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": "Given three equilateral triangles $\\triangle HOW$, $\\triangle BOW$, and $\\triangle DAH$ in a plane:\n- $WO = 7$\n- $AH = 2$\n\nIt is also given that the points $D$, $A$, and $B$ are collinear in that order. \n\nFind the length of $BA$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "29201a6c-9ce5-496b-9a01-5765f287c662"
}
|
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 pen costs \\(11\\) € and a notebook costs \\(13\\) €. Find the number of ways in which a person can spend exactly \\(1000\\) € to buy pens and notebooks.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "df61ee30-69bc-466f-8f4d-99f00a114708"
}
|
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 triangle $\\triangle ABC$ is situated on the plane and a point $E$ is given on segment $AC$. Let $D$ be a point in the plane such that lines $AD$ and $BE$ are parallel. Suppose that $\\angle EBC = 25^\\circ$, $\\angle BCA = 32^\\circ$, and $\\angle CAB = 60^\\circ$. Find the smallest possible value of $\\angle DAB$ in degrees.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "63",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b0b34e7d-cec3-4926-8972-1acd49b1bc3a"
}
|
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 will say that a list of positive integers is *admissible* if all its numbers are less than or equal to $100$ and their sum is greater than $1810$. Find the smallest positive integer $d$ such that from each admissible list, some numbers can be crossed out such that the sum of the numbers left uncrossed is greater than or equal to $1810-d$ and less than or equal to $1810+d$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f48e4691-e4b8-4c90-80e9-9c1db204a978"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "What is the smallest non-square positive integer that is the product of four prime numbers (not necessarily distinct)?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "24",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "84e7fdc1-7adb-4fa5-99f3-790a5d9843b1"
}
|
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 f(x)=\\ln(1+x)(x>0)，定义函数序列 \\{f_n(x)\\}，其中 f_n 是 f 的 n 次迭代. 求所有的 x，使得 \\displaystyle\\lim_{n\\rightarrow+\\infty}nf_n(x) 存在，并求出该极限.\n让我们一步一步地思考。请以“Answer: \\boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "23c85a97-0d82-432c-a9e1-8cc591c13658"
}
|
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 $a$ and $b$ be real numbers such that $$\\left(8^a+2^{b+7}\\right)\\left(2^{a+3}+8^{b-2}\\right)=4^{a+b+2}.$$ The value of the product $ab$ can be written as $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "271",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "ab59beb5-a718-4e87-b791-d2f0c35e0558"
}
|
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 $ABC$ be a triangle such that $AB = 7$, $BC = 8$, and $CA = 9$. There exists a unique point $X$ such that $XB = XC$ and $XA$ is tangent to the circumcircle of $\\triangle ABC$. If $XA = \\frac{a}{b}$, where $a$ and $b$ are coprime positive integers, find $a + b$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "61",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "995a4c9e-fc34-4d46-9df9-8eecee95cc46"
}
|
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 sum of the real roots of the polynomial:\n\\[ \\prod_{k=1}^{100} \\left( x^2-11x+k \\right) = \\left( x^2-11x+1 \\right)\\left( x^2-11x+2 \\right)\\dots\\left(x^2-11x+100\\right). \\]\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "330",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "86196489-a5ce-446c-b6e4-57d4dbc6b9c4"
}
|
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 integer $n$ such that $n$ is divisible by every integer less than $\\sqrt[3]{n}$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "420",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "bb265a3a-3768-49c4-8c1e-e6f75e2447b4"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Two cubical dice each have removable numbers $1$ through $6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is $7$? Express your answer in the form \\(\\frac{k}{m}\\), where k and m are coprime, and find the value of k + m.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "4a4936ba-a40e-4a4e-9fd4-e01b3ad572bd"
}
|
