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": "Find the smallest possible value of $x + y$ where $x, y \\ge 1$ and $x$ and $y$ are integers that satisfy $x^2 - 29y^2 = 1$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "11621",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "0f81be76-1260-473b-a32c-0ce34389bca5"
}
|
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 a soccer tournament, n teams take part. Every team plays with the other one only once. Find the greatest difference between the final scores of the team with neighboring final positions, and express this difference as a function of n. Please provide the value of this difference when n = 10.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "1427ad77-388f-4880-87b5-5513fad40b14"
}
|
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": "已知数列 $\\left\\{a_{n}\\right\\},\\left\\{b_{n}\\right\\}$ 满足:\n$a_{1}=a_{2}=2019, b_{1}=b_{2}=2017$ ，\n且对于一切正整数 $n \\geqslant 2$ ，均有\n$$\n\\begin{aligned}\n& a_{n+1}=\\left(a_{n-1}-1\\right)\\left(b_{n}+1\\right) \\\\\n& b_{n+1}=a_{n} b_{n-1}-1\n\\end{aligned}\n$$\n求 $\\sum_{n=1}^{\\infty} b_{n}\\left(\\frac{1}{a_{n+1}}-\\frac{1}{a_{n+3}}\\right)$ 的值.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": "8148685",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "0d5ceba8-6baf-4230-8cc7-a0c001d0aeea"
}
|
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 real numbers are roots of the polynomial \\[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x?\\]\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "13eaaf18-1ede-41b3-a5c2-3cb592101c68"
}
|
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 $\\triangle A B C$ 的内角 $A, B, C$ 的对边分别为 $a, b, c$, 且 $A-C=\\frac{\\pi}{2}, a, b, c$ 成等差数列, 则 $\\cos B$的值是 $\\qquad$.\n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": "7",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "92367c19-5bdb-411a-bc30-5a1cb846b4aa"
}
|
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 $P_1$ be a regular $r~\\mbox{gon}$ and $P_2$ be a regular $s~\\mbox{gon}$ $(r\\geq s\\geq 3)$ such that each interior angle of $P_1$ is $\\frac{59}{58}$ as large as each interior angle of $P_2$. What's the largest possible value of $s$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "117",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b1013d7b-1998-4312-a527-05c990aeb181"
}
|
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 $k = \\frac{1}{1+2x}$, where $x$ is an integer greater than $1$ and $k$ can be represented as a terminating decimal, find the sum of all possible values of $k$.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": "577ba753-40cd-4b80-9ecf-f7a8775008fa"
}
|
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 primes are in the row of Pascal's Triangle that starts with a $1$ followed by a $6$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b55d7248-6c13-43c1-97aa-bb112b0ecbb1"
}
|
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 five solutions to the equation $$(z-1)(z^2+2z+4)(z^2+4z+6)=0$$ may be written in the form $x_k+y_ki$ for $1 \\le k \\le 5,$ where $x_k$ and $y_k$ are real. Let $\\mathcal E$ be the unique ellipse that passes through the points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4),$ and $(x_5,y_5)$. The eccentricity of $\\mathcal E$ can be written in the form $\\sqrt{\\frac mn}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "06bd8953-9d6c-4ccd-bbeb-6c7a250e6ae9"
}
|
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": "$M$ is the midpoint of the side $AB$ in an equilateral triangle $\\triangle ABC$. The point $D$ on the side $BC$ is such that $BD : DC = 3 : 1$. On the line passing through $C$ and parallel to $MD$, there is a point $T$ inside the triangle $\\triangle ABC$ such that $\\angle CTA = 150^\\circ$. Find the $\\angle MTD$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "120",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "02da3f8c-1a58-430f-8930-b23b856241f2"
}
|
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": "We say that an integer $a$ is a quadratic, cubic, or quintic residue modulo $n$ if there exists an integer $x$ such that $x^2 \\equiv a \\pmod{n}$, $x^3 \\equiv a \\pmod{n}$, or $x^5 \\equiv a \\pmod{n}$, respectively. Further, an integer $a$ is a primitive residue modulo $n$ if it is exactly one of these three types of residues modulo $n$.\n\nHow many integers $1 \\le a \\le 2015$ are primitive residues modulo $2015$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "694",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f4a829c8-5da4-4759-b2e7-faa4fb98781b"
}
|
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 3-digit numbers have the property that the units digit is at least twice the tens digit?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "270",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "9de124ba-3b42-4596-9586-553c452f5c00"
}
|
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": "This year, February $29$ fell on a Sunday. In which year will February $29$ next fall on a Sunday?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2032",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "e5a9e049-d097-41dc-ab07-291f33687b5b"
}
|
