to zero and satisfies the equation \\[2^{s(n)} = s(n^2).\\]\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1111",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "0d6f1115-7adb-483b-be40-deb86772ec0c"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "In the figure, \\(ABCD\\) is a square of side length \\(1\\). The rectangles \\(JKHG\\) and \\(EBCF\\) are congruent. The original answer is in the form \\(k - \\sqrt{m}\\). Find the value of \\(k + m\\).\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "2804aff0-d56f-46e3-8c5a-aa7e6b981790"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "设函数\n$f(x)= \\begin{cases}\\frac{1}{p}, & x=\\frac{q}{p} ; \\\\ 0, & x \\neq \\frac{q}{p},\\end{cases}$\n其中, $p 、 q$ 互质, 且 $p \\geqslant 2$. 则满足 $x \\in[0,1]$,且 $f(x)>\\frac{1}{5}$ 的 $x$ 值的个数是 $\\qquad$.\n\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "d483b615-23bc-4252-b7ca-fdd1688ca1b6"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Carlos has chosen twelve different compact discs (CDs) he would like to buy. Four are rap music, five are country music, and three are heavy metal music. Carlos then randomly selects five of the 12 CDs to purchase. What is the probability that his purchase includes at least one CD from each of the three categories? 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\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "691",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "1ff908a9-34c6-48b8-9dde-ff33b7d806b2"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "A bag contains $8$ green candies and $4$ red candies. You randomly select one candy at a time to eat. If you eat five candies, there are relatively prime positive integers $m$ and $n$ so that $\\frac{m}{n}$ is the probability that you do not eat a green candy after you eat a red candy. Find $m+n$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "472346a6-d64f-4c69-a7b5-4c5c2e659d06"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "在半径为 10 的圆周 C 上任给 63 个点，设以这些点为顶点且三边长都大于 9 的三角形的个数为 S. 求 S 的最大值.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "23121",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "2f655106-d6a5-4898-81fd-13706a7a9650"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Let $ABCD$ be an isosceles trapezoid with $\\overline{AD} \\parallel \\overline{BC}$. The incircle of $\\triangle ABC$ has center $I$ and is tangent to $\\overline{BC}$ at $P$. The incircle of $\\triangle ABD$ has center $J$ and is tangent to $\\overline{AD}$ at $Q$. If $PI = 8$, $IJ = 25$, and $JQ = 15$, compute the greatest integer less than or equal to the area of $ABCD$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1728",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "06548175-8fde-4baa-9ae0-a2e1e26ab2e3"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a - (b - (c - (d + e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for $a$, $b$, $c$, and $d$ were $1$, $2$, $3$, and $4$, respectively. What number did Larry substitute for $e$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f094e99c-63b3-4b7f-9bfe-a242ca3490ed"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. The original answer is in the form $\\frac{a\\sqrt{b}-c}{d}$, please give the value of a + b + c + d.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b2471a26-405a-4a29-af32-f8f600d9a548"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "The diagram below shows rectangle $ABDE$ where $C$ is the midpoint of side $\\overline{BD}$, and $F$ is the midpoint of side $\\overline{AE}$. If $AB=10$ and $BD=24$, find the area of the shaded region.\n\n\\[\\text{[asy]}\n\\text{size(300);}\n\\text{defaultpen(linewidth(0.8));}\n\\text{pair A = (0,10), B=origin, C=(12,0), D=(24,0), E=(24,10), F=(12,10), G=extension(C,E,D,F);}\n\\text{filldraw(A--C--G--F--cycle, gray(0.7));}\n\\text{draw(A--B--D--E--F^^E--G--D);}\n\\text{label(\"$A$\", A, NW);}\n\\text{label(\"$B$\", B, SW);}\n\\text{label(\"$C$\", C, S);}\n\\text{label(\"$D$\", D, SE);}\n\\text{label(\"$E$\", E, NE);}\n\\text{label(\"$F$\", F, N);}\n\\text{[/asy]}\n\\]\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "90",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "4d780b6d-a8d3-4d01-87d3-c966d028618b"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "If $\\frac{3x^2-4x+1}{x-1}=m$, and $x$ can be any real number except $1$, what real values can $m$ NOT have?