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": "数列 \\{a_n\\} 满足 a_1=1,\\df{a_{n+1}-a_n}{a_n}=\\df{a_{n+2}-a_{n+1}}{a_{n+2}} (n\\in\\bN^\\ast). 若 a_1a_2+a_2a_3+\\cdots+a_6a_7=3，则 a_{2024}=__________.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": "2035",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "4668ef21-33ce-4628-bbd8-6c6d6c52f893"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Let $k$ be a real number such that the product of real roots of the equation $$X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0$$ is $-2013$. Find the sum of the squares of these real roots.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "4027",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "53e34e17-6571-4281-8778-d107f2dd848d"
}
|
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": "$ABCDEF GH$ is a regular octagon with a side length of $10$ units. Consider a circle with center $A$ and radius $AC$, which intersects another circle with center $D$ and radius $CD$ at a point $I$, different from $C$. What is the length of the segment $IF$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "dd2b3863-404c-4fff-a9e3-4473e58d50c6"
}
|
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": "Convex quadrilateral $ABCD$ has perpendicular diagonals and satisfies the conditions $\\angle B = \\angle C = 90^\\circ$, $BC = 20$, and $AD = 30$. Compute the square of the area of a triangle with side lengths equal to $CD$, $DA$, and $AB$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "30000",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b09321ad-4ece-45f1-b6c7-9fd1c0e5096e"
}
|
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": "$[2 \\pi-6,4-\\pi]$\n\n设 $f(x)$ 是定义在 $\\mathbf{R}$ 上的以 2 为周期的偶函数, 在区间 $[1,2]$ 上严格递减, 且满足 $f(\\pi)=1, f(2 \\pi)=0$, 则不等式组 $\\left\\{\\begin{array}{l}0 \\leqslant x \\leqslant 1, \\\\ 0 \\leqslant f(x) \\leqslant 1\\end{array}\\right.$ 的解集为. 请给出区间左端点与两倍右端点之和的值。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f815f4dd-bf11-4b86-803d-45f6dda84ff7"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Given that $(1+\\sin t)(1+\\cos t)=5/4$ and\n\n$(1-\\sin t)(1-\\cos t)=\\frac mn-\\sqrt{k},$\nwhere $k, m,$ and $n_{}$ are positive integers with $m_{}$ and $n_{}$ relatively prime, find $k+m+n.$\n\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "27",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "3dd7a812-2c4f-478f-bee8-ab8a03ca8aa1"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "There are three men and eleven women taking a dance class. In how many different ways can each man be paired with a woman partner and then have the eight remaining women be paired into four pairs of two?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "103950",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "672a499b-629b-43c9-adff-82ead0190caa"
}
|
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 the set $S$ of the eight points $(x,y)$ in the Cartesian plane satisfying $x,y \\in \\{-1, 0, 1\\}$ and $(x,y) \\neq (0,0)$. How many ways are there to draw four segments whose endpoints lie in $S$ such that no two segments intersect, even at endpoints?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "1e2e5dce-3413-45a8-b94e-9a54c978390b"
}
|
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 $a_1, a_2, \\dots$ be a sequence defined by $a_1 = a_2=1$ and $a_{n+2}=a_{n+1}+a_n$ for $n\\geq 1$. Find \\[\n\\sum_{n=1}^\\infty \\frac{a_n}{4^{n+1}}.\n\\]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": "12",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "570e2b25-4c56-4b41-aa09-e41db24eb15f"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Square \\(ABCD\\) has side length \\(2\\). A semicircle with diameter \\(\\overline{AB}\\) is constructed inside the square, and the tangent to the semicircle from \\(C\\) intersects side \\(\\overline{AD}\\) at \\(E\\). The length of \\(\\overline{CE}\\) can be expressed in the form \\(\\frac{k}{m}\\), where \\(\\frac{k}{m}\\) is a simplified fraction. Please find the value of \\(k + m\\).\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "7c06ebb1-4191-4901-8242-029a26345f95"
}
|