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 $\\triangle A B C$ 满足 $A B=1, A C=2, \\cos B+\\sin C=1$, 求 $B C$ 边的长。原始答案为 \\frac{m+n\\sqrt{k}}{p} 的形式，请给出 m + n + k + p 的值。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "31",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "884e20bc-0021-42e8-8003-6d6cb60d0012"
}
|
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 remainder of the number $1 \\binom{2008}{0} + 2 \\binom{2008}{1} + \\ldots + 2009 \\binom{2008}{2008}$ when divided by $2008$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "256",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "bcefd7dc-9cd4-4a66-be1c-fc56ebb7f6d0"
}
|
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": "For how many integers $n$ is $\\dfrac{n}{20-n}$ the square of an integer?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "3ef28b0e-0413-4de1-bb69-df24ad40e6ec"
}
|
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 house worth $9000 is sold by Mr. A to Mr. B at a 10\\% loss. Mr. B sells the house back to Mr. A at a 10\\% gain. How much money does Mr. A lose in total from these two transactions?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "810",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b1d6d5dc-7c72-49d9-a9c4-d7f3daebc4c5"
}
|
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 $LOVER$ be a convex pentagon such that $LOVE$ is a rectangle. Given that $OV = 20$ and $LO = VE = RE = RL = 23$, compute the radius of the circle passing through $R$, $O$, and $V$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "23",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "08eefdca-2eb0-4b2b-b544-a178815760dd"
}
|
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 $t(x) = 3x-8$ and $s(t(x)) = x^2 + 3x - 2$. Find $s(1)$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "16",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "6bdb7774-ea3b-471f-805f-6557ef0ad5f9"
}
|
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 a polynomial of degree $3$ with integer coefficients such that $f(0) = 3$ and $f(1) = 11$.\nIf f has exactly $2$ integer roots, how many such polynomials $f$ exist?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "0bf185b0-fce8-4816-8cd9-d8874399cbf6"
}
|
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": "Each of the integers $1, 2, \\ldots, 729$ is written in its base-$3$ representation without leading zeroes. The numbers are then joined together in that order to form a continuous string of digits: $12101112202122\\ldots$ How many times in this string does the substring $012$ appear?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "148",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "dc86d3d9-96a5-4cf1-bfa5-bd60f3d5fb30"
}
|
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 certain integer has $4$ digits when written in base $8$. The same integer has $d$ digits when written in base $2$. What is the sum of all possible values of $d$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "33",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "883c1a4a-e31b-40fe-b110-c4cb005ee944"
}
|
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 $4$-foot by $8$-foot rectangular piece of plywood will be cut into $4$ congruent rectangles with no wood left over and no wood lost due to the cuts. What is the positive difference, in feet, between the greatest possible perimeter of a single piece and the least possible perimeter of a single piece?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "1a4a707b-a4c8-4b21-a7e5-feabeeea79db"
}
|
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 lattice points that the line $19x + 20y = 1909$ passes through in Quadrant I.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "fcb7fe41-aa1b-4098-8b25-917802b1b0f7"
}
|
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 integer $n$ satisfying the following conditions: \n$\\mathrm{(i)}$ $n^2$ can be expressed as the difference of two consecutive cubes; \n$\\mathrm{(ii)}$ $2n + 79$ is a perfect square.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "181",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "77480081-a576-4a44-beb1-a36a8c6ed159"
}
|
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 quadrilateral $ABCD$, $\\angle{BAD} \\cong \\angle{ADC}$ and $\\angle{ABD} \\cong \\angle{BCD}$, $AB = 8$, $BD = 10$, and $BC = 6$. The length $CD$ may be written in the form $\\frac {m}{n}$, 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": "69",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "d08d5be2-1c1b-4486-992f-6635a4200a00"
}
|
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 integer value of $n < 1000$ such that $\\phi(36n) = \\phi(25n) + \\phi(16n)$, where $\\phi(n)$ denotes the number of positive integers less than $n$ that are relatively prime to $n$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "996",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "9a10960e-4442-446a-87b8-d8c6c480eb5d"
}
|
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": "Samantha is about to celebrate her sweet 16th birthday. To celebrate, she chooses a five-digit positive integer of the form \\( SWEET \\), in which the two \\( E \\)'s represent the same digit but otherwise the digits are distinct. (The leading digit \\( S \\) can't be 0.) How many such integers are divisible by 16?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "282",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f4dc9748-9db6-492d-a539-f60cda26bd57"