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "682f8f26-d68c-4703-9346-0c0e062f304b"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Find the remainder when $26!^{26} + 27!^{27}$ is divided by $29$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "0afdae4e-2e85-4b13-8e37-537ac5e4e135"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Determine the number of ordered pairs of positive integers $(x, y)$ with $y < x \\le 100$ such that $x^2 - y^2$ and $x^3 - y^3$ are relatively prime. (Two numbers are relatively prime if they have no common factor other than $1$.)\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "99",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "87a4b90d-29d5-40ec-9bd9-2c921a150062"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Given that the base-17 integer $\\overline{8323a02421_{17}}$ (where $a$ is a base-17 digit) is divisible by $16_{10}$, find $a$. Express your answer in base $10$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "91593343-0b11-4db1-941a-d564f99857bc"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Define $L(x) = x - \\frac{x^2}{2}$ for every real number $x$. If $n$ is a positive integer, define $a_n$ by\n\\[\n a_n = L \\Bigl( L \\Bigl( L \\Bigl( \\cdots L \\Bigl( \\frac{17}{n} \\Bigr) \\cdots \\Bigr) \\Bigr) \\Bigr),\n\\]where there are $n$ iterations of $L$. For example,\n\\[\n a_4 = L \\Bigl( L \\Bigl( L \\Bigl( L \\Bigl( \\frac{17}{4} \\Bigr) \\Bigr) \\Bigr) \\Bigr).\n\\]As $n$ approaches infinity, what value does $n a_n$ approach?The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "53",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "7ce34d7d-32d9-4536-bfbe-0d14629f68df"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Let $k$ be a positive integer. A sequence $a_0, a_1, \\ldots, a_n$, where $n > 0$, of positive integers satisfies the following conditions:\n\n1. $a_0 = a_n = 1$;\n2. $2 \\leq a_i \\leq k$ for each $i = 1, 2, \\ldots, n-1$;\n3. For each $j = 2, 3, \\ldots, k$, the number $j$ appears $\\phi(j)$ times in the sequence $a_0, a_1, \\ldots, a_n$, where $\\phi(j)$ is the number of positive integers that do not exceed $j$ and are coprime to $j$;\n4. For any $i = 1, 2, \\ldots, n-1$, $\\gcd(a_i, a_{i-1}) = 1 = \\gcd(a_i, a_{i+1})$, and $a_i$ divides $a_{i-1} + a_{i+1}$.\n\nSuppose there is another sequence $b_0, b_1, \\ldots, b_n$ of integers such that $\\frac{b_{i+1}}{a_{i+1}} > \\frac{b_i}{a_i}$ for all $i = 0, 1, \\ldots, n-1$. Find the minimum value of $b_n - b_0$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "4bcea3e5-789b-4505-9381-88ce0c3a20cf"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Find the area of rhombus $ABCD$ given that the radii of the circles circumscribed around triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectively.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "400",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "5cd32910-cfe1-4908-aee5-d488b8a2edc9"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Tasha and Amy both pick a number, and they notice that Tasha's number is greater than Amy's number by $12$. They each square their numbers to get a new number and see that the sum of these new numbers is half of $169$. Finally, they square their new numbers and note that Tasha's latest number is now greater than Amy's by $5070$. What was the sum of their original numbers?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "c6404b78-629e-4c6f-8e12-a9de822e67a1"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to $32$? Provide your answer as an integer.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "568",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f5c0b453-1e83-4655-983e-f712b30587b6"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "A diagonal of a polyhedron is a line segment connecting two non-adjacent vertices. How many diagonals does a pentagonal prism have? [asy]\nimport three;\nsize(100);\ndefaultpen(linewidth(0.7));\ncurrentprojection = orthographic(-0.2,-1,2/3);\n\nvoid drawPentagon (real h)\n{\n\npath3 y;\n\ny=(0,0,h)--(3,0,h)--(5,3,h)--(3,6,h)--(0,5,h)--cycle;\n\ndraw(surface(y),white,nolight);\n\ndraw(y);\n}\n\nvoid drawRectangle(triple a, triple b, real h)\n{\n\npath3 