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": "在平面直角坐标系中, 点 $A, B, C$ 在双曲线 $x y=1$ 上, 满足 $\\triangle A B C$ 为等腰直角三角形。求 $\\triangle A B C$ 的面积的最小值.\n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": "6",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "16b804ab-bdb1-4511-8e0b-9941a9777e43"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Given that three roots of \\( f(x) = x^4 + ax^2 + bx + c \\) are \\( 2 \\), \\( -3 \\), and \\( 5 \\), what is the value of \\( a + b + c \\)?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "79",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "90a0bac3-d854-42ad-905b-1c6f69e6dc85"
}
|
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 $a>0$, 函数 $f(x)=x+\\frac{100}{x}$ 在区间 $(0, a]$ 上的最小值为 $m_{1}$, 在区间 $[a,+\\infty)$ 上的最小值为 $m_{2}$, 若 $m_{1} m_{2}=2020$, 则 $a$ 的值为 $\\qquad$。请提供所有可能的 $a$ 的值的和。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "101",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b3e7d7d7-89bc-4a64-9a65-bb47f3acde4c"
}
|
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 $f(x)=\\frac{1-2 x}{\\left(1-x+x^{2}\\right)^{2}}$, 则 $\\left[\\frac{f^{(2022)}(0)}{2022!}\\right]=$ $\\qquad$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2023",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "610b004a-9008-487b-a99a-502df670c7b7"
}
|
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 $\\text{lcm} (a,b)$ denote the least common multiple of $a$ and $b$. Find the sum of all positive integers $x$ such that $x \\le 100$ and $\\text{lcm}(16,x) = 16x$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2500",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "7e9596bb-2fcc-45d2-a7ef-32f2ff205283"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Define the determinant $D_1 = |1|$, the determinant $D_2 = \\begin{vmatrix} 1 & 1 \\\\ 1 & 3 \\end{vmatrix}$, and the determinant $D_3 = \\begin{vmatrix} 1 & 1 & 1 \\\\ 1 & 3 & 3 \\\\ 1 & 3 & 5 \\end{vmatrix}$.\n\nIn general, for a positive integer $n$, let the determinant $D_n$ have 1s in every position of its first row and first column, 3s in the remaining positions of the second row and second column, 5s in the remaining positions of the third row and third column, and so forth. Find the least $n$ such that $D_n \\geq 2015$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "71282c14-698a-4795-8f95-fafc9f771c7c"
}
|
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\"面包圈\"是指从 (a+2)\\times(b+2) 矩形的中间去掉 a\\times b 矩形后得到的由 2a+2b+4 个单位方格构成的环，其中 a,b 是任意正整数.考虑无限大的方格表. 对每个偶数 n\\geq8，一个\"n-阶面包店\"是指由有限个方格构成的集合 S，满足对任意一个由 n 个方格构成的面包圈 B，存在与 B 全等（可平移和旋转）的面包圈，其方格都属于 S. 用 f(n) 表示 n-阶面包店中方格数的最小可能值.求实数 \\alpha，使得对任意充分大的偶数 n，均有\\df{1}{100}<\\df{f(n)}{n^\\alpha}<100.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": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "c0a5cb2e-95c1-4bea-bc40-a4a18ce92756"
}
|
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": "Steve has three quarters, three nickels and three pennies. If Steve selects three coins at random and without replacement, what is the probability that the total value is exactly 35 cents? Express your answer as a common fraction.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "31",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "613df862-6d5f-4d06-8556-18726dc55254"
}
|
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 $n \\in \\mathbb{N}$, consider non-negative valued functions $f$ on $\\{1,2, \\cdots , n\\}$ satisfying $f(i) \\geq f(j)$ for $i > j$ and $\\sum_{i=1}^{n} (i + f(i)) = 2023.$ Choose $n$ such that $\\sum_{i=1}^{n} f(i)$ is minimized. How many such functions exist in that case?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "15",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "07915631-0d1a-49a9-a978-2e8e1169ea9b"
}
|
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": "As in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side (i.e. $C$ is between $B$ and $E$ and $G$ is between $C$ and $D$). If $CE = 14$ and $AB > 14$, compute the minimal area of $\\triangle AEG$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "98",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "49d10c1c-56bc-40cd-9791-3d416872aa6d"
}
|