}
|
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 real numbers such that\n$$12x - 9y^2 = 7$$\n$$6y - 9z^2 = -2$$\n$$12z - 9x^2 = 4$$\nFind $6x^2 + 9y^2 + 12z^2$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "1aa3be8a-3011-4379-b7c8-24d5a2cf7d8d"
}
|
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 library has six identical copies of a certain book. At any given time, some of these copies are at the library and some are checked out. How many different ways are there for some of the books to be in the library and the rest to be checked out if at least one book is in the library and at least one is checked out? (The books should be considered indistinguishable.)\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "9c05738b-cee9-4b66-a7f9-08f28af23ae6"
}
|
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 decimal expression of a positive number, a, all decimals beginning with the third after the decimal point, are deleted (i.e., we take an approximation of a with accuracy to 0.01 with deficiency). The number obtained is divided by a and the quotient is similarly approximated with the same accuracy by a number b. What numbers b can be thus obtained? Please provide the sum of all possible values of b.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "51",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "50ccb563-5170-41db-915e-88ce299be81d"
}
|
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 cross-section of a prism with an index of refraction $1.5$ is an equilateral triangle, as shown in the diagram. A ray of light enters horizontally from the air into the prism and exits at an angle $\\theta$ with respect to the surface of the triangle. Find $\\theta$ in degrees and round to the nearest whole number.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "e32d3727-7cdb-46a7-a8ac-f4790a376fbb"
}
|
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 $f(n) = 1 \\times 3 \\times 5 \\times \\cdots \\times (2n-1)$. Compute the remainder when $f(1) + f(2) + f(3) + \\cdots + f(2016)$ is divided by $100$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "24",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "ab40f652-760d-4ba3-b6bb-bc3594359c46"
}
|
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 diagram, each of the three identical circles touch the other two. The circumference of each circle is 36. What is the perimeter of the shaded region? [asy]\n\ndefaultpen(1);\n\npath p = (1, 0){down}..{-dir(30)}dir(-60){dir(30)}..{dir(-30)}((2, 0) + dir(-120)){-dir(-30)}..{up}(1, 0)--cycle;\nfill(p, gray(0.75));\n\ndraw(unitcircle);\ndraw(shift(2 * dir(-60)) * unitcircle);\ndraw(shift(2) * unitcircle);\n[/asy]\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "18",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "0e8d48da-1b04-44c3-bc91-49f66ca17a29"
}
|
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 roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by 2 units. What is the volume of the new box?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "932b7ba3-83cc-4ba8-a05d-451b5799eec6"
}
|
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 1,2,3,4,5,6 随机排成一行，记为 a, b, c, d, e, f，则 abc+def 是偶数的概率为________.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": "19",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "ff6b1e0b-c71f-4ca3-bec4-21633fe31d2e"
}
|
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": "求最大的实数 $c$ ，满足：对任意正整数 $n>1$ ，均存在 $\\{1,2, \\cdots, n\\}$ 的子集 $A$ ，使得 $|A| \\geqslant_{c n}$ ，且对任意的 $x 、 y_{、} z \\in A$ （允许相同），均有 $n$ 不是 $x+y-z$ 的约数。请给出m + n的值，其中原始答案为\\frac{m}{n}。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "5b4bd54c-f99b-4e8a-8357-a6b6517e76f6"
}
|
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 $33$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $33$ vertices equals $99$. The midpoints of the sides of $P_1$ form a second $33$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $33$-gon, $P_3$. Find the sum of the $x$-coordinates of the vertices of $P_3$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "99",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f7ae13d7-b69a-451d-9b0d-61063d6f691f"
}
|
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 $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S,$ the probability that it is divisible by $9$ is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$\n\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "-1",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "bac74bb3-d12d-4236-9e9c-08077c44994b"
}
|
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 $ATHEM$ be a convex pentagon with $AT = 14$, $TH = MA = 20$, $HE = EM = 15$, and $\\angle THE = \\angle EMA = 90^{\\circ}$. Find the area of $ATHEM$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "468",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "c6e4e090-162b-4ade-a1e6-fd20e3156954"
}
|