y;\n\ny=a--b--b+(0,0,h)--a+(0,0,h)--cycle;\n\ndraw(surface(y),white,black,nolight);\n}\n\ndrawRectangle((0,0,0),(0,5,0),4);\ndrawRectangle((0,0,0),(3,0,0),4);\ndrawRectangle((3,0,0),(5,3,0),4);\ndrawPentagon(4);\n\n//import three; currentprojection = orthographic(25,150,100); //defaultpen(linewidth(0.8)); size(100);\n//void nGon(int numSides, real height){\n//real angle = 360/numSides;\n//for(int i = 0; i < numSides; ++i){\n//draw(dir(angle*i,0)--dir(angle*(i + 1),0));\n//draw((dir(angle*i,0) + (0,0,height))--(dir(angle*(i + 1),0) + (0,0,height)));\n//draw(dir(angle*i,0)--(dir(angle*i,0) + (0,0,height)));\n// }\n//}\n//nGon(5,2);\n[/asy]\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "7e3a5ddb-48c5-4e3c-a207-e2e77cf96485"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Let $d_1, d_2, \\ldots , d_{k}$ be the distinct positive integer divisors of $6^8$. Find the number of ordered pairs $(i, j)$ such that $d_i - d_j$ is divisible by $11$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "665",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "a2a3d2de-e34e-487e-8cd9-adb3393c525f"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "What is the sum of all the solutions of $x = \\left|2x-|60-2x|\\right|$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "92",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "29ec3711-2c6a-4b0c-aa22-34436ce84402"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "The \\textit{arithmetic derivative} $D(n)$ of a positive integer $n$ is defined via the following rules:\n\\begin{itemize}\n \\item $D(1) = 0$;\n \\item $D(p) = 1$ for all primes $p$;\n \\item $D(ab) = D(a)b + aD(b)$ for all positive integers $a$ and $b$.\n\\end{itemize}\nFind the sum of all positive integers $n$ below $1000$ satisfying $D(n) = n$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "31",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "10668c6b-0239-4e91-b035-1b98da867ce6"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "已知函数\n$f(x)=2\\left(\\sin ^{4} x+\\cos ^{4} x\\right)+m(\\sin x+\\cos x)^{4}$\n在 $x \\in\\left[0, \\frac{\\pi}{2}\\right]$ 有最大值 5. 求实数 $m$ 的值.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "727ae271-cb42-4375-9b19-e98780e2be3e"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Given the polynomial $a_0x^n + a_1x^{n-1} + \\cdots + a_{n-1}x + a_n$, where $n$ is a non-negative integer, and $a_0$ is a positive integer. The remaining coefficients $a_1, a_2, \\ldots, a_n$ are integers or zero. Define $h = n + a_0 + |a_1| + |a_2| + \\cdots + |a_n|$. Find the number of polynomials for which $h = 3$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "74186e8a-00a1-4de6-a52c-acfa99522b4d"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Let $k$ be the smallest positive integer with the following property:\nThere are distinct integers $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ such that the polynomial\n$$\np(x)=\\left(x-m_{1}\\right)\\left(x-m_{2}\\right)\\left(x-m_{3}\\right)\\left(x-m_{4}\\right)\\left(x-m_{5}\\right)\n$$\nhas exactly $k$ nonzero coefficients.\nFind, with proof, a set of integers $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ for which this minimum $k$ is achieved.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "5c02674f-9ec5-47d5-812b-20edff468a50"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "If $x$ and $y$ are numbers such that $(x + 3)(y-5) = 0$, what is the smallest possible value of $x^2+ y^2$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "657a4281-9eb3-464c-ba61-d496268a13a3"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Three uncoordinated aliens launch a three-day attack on four galaxies. Each day, each of the three aliens chooses a galaxy uniformly at random from the remaining galaxies and destroys it. They make their choice simultaneously and independently, so two aliens could destroy the same galaxy. If the probability that every galaxy is destroyed by the end of the attack can be expressed as \\( \\frac{m}{n} \\) for relatively prime positive integers \\( m \\) and \\( n \\), what is \\( m+n \\)?