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 $u_0, u_1, u_2, \\ldots$ be integers such that $u_0 = 100$; $u_{k+2} \\geq 2 + u_k$ for all $k \\geq 0$; and $u_{\\ell+5} \\leq 5 + u_\\ell$ for all $\\ell \\geq 0$. Find all possible values for the integer $u_{2023}$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2123",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "6bcc686f-8cf0-4888-a917-6c07fd2ba79d"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Find all integers $k$ such that all roots of the following polynomial are also integers:\n\n$$f(x) = x^3 - (k-3)x^2 - 11x + (4k-8).$$\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b7119f82-4161-4e79-8194-93596b4f2fa3"
}
|
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": "一珍稀物种出现在地球，对每个珍稀生物，每天有如下事件发生：有 $p(0 \\leq p \\leq 1)$ 的概率消失，有 $\\frac{1-p}{3}$的概率保持不变，有 $\\frac{1-p}{3}$ 的概率分裂成两个，有 $\\frac{1-p}{3}$ 的概率分裂成三个.对所有新产生的生物每天也会发生上述事件.假设开始只有一个这样的珍稀生物，若希望最终这种生物灭绝的概率不超过 $\\frac{1}{2}$ ，则 $p$ 至多为 $\\qquad$.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": "22",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "9cd66adc-8d56-4ea4-b17f-30775f056e62"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "An *iterative average* of the numbers \\(1\\), \\(2\\), \\(3\\), \\(4\\), and \\(5\\) is computed in the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. Find the difference between the largest and smallest possible values that can be obtained using this procedure. The answer is in the form \\(\\frac{k}{m}\\). Please provide the value of \\(k + m\\).\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "25",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b4155cb3-e1db-4397-94fb-f6e4841b06b2"
}
|
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": "已知正数 a,b,c 满足 a+b+c=13. 则 \\df{a^2+b^3+c^4+2019}{10b+123c+26} 的最小值为__________.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "815e1bff-f9ae-45ac-80d8-cfc9eec8bbb2"
}
|
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 $x$-intercepts on the graph of $y = \\sin \\frac{1}{x}$ (evaluated in terms of radians) in the interval $(0.0001, 0.001).$\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2865",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "92d3047f-26d2-4c97-9588-7f0613ba9440"
}
|
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": "Mr. Smith brings home 7 animals for his 7 children. Each child will adopt a pet to be her or his own. There are 4 different cats (a Siamese, a Persian, a Calico, and a Minx), 2 different dogs (a Poodle and a Golden Retriever), and a goldfish. Anna and Betty refuse to take care of the goldfish, and Charlie and Danny insist on having cats. The other 3 kids are easier to please -- they'll take anything. In how many ways can Mr. Smith give the children pets they'll want?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "864",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "d9f6707d-ce57-4a11-adfb-ea31903fd62d"
}
|
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 $\\varphi$ be the positive solution to the equation $x^2 = x + 1$. For $n \\ge 0$, let $a_n$ be the unique integer such that $\\varphi^n - a_n \\varphi$ is also an integer. Compute $\\sum_{n=0}^{10} a_n$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "143",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "bdfe8e99-9ab4-45f6-a285-9c44ef939193"
}
|
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 convex quadrilateral $ABCD, \\angle A \\cong \\angle C, AB = CD = 180,$ and $AD \\neq BC.$ The perimeter of $ABCD$ is $640$. Find $\\lfloor 1000 \\cos A \\rfloor.$ (The notation $\\lfloor x \\rfloor$ means the greatest integer that is less than or equal to $x.$)\n\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "777",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "54ecd939-9bc4-4030-b4d1-4f0d034f6495"
}
|
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": "Andrew writes down all of the prime numbers less than $50$. How many times does he write the digit $2$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "14149baa-26ca-4034-b030-44745f3a8601"
}
|
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": "一个三角形的一条高长为 $2,$ 另一条高长为 $4,$ 则这个三角形内切圆半径取值范围是 \\qquad. 答案的标准形式为[\\frac{a}{b},c]，请给出区间两端a+b+c的值。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "89069521-f666-4841-930b-3b41bbee1820"
}
|
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 value of $t$ such that \\[\\frac{13t^2 - 34t + 12}{3t - 2 } + 5t = 6t - 1.\\]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": "7",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "83fe2af4-f872-44df-90ad-d473c6073a04"