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 points $A$ = $(0 ,0 ,0)$, $B$ = $(1, 0, 0)$, $C$ = $(0, 2, 0)$, and $D$ = $(0, 0, 3)$. Points $E$, $F$, $G$, and $H$ are midpoints of line segments $\\overline{BD},\\text{ } \\overline{AB}, \\text{ } \\overline {AC},$ and $\\overline{DC}$ respectively. Find the area of quadrilateral $EFGH$. The original answer is in the form \\frac{k\\sqrt{m}}{n}, please give the value of k + m + n.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b0bcdaa7-1c5b-4970-899b-d3e6e74e0ee0"
}
|
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": "函数 $y=\\tan x-\\frac{2}{\\cos x}$ 的最大值是多少？请以$-\\sqrt{m}$的格式给出答案，并求出m的值。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "40ff6954-ad95-4929-ba29-cfeabd81e7c6"
}
|
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 $EFGH$, $EFDC$, and $EHBC$ be three adjacent square faces of a cube, for which $EC = 8$, and let $A$ be the eighth vertex of the cube. Let $I$, $J$, and $K$, be the points on $\\overline{EF}$, $\\overline{EH}$, and $\\overline{EC}$, respectively, so that $EI = EJ = EK = 2$. A solid $S$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\\overline{AE}$, and containing the edges, $\\overline{IJ}$, $\\overline{JK}$, and $\\overline{KI}$. The surface area of $S$, including the walls of the tunnel, is $m + n\\sqrt {p}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find $m + n + p$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "417",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "9e1a9f36-c2c8-43ef-929d-1953642be553"
}
|
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 sum of the following seven numbers is exactly 19: \n\\[a_1=2.56,\\ a_2=2.61,\\ a_3=2.65,\\ a_4=2.71,\\ a_5=2.79,\\ a_6=2.82,\\ a_7=2.86.\\] \nIt is desired to replace each $a_i$ by an integer approximation $A_i$, where $1 \\le i \\le 7$, so that the sum of the $A_i$'s is also 19 and the maximum of the \"errors\" $|A_i - a_i|$, denoted as $M$, is minimized. For this minimum $M$, what is $100M$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "61",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "34f3f4a8-be60-4d6d-8336-281e96e94b80"
}
|
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 largest positive integer $n < 1000$ for which there is a positive integer $m$ satisfying \\[ \\text{lcm}(m,n) = 3m \\times \\gcd(m,n)? \\]\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "972",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "598e92ee-ec95-40f8-abd3-f959b4001d48"
}
|
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": "Suppose $a,b$ are nonzero integers such that two roots of $x^3+ax^2+bx+9a$ coincide, and all three roots are integers. Find $|ab|$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1344",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "dceb69ae-2ce3-43fc-885e-be0bd858b93a"
}
|
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": "设集合 $T=\\left\\{(a, b, c) \\mid a, b, c \\in \\mathbf{N}^{*}\\right.$ ，且 $a, b, c$ 可构成某个三角形的边长 $\\}$ 。则 $\\sum_{(a, b, c) \\in T} \\frac{2^{a}}{3^{b} \\cdot 5^{c}}$ 的值为 \\frac{m}{n} 格式，请给出 m + n 的值。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "38",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "449dc0a8-f5a9-4716-9cf6-7a2cf38f1b07"
}
|
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": "对于每个自然数 \\(n\\)，抛物线 \\(y=\\left(n^{2}+n\\right) x^{2}-(2 n+1) x+1\\) 与 \\(x\\) 轴交于 \\(A_{n}, B_{n}\\)两点，以 \\(\\left|A_{n} B_{n}\\right|\\) 表示该两点的距离，求 \\(\\sum_{i=1}^{1992}\\left|A_{i} B_{i}\\right|\\) 的值。答案的形式为\\(\\frac{k}{m}\\)，请给出\\(k + m\\)的值。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "3985",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "07a08004-77a2-4a36-90c8-b1378995315a"
}
|
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": "Consider a triangle $ABC$ where $AB = AC = 15$ and $BC = 18$. Points $D$ and $E$ are chosen on $CA$ and $CB$, respectively, such that $CD = 5$ and $CE = 3$. The point $F$ is chosen on the half-line $\\overrightarrow{DE}$ so that $EF = 8$. If $M$ is the midpoint of $AB$ and $N$ is the intersection of $FM$ and $BC$, what is the length of $CN$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "2dd26053-d481-45c6-a1a8-54b819aaf942"
}
|
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": "Each outcome on the spinner below has equal probability. If you spin the spinner three times and form a three-digit number from the three outcomes, such that the first outcome is the hundreds digit, the second outcome is the tens digit and the third outcome is the units digit, what is the probability that you will end up with a three-digit number that is divisible by 4? Express your answer as a common fraction.\n\n[asy]\ndraw(Circle((0,0),10));\ndraw((0,0)--(8.7,-5));\ndraw((0,0)--(-8.7,-5));\ndraw((0,0)--(0,10));\nlabel(\"1\",(7,7),SW);\nlabel(\"3\",(-7,7),SE);\nlabel(\"2\",(0,-2),S);\ndraw((0,0)--(3,5),Arrow);\n[/asy]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": "11",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "c6139d31-c0ff-46e0-9edd-648e6f568668"
}
|