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1151",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "6c2e988d-7f11-4536-bac8-fd781c21eae1"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "An \"n-pointed star\" is formed as follows: the sides of a convex polygon are numbered consecutively $1, 2, \\ldots, k, \\ldots, n$, where $n \\geq 5$. For all $n$ values of $k$, sides $k$ and $k+2$ are non-parallel, with sides $n+1$ and $n+2$ being respectively identical with sides $1$ and $2$. Prolong the $n$ pairs of sides numbered $k$ and $k+2$ until they meet. (A figure is shown for the case $n=5$)\n\nLet $S$ be the degree-sum of the interior angles at the $n$ points of the star; then find the expression for $S$ in terms of $n$. The original answer is in the form $a(n-b)$, where $a$ and $b$ are integers. Please find the value of $a + b$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "184",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "fb99e084-9eac-4f4c-aeb0-87b04ceb6e08"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "在三棱锥 $P-A B C$ 中, 三条棱 $P A$ 、 $P B 、 P C$ 两两垂直，且 $P A=1, P B=P C=2$. 若 $Q$ 为三棱锥 $P-A B C$ 外接球的球面上任一点, 则 $Q$ 到面 $A B C$ 距离的最大值减去$\\frac{\\sqrt{6}}6$的结果的2倍为 $\\qquad$. \n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "838c5797-0780-4a91-b589-bd7a793b1479"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "由 $1,2, \\cdots, 9$ 这九个正整数构成的所有圆排列中, 任意相邻两数之积均不超过 60 的圆排列的个数为 $\\qquad$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "21600",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "448f95b9-a9ab-40e3-a135-005e17000836"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Let $f$ be the unique function defined on the positive integers such that \\[\\sum_{d\\mid n}d\\cdot f\\left(\\frac{n}{d}\\right)=1\\] for all positive integers $n$. What is the value of $f(2023)$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "96",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "9af74819-3561-49ac-b0aa-cee1fde3982f"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Let set $T = \\{1, 2, 3, 4, 5, 6, 7, 8\\}$. Find the number of all nonempty subsets $A$ of $T$ such that $3 \\mid S(A)$ and $5 \\nmid S(A)$, where $S(A)$ is the sum of all the elements in $A$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "70",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "0eca3b48-9103-4f4a-ac13-308ef5c2cf83"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "$ABC$ is a triangle: $A=(0,0), B=(36,15)$ and both the coordinates of $C$ are integers. What is the minimum area $\\triangle ABC$ can have?\n$\\textbf{(A)}\\ \\frac{1}{2} \\qquad \\textbf{(B)}\\ 1 \\qquad \\textbf{(C)}\\ \\frac{3}{2} \\qquad \\textbf{(D)}\\ \\frac{13}{2}\\qquad \\textbf{(E)}\\ \\text{there is no minimum}$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "3a8f1cd9-7a36-4100-8e3e-0b9a11b8b6f7"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "How many integers between $2$ and $100$ inclusive cannot be written as $m \\cdot n$, where $m$ and $n$ have no common factors and neither $m$ nor $n$ is equal to $1$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "35",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "6e100349-6666-4f7e-8722-75766a0577ea"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "22 football players took part in the football training. They were divided into teams of equal size for each game (11 vs 11). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "052bac5f-d601-4258-bd0c-8ae47eff8ce6"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "将一个 $3 \\times 3$ 的正方形的四个角上各去掉一个单位正方形所得到的图形称为\"十字形\"。在一个 $10 \\times 11$ 的棋盘上，最多可以放置多少个互不重叠的 \"十字形\"(每个\"十字形\"恰好盖住棋盘上的 5 个小方格）？\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "15",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "696d347e-5116-4264-b9e2-b7a48a728690"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Given that $A^4=75600\\times B$. If $A$ and $B$ are positive integers, find the smallest value of $B$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "25725",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "4b3af2de-d387-4f6f-ada3-6226ddc96850"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Square $ABCD$ has side length $2$. A semicircle with diameter $\\overline{AB}$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $\\overline{AD}$ at $E$. What is the length of $\\overline{CE}$? The original answer is in \\(\\frac{k}{m}\\) format, please give the value of k + m.