}
|
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 a>b>1，若 \\log_ab+\\log_ba=\\df{5}{2}，则 \\df{b}{a+4} 的最大值为__________.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": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "aad1cb22-1e24-4227-b6e2-48e20f703c30"
}
|
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 numbers $1, 2, 3, 4, 5, 6, 7,$ and $8$ are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where $8$ and $1$ are considered to be consecutive, are written on faces that share an edge is $m/n,$ 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": "85",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "14dcdc1f-35ed-427b-92ee-e6ae0a8c3281"
}
|
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": "桌上有两个盒子，初始时，其中一个是空的，另一个有 29 个不同颜色的石子. 下面对这两个盒子轮流进行如下操作，从满的盒子开始. 在每次操作中，将当前盒子中的至少一个石子移动到另一个盒子中. 如果各次操作移动的石子不完全相同，问：至多能进行多少次操作？\n让我们一步一步地思考。请以“Answer: \\boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "29",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "cb08fad6-f3ea-48b3-b679-58a2f164a786"
}
|
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": "Triangle $ABC$ is an isosceles triangle with $\\overline{AB}=\\overline{BC}$. Point $D$ is the midpoint of both $\\overline{BC}$ and $\\overline{AE}$, and $\\overline{CE}$ is $11$ units long. Triangle $ABD$ is congruent to triangle $ECD$. If the length of $\\overline{BD}$ is $x$ units, what is the value of $10x$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "55",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "87f0ed72-0834-4a50-a821-1d82b0436ca6"
}
|
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": "\\( \\triangle ABC \\) has \\( AB = 4 \\) and \\( AC = 6 \\). Let point \\( D \\) be on line \\( AB \\) so that \\( A \\) is between \\( B \\) and \\( D \\). Let the angle bisector of \\( \\angle BAC \\) intersect line \\( BC \\) at \\( E \\), and let the angle bisector of \\( \\angle DAC \\) intersect line \\( BC \\) at \\( F \\). Given that \\( AE = AF \\), find the square of the circumcircle's radius' length.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "c83617d8-7d93-4b68-bc6f-f9b9c5183f48"
}
|
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 unit price of a screw is listed as $0.2$ cents. When a group of screws is sold to a customer, the total cost of the screws is computed using the listed price and then rounded to the nearest cent. If Al has $50$ cents and wishes to make only one purchase, what is the maximum possible number of screws he can buy?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "252",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "5cbf07aa-b777-4600-b9ce-ba150113a474"
}
|
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 $a, b, c$ be real nonzero numbers such that $a + b + c = 12$ and \\[ \\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} + \\frac{1}{abc} = 1. \\] Compute the largest possible value of $abc - \\left(a + 2b - 3c\\right)$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "56",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "45a96338-f6c2-4187-a320-5fa311f459ca"
}
|
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 digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "23",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "5ec1a0af-f0e6-4054-abd5-56b7ce7b5b7f"
}
|
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 $\\sin ^{4} 10^{\\circ}+\\sin ^{4} 50^{\\circ}+\\sin ^{4} 70^{\\circ}$ 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": "17",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "fbd74d8a-3760-4c30-916a-81174aac1c5d"
}
|
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 complex numbers of the form $x + yi,$ where $x$ and $y$ are real numbers, such that\n\\[\\frac{\\sqrt{2}}{2} \\le x \\le \\frac{\\sqrt{3}}{2}.\\]Find the smallest positive integer $m$ such that for all positive integers $n \\ge m,$ there exists a complex number $z \\in S$ such that $z^n = 1.$\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "16",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "620654c0-1caf-484c-a04e-7018b6ba5675"
}
|