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": "A5. Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. By rule, a tile must not cover a square that is already covered by another tile. The game ends when no tile can be placed according to this rule. Alice's goal is to maximize the number of uncovered squares when the game ends; Bob's goal is to minimize it. What is the greatest number of uncovered squares that Alice can ensure at the end of the game, no matter how Bob plays?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "290",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "879f3fa7-498b-498f-ab89-7c6e12e6cd38"
}
|
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": "Q. A light source at the point $(0, 16)$ in the coordinate plane casts light in all directions. A disc (circle along with its interior) of radius $2$ with center at $(6, 10)$ casts a shadow on the X-axis. The length of the shadow can be written in the form $m\\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is square-free. Find $m + n$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b96a23be-617d-4047-b651-0d5ba7223d0d"
}
|
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 a_1,a_2,\\cdots,a_{21} 为 1,2,\\cdots,21 的排列，满足|a_{20}-a_{21}|\\geq|a_{19}-a_{21}|\\geq\\cdots\\geq|a_1-a_{21}|.这样的排列的个数为__________.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "3070",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "9def6462-a0e6-49bb-8b11-02749d257490"
}
|
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 real solutions $(x,y,z,w)$ of the simultaneous equations\n\\begin{align*}\n2y &= x + \\frac{17}{x}, \\\\\n2z &= y + \\frac{17}{y}, \\\\\n2w &= z + \\frac{17}{z}, \\\\\n2x &= w + \\frac{17}{w}.\n\\end{align*}\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "fd6618b6-ae7a-4911-86d1-06874821595b"
}
|
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 $x_{1}, x_{2}, x_{3}$ 是非负实数, 满足 $x_{1}+x_{2}+x_{3}=1$, 求 $\\left(x_{1}+3 x_{2}+5 x_{3}\\right)\\left(x_{1}+\\frac{x_{2}}{3}+\\frac{x_{3}}{5}\\right)$的最小值和最大值。已知最小值为整数，最大值为\\frac{m}{n}的格式，请给出m + n+最小值的值。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "15",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "5591558e-1d2d-429a-b6c4-3a55ac34e41c"
}
|
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 greatest possible value of $\\sum_{i=1}^{10}\\cos(3x_i)$ for real numbers $x_1,x_2,\\ldots,x_{10}$ satisfying $\\sum_{i=1}^{10}\\cos(x_i)=0.$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": "529",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "7235ddb1-8f5b-4089-b8a6-0b9c7f12e971"
}
|
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": "使得 n^2+2023n 为平方数的正整数 n 的最小值是__________.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "425",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "d79e6975-b3e4-463b-92cc-942c99def3ac"
}
|
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 是 10 阶实矩阵, 且 $A^{5}=O$, 则 $\\operatorname{rank}(A)$ 的所有可能值之和为 $\\qquad$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "36",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "c1f93952-9b04-46cb-82fc-a4101446bcc5"
}
|
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 triples \\((a, b, c)\\) of positive integers such that \\(a + ab + abc = 11\\).\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "78d3bd6d-0057-4c52-8f5d-da7bfa5ce2bc"
}
|
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 $\\mathbf{a}$ and $\\mathbf{b}$ be two non-zero vectors such that\n\\[\\mathbf{a} \\times (\\mathbf{a} \\times \\mathbf{b}) = -3 \\mathbf{b}.\\]Find $\\|\\mathbf{a}\\|.$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": "4",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "0808daf1-95ba-483c-8275-509934eb19e0"
}
|
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 $x_0, x_1, \\ldots$ be a sequence of real numbers such that $x_n = \\frac{1 + x_{n -1}}{x_{n - 2}}$ for $n \\geq 2$. Find the number of ordered pairs of positive integers $(x_0, x_1)$ such that the sequence gives $x_{2018} = \\frac{1}{1000}$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "128",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "6616aed5-0fcf-409a-94b4-98c5ac3e257c"
}
|
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": "Suppose that all four of the numbers \\[1-\\sqrt{2}, \\;3+\\sqrt{7},\\;1+\\sqrt{2},\\; 5\\]are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "ef1f4ab2-61bf-4be9-b10e-061a7314515e"
}
|
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 class of $50$ students took an exam with $4$ questions. At least $1$ of any $40$ students gave exactly $3$ correct answers, at least $2$ of any $40$ gave exactly $2$ correct answers, and at least $3$ of any $40$ gave exactly $1$ correct answer. At least $4$ of any $40$ students gave exactly $4$ wrong answers. What is the least number of students who gave an odd number of correct answers?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "24",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "7ed7c245-31d4-497a-822b-0ec0d5d6bfb8"
}
|
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 coordinate space, a particle starts at the point $(2,3,4)$ and ends at the point $(-1,-3,-3),$ along the line connecting the two points. Along the way, the particle intersects the unit sphere centered at the origin at two points. Then the distance between these two points can be expressed in the form $\\frac{a}{\\sqrt{b}},$ where $a$ and $b$ are positive integers, and $b$ is not divisible by the square of a prime. Find $a + b.$\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "59",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "324c1cb0-3ef8-4545-ace8-a7299a695884"