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "22ed5c2f-0b12-493e-aca5-17ddf7e6bbd0"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "A sequence of integers $x_1, x_2, \\ldots$ is called \"double-dipped\" if it satisfies the recurrence relation $x_{n+2} = ax_{n+1} + bx_n$ for all $n \\ge 1$, where $a$ and $b$ are fixed integers.\n\nRi starts forming a sequence by randomly picking three integers from the set $\\{1, 2, \\ldots, 12\\}$, with replacement. It is known that if Ri adds a fourth term by picking another element at random from $\\{1, 2, \\ldots, 12\\}$, there is at least a $\\frac{1}{3}$ chance that the resulting four-term sequence forms the beginning of a double-dipped sequence.\n\nGiven this information, how many distinct three-term sequences could Ri have initially picked?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "48",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "4aec9727-61c8-43f8-957d-c3f1a2619f15"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "若三个正整数 $a, b, c$ 的位数之和为 8, 且组成 $a, b, c$ 的 8 个数码能排列为 $2,0,2,4,0,9,0,8$ ，则称 $(a, b, c)$ 为 \"幸运数组\"，例如 $(9,8,202400)$ 是一个幸运数组. 满足 $10<a<b<c$ 的幸运数组 $(a, b, c)$ 的个数为 $\\qquad$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "591",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "90376c63-688b-4500-b1fb-85a827f9d940"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Two fair six-sided dice are rolled. The probability that the positive difference between the two rolls is at least $4$ can be written in simplest form as $\\frac{m}{n}$. Compute $m + n$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "d321e61d-e4a1-4ee3-9a2c-2fb0ff705135"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Juan wrote a natural number. Maria then added a digit $1$ to the left and a digit $1$ to the right of Juan's number. As a result, Maria's number exceeds Juan's number by $14789$. Find the original number that Juan wrote.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "532",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "c6fc994b-3a37-404c-8896-632b131e340e"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \\frac{3}{7}$, and\n\\[\na_n = \\frac{a_{n-2} \\cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\n\\]\nfor all $n \\geq 3$. Then $a_{2019}$ can be written as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the value of $p+q$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "8078",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "fb4f15e1-5e28-43e6-bc4a-1a088bedd41a"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Find the number of positive divisors $d$ of $15! = 15 \\cdot 14 \\cdot \\cdots \\cdot 2 \\cdot 1$ such that $\\gcd(d, 60) = 5$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "36",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "97fcabc9-323b-41a0-82cd-341443e48135"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ be three unit vectors, such that the angle between any of them is the acute angle $\\theta.$ The volume of the tetrahedron generated by these three vectors is $\\frac{1}{\\sqrt{360}}.$ Find\n\\[3 \\cos^2 \\theta - 2 \\cos^3 \\theta.\\]The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "19",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "34e67d31-3e5d-4d0c-9112-20df3bfe463a"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "There are $22$ chairs around a round table. Find the minimum number $n$ such that for any group of $n$ people sitting at the table, there will always be two people with exactly $2$ or $8$ chairs between them.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "c5bc2b39-20fd-49eb-9095-5c527cb6cfd6"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "在正方体 \\(A B C D-A_{1} B_{1} C_{1} D_{1}\\) 中，求 \\(B C_{1}\\) 与截面 \\(B B_{1} D_{1} D\\) 所成的角。如果答案是 \\(\\frac{k}{m}\\) 的形式，请以 \\(\\pi\\) 为单位给出，并求出 \\(k + m\\) 的值。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "823b646e-94ac-4d20-b51d-b8aa0358bcbe"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Let $ f(x) = x^3 + x + 1$. Suppose $ g$ is a cubic polynomial such that $ g(0) = - 1$, and the roots of $ g$ are the squares of the roots of $ f$. Find $ g(9)$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "899",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "cc83e6ef-bd01-4327-8571-80f09e306908"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Find the greatest prime divisor of $29! + 33!