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 product of the real roots of the equation $x^4+3x^3+5x^2 + 21x -14=0$. The answer is an integer. Please provide the integer value.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "-2",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "dc0272d8-cd32-4c4b-b336-5b2d188b5ee9"
}
|
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": "$a_{n}$ 是离 $\\sqrt{n}$ 最近的整数, 则 $S_{2021}=$\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "60585",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "594f78c5-e3b0-4c08-a4d0-77ca13459814"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Three rods of lengths $1396$, $1439$, and $2018$ millimeters have been hinged from one tip on the ground. What is the smallest value for the radius of the circle passing through the other three tips of the rods in millimeters?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "311",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "81621120-d8e4-4ed7-add9-1480224a204d"
}
|
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 equation $x^2+14x=33$ has two solutions. The positive solution has the form $\\sqrt{a}-b$ for positive natural numbers $a$ and $b$. What is $a+b$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "89",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b20df18b-db74-40c5-a183-ba8e6905b778"
}
|
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 square with side length \\(3\\) is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length \\(2\\) has two vertices on the other square and the other two on sides of the triangle, as shown. What is the area of the triangle in the form of a fraction \\(\\frac{k}{m}\\)? Please provide the value of \\(k + m\\).\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "85",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "7c304c37-f660-4727-af8c-fac94c230cda"
}
|
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 each positive integer $k$, let $S_k$ denote the increasing arithmetic sequence of integers whose first term is 1 and whose common difference is $k$. For example, $S_3$ is the sequence $1,4,7,\\ldots$. For how many values of $k$ does $S_k$ contain $2005$ as a term?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f35fb17a-b502-4711-95fc-fa5aaf706bda"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Given that $\\displaystyle {{\\left((3!)!\\right)!}\\over{3!}}= k\\cdot\nn!$, where $k$ and $n$ are positive integers and $n$ is as large as possible, find $k+n$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "839",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "9150b001-c379-4b1b-b529-d8dbfd6b632d"
}
|
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 area of rectangle $ABCD$ is 72. If point $A$ and the midpoints of $\\overline{BC}$ and $\\overline{CD}$ are joined to form a triangle, what is the area of the triangle?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "27",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "0a3d137b-59d8-49d0-9601-04e2325a2f2e"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Let $p(x)$ be the polynomial of degree at most 8 such that\n\\[p(n) = \\frac{1}{n}\\]for $n = 1,$ 2, 3, $\\dots,$ 9. Find $p(10).$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": "6",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "1a2883a8-3661-43d3-8657-38fafa502265"
}
|
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$(x-1)(x-2)(x-3)(x-4)$的最小值，其中$x$为实数。请给出最小值的整数部分。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "-1",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "c6d73047-6f58-45f3-835e-3dfbd65be30c"
}
|
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 $n$ for which there exists a sequence $(a_0, a_1, \\ldots, a_n)$ of non-zero digits such that, for each $k$, $1 \\le k \\le n$, the $k$-digit number $\\overline{a_{k-1} a_{k-2} \\ldots a_0} = a_{k-1} 10^{k-1} + a_{k-2} 10^{k-2} + \\cdots + a_0$ divides the $(k+1)$-digit number $\\overline{a_{k} a_{k-1}a_{k-2} \\ldots a_0}$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "255d1c07-7bca-4c6d-b8e2-6ac0e1871a59"
}
|
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": "设数列 $\\left\\{a_{n}\\right\\}$ 是等差数列，数列 $\\left\\{b_{n}\\right\\}$ 满足 $b_{n}=a_{n+1} a_{n+2}-a_{n}^{2}, n=1,2, \\cdots$ 。\n（2）设数列 $\\left\\{a_{n}\\right\\},\\left\\{b_{n}\\right\\}$ 的公差均是 $d \\neq 0$ ，并且存在正整数 $s, t$ ，使得 $a_{s}+b_{t}$ 是整数，求 $\\left|a_{1}\\right|$ 的最小值。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": "19",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "7214b03e-1a2b-40de-9ca2-1c6a42c13992"