}
|
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 a parallelogram $ABCD$, a point $P$ on the segment $AB$ is taken such that $\\frac{AP}{AB} = \\frac{61}{2022}$, and a point $Q$ on the segment $AD$ is taken such that $\\frac{AQ}{AD} = \\frac{61}{2065}$. If $PQ$ intersects $AC$ at $T$, find $\\frac{AC}{AT}$ to the nearest integer.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "67",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "9a27631d-4ee5-4620-bfa6-29ae48adaa8b"
}
|
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": "双曲线 $\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{16}=1$ 的左、右焦点分别为 $A, B, P$ 是双曲线上一点, 若 $\\triangle P A B$ 的内切圆圆心为 $(3,1)$, 则 $\\triangle P A B$ 外接圆的半径为 $\\qquad$.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": "77",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "9c1c342f-c0c2-46bf-b866-10e8726054a3"
}
|
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": "Eight identical cubes with of size $1 \\times 1 \\times 1$ each have the numbers $1$ through $6$ written on their faces with the number $1$ written on the face opposite number $2$, number $3$ written on the face opposite number $5$, and number $4$ written on the face opposite number $6$. The eight cubes are stacked into a single $2 \\times 2 \\times 2$ cube. Add all of the numbers appearing on the outer surface of the new cube. Let $M$ be the maximum possible value for this sum, and $N$ be the minimum possible value for this sum. Find $M - N$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "40",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "68362697-6d8d-45f9-bd6d-6b7596049856"
}
|
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": "For how many positive integer values of $x$ is the sum $x^2+4x+4$ less than 20?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "7a02bedf-c3e5-4eb2-9be2-94f95f289149"
}
|
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 $5 \\times 5$ 矩阵中, 每个元素都为 0 或 1, 且满足：五行的元素之和都相等,但五列的元素之和两两不等.这样的矩阵的个数为 $\\qquad$ （答案用数值表示）.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "26400",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "1972040d-1932-4a57-bc68-55e482fc019a"
}
|
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": "给定整数 $m 、 n(2 \\leqslant m<n,(m, n)=1)$.求最小的整数 $k$ ，满足对集合 $\\{1,2, \\cdots, n\\}$ 的任意 $m$ 元子集 $I$, 若 $\\sum_{i \\in I} i>k$, 则存在 $n$ 个实数 $a_{1} \\leqslant a_{2} \\leqslant \\cdots \\leqslant a_{n}$, 使得\n$$\n\\frac{1}{m} \\sum_{i \\in I} a_{i}>\\frac{1}{n} \\sum_{i=1}^{n} a_{i}\n$$\n若$m=3,n=5$，求$k$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "c100b6db-7342-435d-88a6-0d46eb0f324b"
}
|
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 remainder when 1,234,567,890 is divided by 99?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "72",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "0eae7d97-cf06-4223-b3fe-7c992935b0bc"
}
|
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 less than $1000$ that can be expressed as the difference of two integral powers of $2.$\n\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "50",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "9aac97f3-c2ff-490b-84ee-137ab06e3449"
}
|
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": "11. Let $f(z)=\\frac{a z+b}{c z+d}$ for $a, b, c, d \\in \\mathbb{C}$. Suppose that $f(1)=i, f(2)=i^{2}$, and $f(3)=i^{3}$. If the real part of $f(4)$ can be written as $\\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m^{2}+n^{2}$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "34",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "0b90661f-1197-4750-b274-0a3552fe1eaa"
}
|
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": "Joshua and Wendy are cleaning the storage shed to earn some spending money for the family vacation. They set aside five boxes to take to their bedrooms. Each box is a cube. The four smaller boxes are of equal size and, when stacked, match the height of the larger box.\n\nGiven that the volume of one smaller box is $216$ cubic inches, find the sum of the volumes of all five boxes (in cubic inches).\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "14688",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "761b851c-55a1-49af-9c8d-a2bf4386c0df"
}
|
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": "Add one pair of brackets to the expression $1+2\\times 3+4\\times 5+6$ so that the resulting expression has a valid mathematical value. What is the largest possible value that one can make?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "77",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "875a80b5-9729-493c-86d4-5dfeabdfe965"
}
|
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 Princeton University Band plays a setlist of 8 distinct songs, 3 of which are tiring to play. If the Band can't play any two tiring songs in a row, how many ways can the band play its 8 songs?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "14400",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "c1075062-aaf7-48e2-8ac9-e3a731ddfe71"