$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "991",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "eeeb85b9-ce6b-4d32-9494-ed849a6c1bab"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Find the largest positive value attained by the function\n\\[ f(x)=\\sqrt{8x-x^2}-\\sqrt{14x-x^2-48}, \\ x\\ \\text{a real number} \\]\nIf the answer is in the form $a\\sqrt{b}$, where $a$ and $b$ are integers, find the value of $a$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "79ffe5ae-35e8-4205-8974-3359017a633d"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "将一个 99\\times101 的方格表中若干个小方格涂黑，其余小方格涂白. 如果每个黑格至多和一个黑格相邻（有公边），最多能涂黑多少个小方格？\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5017",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "24c5478b-43e8-4618-a89f-ee589387164b"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "过抛物线 $E: y^{2}=2 x$ 的焦点 $F$ 作两条斜率之积为 $-\\frac{1}{2}$ 的直线 $l_{1} 、 l_{2}$, 其中, $l_{1}$ 与抛物线 $E$ 交于 $A 、 C$ 两点, $l_{2}$ 与抛物线 $E$ 交于 $B 、 D$两点.则 $|A C|+2|B D|$ 的最小值为 $\\qquad$. 原始的答案是k\\sqrt{m}+n的形式，请给出k + m + n的值。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "16",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "69a92b90-c174-4752-9b0e-7013d45f8a21"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "If $y = \\sqrt{x^2 - 2x + 1} + \\sqrt{x^2 + 2x + 1}$, find the expression for $y$ in terms of $x$. The original answer is in the form |ax + b| + |cx + d|. Please give the value of a + b + c + d.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "6c833901-e0b2-4063-9270-29f628a9a1dd"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Two distinct positive integers $x$ and $y$ are factors of 36. If $x\\cdot y$ is not a factor of 36, what is the smallest possible value of $x\\cdot y$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "51466923-e493-4fc3-89f9-74c4c77c3eb3"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Find all integers $n$ such that $\\frac{5^n - 1}{3}$ is a prime or a perfect square of an integer.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "058d0f0c-94ca-4653-93bc-9463df11d7e1"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Let $P(x) = x^3 + ax^2 + bx + c$ be a polynomial where $a, b, c$ are integers and $c$ is odd. Let $p_i$ be the value of $P(x)$ at $x = i$. Given that $p_1^3 + p_2^3 + p_3^3 = 3p_1p_2p_3$, find the value of $p_2 + 2p_1 - 3p_0$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "18",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "18335e02-fc67-4d78-9992-8dead3dc2a0b"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "In triangle $ABC$, where $AB = 13$, $BC = 14$, and $CA = 15$, a circle with radius $r$ passes through point $A$ and is tangent to line $BC$ at point $C$. If $r = \\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find the value of $100m + n$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "7508",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "ab9c6ac0-d0d2-467f-bdca-a9b29b2bb8d7"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "The figure shows a convex polygon with nine vertices. The six diagonals drawn dissect the polygon into the following seven triangles: \n- $P_{0}P_{1}P_{3}$\n- $P_{0}P_{3}P_{6}$\n- $P_{0}P_{6}P_{7}$\n- $P_{0}P_{7}P_{8}$\n- $P_{1}P_{2}P_{3}$\n- $P_{3}P_{4}P_{6}$\n- $P_{4}P_{5}P_{6}$\n\nIn how many ways can these triangles be labeled with the names $\\triangle_{1}$, $\\triangle_{2}$, $\\triangle_{3}$, $\\triangle_{4}$, $\\triangle_{5}$, $\\triangle_{6}$, $\\triangle_{7}$ such that $P_{i}$ is a vertex of triangle $\\triangle_{i}$ for $i = 1, 2, 3, 4, 5, 6, 7$? Justify your answer.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "07d77a91-25a9-4326-a436-3c4efd291499"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. Find the probability that the dart lands within the center square. The original answer is in the form \\(\\frac{\\sqrt{k} - 1}{m}\\). Please give the value of k + m.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "52721d50-3904-4948-82ef-29a754a24b12"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Find the largest possible value in the real numbers of the term $$\\frac{3x^2 + 16xy + 15y^2}{x^2 + y^2}$$ with $x^2 + y^2 \\neq 0$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "19",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "7831e846-3039-4301-8190-cc17846efc33"
}
|
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