}
|
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 pairs \\((a, b)\\) of positive integers such that the greatest common divisor (GCD) of \\(a\\) and \\(b\\) is equal to \\(1 \\cdot 2 \\cdot 3 \\cdot \\ldots \\cdot 50\\), and the least common multiple (LCM) of \\(a\\) and \\(b\\) is \\(1^2 \\cdot 2^2 \\cdot 3^2 \\cdot \\ldots \\cdot 50^2\\).\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "32768",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "4271424c-82a1-428f-928a-802e41e3b25f"
}
|
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 $\\ell_1$ and $\\ell_2$ be two parallel lines, a distance of 15 apart. Points $A$ and $B$ lie on $\\ell_1$ while points $C$ and $D$ lie on $\\ell_2$ such that $\\angle BAC = 30^\\circ$ and $\\angle ABD = 60^\\circ$. The minimum value of $AD + BC$ is $a\\sqrt b$, where $a$ and $b$ are integers and $b$ is squarefree. Find $a + b$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "23",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "5cb94761-5b69-466d-b5b1-f8a8031035e0"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "已知函数 $f(x)=|2 x-4|-3 x+8$ $(x \\in \\mathbf{R})$. 求 $f(x)$ 的反函数 $f^{-1}(x)$ 的解析式。请将答案写成如下分段函数的形式：\\begin{cases}-ax+b,\\:x\\leqslant2\\\\-\\frac{c}{d}x+\\frac{e}{f},\\:x>2\\end{cases}，其中a, b, c, d, e, f是待定系数。请给出a + b + c + d + e + f的值。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "28",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f97f24cc-678a-4645-997f-514b283bfa70"
}
|
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 infinite geometric series of positive reals $a_1, a_2, \\dots$ satisfies the equation:\n\\[ 1 = \\sum_{n=1}^\\infty a_n = -\\frac{1}{2013} + \\sum_{n=1}^{\\infty} \\text{GM}(a_1, a_2, \\dots, a_n) = \\frac{1}{N} + a_1 \\]\nwhere $\\text{GM}(x_1, x_2, \\dots, x_k) = \\sqrt[k]{x_1x_2\\cdots x_k}$ denotes the geometric mean. Compute $N$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "4052169",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "5b7be1d0-e341-4ab6-95d4-4195aa400551"
}
|
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 park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point?\n$\\textbf{(A)}\\ \\sqrt{13}\\qquad \\textbf{(B)}\\ \\sqrt{14}\\qquad \\textbf{(C)}\\ \\sqrt{15}\\qquad \\textbf{(D)}\\ \\sqrt{16}\\qquad \\textbf{(E)}\\ \\sqrt{17}$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": "14",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "c14fbb54-23d4-4757-b246-d706c25cacc1"
}
|
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 real numbers $a_0, a_1, \\dots, a_{2013}$ and $b_0, b_1, \\dots, b_{2013}$ satisfy the recurrence relations:\n\\[\na_{n} = \\frac{1}{63} \\sqrt{2n+2} + a_{n-1} \\quad \\text{and} \\quad b_{n} = \\frac{1}{96} \\sqrt{2n+2} - b_{n-1}\n\\]\nfor every integer $n = 1, 2, \\dots, 2013$. Given the initial conditions $a_0 = b_{2013}$ and $b_0 = a_{2013}$, compute the following sum:\n\\[\n\\sum_{k=1}^{2013} \\left( a_kb_{k-1} - a_{k-1}b_k \\right).\n\\]\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "671",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "fabac893-6189-405f-9c53-1b21d69b432b"
}
|
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_n\\) denote the angle opposite to the side of length \\(4n^2\\) units in an integer right-angled triangle with side lengths \\(4n^2\\), \\(4n^4+1\\), and \\(4n^4-1\\), where \\(n \\in \\mathbb{N}\\). Find the value of:\n\\[\n\\lim_{p \\to \\infty} \\sum_{n=1}^p a_n\n\\]\nThe original answer is in the format \\(\\frac{k\\pi}{m}\\). Please give the value of \\(k + m\\).\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "5f95a343-d23f-4e3c-b17c-c14e9e26dcb8"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Find the minimum value of\n\\[\\frac{\\sin^6 x + \\cos^6 x + 1}{\\sin^4 x + \\cos^4 x + 1}\\]over all real values $x.$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": "feb41d05-4db3-4c6c-a391-2279a6fe1e7c"
}
|