}
|
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 2, 3, 5, 6, 7, 10, 11, $\\ldots$ contains all the positive integers from least to greatest that are neither squares nor cubes nor perfect fifth powers (in the form of $x^{5}$, where $x$ is an integer). What is the $1000^{\\mathrm{th}}$ term of the sequence?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1041",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "3ed55b63-4b1e-4cfa-970c-f9a6f492f268"
}
|
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": "Suppose that $x$ and $y$ are real numbers satisfying \\begin{align*}\n4y - 4x^2 &= 1 \\\\\n4x - 4y^2 &= 1.\n\\end{align*} What is $\\dfrac{1}{x^3 + y^3}$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f854f853-b113-4c70-ac97-3c2048e839f2"
}
|
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": "一个均匀的正方体骰子的各面上分别标有数字 $1,2, \\cdots, 6$ ，每次投掷这样两个相同的骰子，规定向上的两个面的数字之和为这次投掷的点数.那么投郑 3 次所得 3 个点数之积能被 14 整除的概率是 $\\qquad$.(用最简分数表示)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": "4",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "9da815e7-1ad5-4b00-809d-e11126ed6107"
}
|
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 a convex quadrilateral with $BC=2$ and $CD=6.$ Suppose that the centroids of $\\triangle ABC,\\triangle BCD,$ and $\\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$? Express your answer in the form $a + b\\sqrt{3}$, where $a$ and $b$ are integers. Please provide the value of a + b.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "22",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "cfe99105-41e0-46d4-8cc4-dddaceb658c5"
}
|
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": "Five points $A$, $B$, $C$, $D$, and $O$ lie on a flat field. $A$ is directly north of $O$, $B$ is directly west of $O$, $C$ is directly south of $O$, and $D$ is directly east of $O$. The distance between $C$ and $D$ is 140 m. A hot-air balloon is positioned in the air at $H$ directly above $O$. The balloon is held in place by four ropes $HA$, $HB$, $HC$, and $HD$. Rope $HC$ has length 150 m and rope $HD$ has length 130 m. How high is the balloon above the field (that is, the length of $OH$)? [asy]\nsize(200);\npair A, B, C, D, O, H, W, X, Y, Z;\nO=(0,0);\nA=(1,1);\nD=(1.5,-.3);\nB=(-1.5,.3);\nC=(-1,-1);\nH=(0,2.5);\nW=(5/3)*(A+D);\nX=(5/3)*(A+B);\nY=(-1)*(W);\nZ=(-1)*(X);\ndraw(W--X--Y--Z--W);\ndraw(A--C);\ndraw(B--D);\ndraw(O--H, linewidth(1));\ndraw(A--H, dashed);\ndraw(B--H, dashed);\ndraw(C--H, dashed);\ndraw(D--H, dashed);\ndot(A);\ndot(B);\ndot(C);\ndot(D);\ndot(O);\ndot(H);\nlabel(\"A\", A, NE);\nlabel(\"B\", B, SW);\nlabel(\"C\", C, SE);\nlabel(\"D\", D, NE);\nlabel(\"O\", O, SE);\nlabel(\"H\", H, NW);\n[/asy]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": "41",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "ca6284b8-5127-4cad-ab75-8f19d4597886"
}
|
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": "By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called nice if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "182",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "7da7dc45-5a88-4a13-a326-21f3d6505bde"
}
|
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 $a_1, a_2, \\ldots$ be a sequence defined by $a_1 = 1$ and for $n \\ge 1$, $a_{n+1} = \\sqrt{a_n^2 - 2a_n + 3} + 1$. Find $a_{513}$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "33",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "50aaf796-b3d3-4a48-b57a-0066121aea95"
}
|
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 $z_1,z_2,z_3,\\dots,z_{12}$ be the $12$ zeroes of the polynomial $z^{12}-2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $i z_j$. Then the maximum possible value of the real part of $\\sum\\limits_{j=1}^{12} w_j$ can be written as $m+\\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m+n$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "784",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "01b81c1c-17c9-412e-aa2e-9a8b55c786a0"
}
|
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 coefficient of $x^{49}$ in the expansion of\n\\[(x + 1)(x + 3)(x + 5) \\dotsm (x + 95)(x + 97)(x + 99).\\]\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2500",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "e1642361-a528-48dd-8f99-253787c19ccb"
}
|
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": "已知双曲线 $C: \\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1(a 、 b>0)$的左、右焦点分别为 $F_{1} 、 F_{2}$, 双曲线 $C$ 与圆 $x^{2}+y^{2}=r^{2}(r>0)$ 的一个交点为 $P$ 。若 $\\frac{\\left|P F_{1}\\right|+\\left|P F_{2}\\right|}{r}$ 的最大值为 $4 \\sqrt{2}$, 则双曲线 $C$的离心率为 $\\qquad$。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": "4",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "ffa7f123-37bb-46b4-abbf-56035d671259"
}
|
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 $\\mathbb{Z}^{\\geq 0}$ be the set of all non-negative integers. Consider a function $f: \\mathbb{Z}^{\\geq 0} \\to \\mathbb{Z}^{\\geq 0}$ such that $f(0) = 1$ and $f(1) = 1$, and for any integer $n \\geq 1$, we have:\n\\[\nf(n + 1)f(n - 1) = nf(n)f(n - 1) + (f(n))^2.\n\\]\nDetermine the value of $\\frac{f(2023)}{f(2022)}$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2045254",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "897b1674-7aca-4f09-ab04-50955191559b"
}
|
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 integers $n$ such that \\[1+\\left\\lfloor\\dfrac{100n}{101}\\right\\rfloor=\\left\\lceil\\dfrac{99n}{100}\\right\\rceil.\\]\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "10100",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "99eeeb73-1741-454b-a468-483b530478d1"
}
|
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 edges and diagonals of convex pentagon $ABCDE$ are all colored either red or blue. How many ways are there to color the segments such that there is exactly one monochromatic triangle with vertices among $A$, $B$, $C$, $D$, $E$; that is, triangles whose edges are all the same color?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "260",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "24b1c70f-11d9-44b6-b937-b43edec8423f"
}
|
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 $x$ be a positive integer such that $9x\\equiv 1\\pmod{25}$.\n\nWhat is the remainder when $11+x$ is divided by $25$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "2eac4ecf-9feb-4815-8fd8-a0fbde0b5876"
}
|
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": "Call a positive integer $n \\geq 2$ \"junk\" if there exist two distinct $n$-digit binary strings $a_1a_2\\cdots a_n$ and $b_1b_2\\cdots b_n$ such that:\n- $a_1 + a_2 = b_1 + b_2$,\n- $a_{i-1} + a_i + a_{i+1} = b_{i-1} + b_i + b_{i+1}$ for all $2 \\leq i \\leq n-1$,\n- $a_{n-1} + a_n = b_{n-1} + b_n$.\n\nFind the number of junk positive integers less than or equal to $2016$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "672",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "0fa90487-f85a-4cbf-bad3-8058806ee60e"
}
|
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 given a positive real number $a$ such that: \n$$ \\left\\{ \\frac{1}{a} \\right\\} = \\{ a^2 \\} $$\n$$ 2 < a^2 < 3 $$\nFind the value of $a^{12} - \\frac{144}{a}$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "233",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "abaa85b1-3447-4e28-a79b-ea1b6d8f8634"
}
|
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 least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "797",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "875f0511-8707-4057-b228-0a5239b501cb"
}
|
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 $a\\%b$ denote the remainder when $a$ is divided by $b$. Find $\\Sigma_{i=1}^{100}(100\\%i)$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1701",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "6c7550ba-1ad2-4e5e-bb08-e3d0eb43bd0d"
}
|
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": "若正方体 $A_{1} A_{2} A_{3} A_{4}-B_{1} B_{2} B_{3} B_{4}$ 的棱长为 1 ，则集合\n$\\left\\{x \\mid x=\\overrightarrow{A_{1} B_{1}} \\cdot \\overrightarrow{A_{i} B_{j}}, i, j \\in\\{1,2,3,4\\}\\right\\}$\n中元素的个数为 $\\qquad$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "aacb5831-e03b-4bf4-b811-35ad7d928a96"
}
|
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": "Suppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite \"word\" $A{ }^{{ }^{1}} B^{n_{2}} A^{n_{3}} \\cdots B^{n_{k}}$, where $n_{1}, \\ldots, n_{k}$ are any integers, and $A^{\\prime \\prime}=B^{\\prime \\prime}=1$ as usual. Also, suppose that $A^{4}=B^{7}=A B A^{-1} B=1, A^{2} \\neq 1$, and $B \\neq 1$.\n\n(a) How many elements of $G$ are of the form $C^{2}$ with $C$ in $G$?\n\n(b) Write each such square as a word in $A$ and $B$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "0f7dcba2-3e0a-4cd8-a1f9-7de63d611f95"
}
|
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 2-digit numbers $n$ having the property:\n\nThe number $n^2$ is a 4-digit number of the form $\\overline{xxyy}$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "88",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "e25f817c-7fd3-4ec2-9de5-ca4d930908a6"
}
|
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 a regular 2007-gon, find the minimal number $k$ such that among every $k$ vertices of the polygon, there always exist 4 vertices forming a convex quadrilateral where 3 sides of the quadrilateral are also sides of the polygon.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1506",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "1bfc790a-0002-4cbc-b666-84fe3b1bd8c3"
}
|
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 $S$ be the set of all positive rational numbers $r$ such that when the two numbers $r$ and $55r$ are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of $S$ can be expressed in the form $\\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "719",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "578d78cc-147e-4951-9cdf-81c1be76d80e"
}
|
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 $n \\in \\mathbf{N}_{+}$，用 $d(n)$ 表示 $n$ 的所有正约数的个数, $\\varphi(n)$ 表示 $1,2, \\cdots, n$ 中与 $n$ 互质的数的个数. 求所有的非负整数 $c$, 使得存在正整数 $n$, 满足$$d(n)+\\varphi(n)=n+c,$$且对这样的每一个 $c$, 求出所有满足上式的正整数 $n$ 。请给出所有满足条件的正整数 $n$ 的和。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "23",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "6f1b13d3-5de5-4b87-a5a3-2b1bc3e3c2cb"
}
|
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