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": "两个半径都是 1 的球 $O_{1}$ 和球 $O_{2}$ 相切，且均与直二面角 $\\alpha-l-\\beta$ 的两个半平面都相切，另有一个半径为 $r(r<1)$ 的小球 $O$与这个二面角的两个半平面也都相切, 同时,与球 $O_{1}$ 和球 $O_{2}$ 都外切。则 $r$ 的值为原始答案为 $n-\\sqrt{k}$ 的形式，请给出n + k的值。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "a6fab060-da83-46f8-9ee1-65901bf5be6f"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "如图3，在直角坐标系 $x O y$中， $\\\\odot O: x^{2}+y^{2}=4$ 与 $x$ 轴的正半轴交于点 $A, \\\\odot A:(x-2)^{2}+y^{2}=r^{2}(r>0)$ 与 \\\\odot O 交于 $B 、 C$ 两点. (1) 求 \\\\overrightarrow{A B} \\\\cdot \\\\overrightarrow{A C} 的最小值; (2) 设 $P$ 为 \\\\odot O 上异于 $B 、 C$ 的任一点，直线 $P B 、 P C$ 与 $x$ 轴分别交于点 $M 、 N$, 求 $S_{\\\\triangle P O M} S_{\\\\triangle P O N}$ 的最大值。请提供两个部分答案之和。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "00d8f228-92a3-41c9-bce0-568bb699a2fe"
}
|
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 a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\\frac{1}{2}$ is $\\frac{a-b\\pi}{c}$, where $a$, $b$, and $c$ are positive integers and $\\gcd(a,b,c)=1$. Find the value of $a+b+c$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "59",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "fe7b9526-a184-4af1-a243-bba80f488b9f"
}
|
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": "Circle $w$ with center $O$ meets circle $\\Gamma$ at $X,Y,$ and $O$ is on $\\Gamma$. Point $Z\\in\\Gamma$ lies outside $w$ such that $XZ=11$, $OZ=15$, and $YZ=13$. If the radius of circle $w$ is $r$, find $r^2$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "82",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "a8a4ae70-c16c-47b2-be6a-b3689481f379"
}
|
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 hundreds digit of $2011^{2011}$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "fe23c54d-16d2-4487-b711-3aac1d177ce0"
}
|
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 what value of $n$ is $i + 2i^2 + 3i^3 + \\cdots + ni^n = 48 + 49i$? Find the value of $n$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "97",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "e6c453c8-d43f-4765-b397-9a792fcc3539"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "In $\\triangle ABC$, $\\angle A = 30^{\\circ}$ and $AB = AC = 16$ inches. Let $D$ lie on segment $BC$ such that $\\frac{DB}{DC} = \\frac{2}{3}$. Let $E$ and $F$ be the orthogonal projections of $D$ onto $AB$ and $AC$, respectively. Find $DE + DF$ in inches.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f709c4b6-e9bb-47a5-b379-e756605739a3"
}
|
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 palindrome is a number that reads the same forwards and backwards such as $3773$ or $42924$. Find the sum of the twelve smallest five digit palindromes.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "126632",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "f723bcdb-781e-42ec-bdf6-b6216546e79a"
}
|
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": "已知正实数 $a, b$ 满足 $a^{b}=(8 a)^{9 b}$, 则 $\\log _{a}\\left(4 a^{2}\\right)$ 的值为 $\\qquad$.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "65",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "56691a7a-e3ca-4ecc-9f58-2db55661c70d"
}
|
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": "集合 $U=\\{1,2, \\cdots, 366\\}$ ，则 $U$ 的互不相交各元素之和为 $17$ 的倍数的二元子集最多有 $\\qquad$个. \n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "179",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "b577a014-0399-45f2-9a57-cb2aed7395f0"
}
|
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 the \"complexity\" of a number is the minimum possible length of a sequence (if any) that begins with 0 and ends with that number, with each subsequent term either equal to half of the previous term or equal to the sum of the previous term and the previous term, which is equal to 1. (The length of the empty sequence is assumed to be 0.) Find the number with the largest \"complexity\" among all numbers of the form $\\frac m {2^{50}}$, where $m = 1, 3, 5, . . . , 2^{50} − 1.$. If the number $\\frac{a}{b}+\\frac{c}{b}\\cdot\\left(\\frac{1}{2}\\right)^{50}$ has the largest \"complexity\", equal to 101, find the value of $a+b+c$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "86859afb-fdc8-44af-bc97-d6a6b384c662"
}
|
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": "正四面体的 4 个表面上分别写有数字 $1,2,3,4$, 将 4 个这样的密度均匀的正四面体同时投掷于桌面上，与桌面接触的 4 个面上的 4 个数的和能被 4 整除的概率是 。请给出答案为\\(\\frac{m}{n}\\)形式时，m + n的值。\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "63291da5-c503-446c-a53b-a9c1fd84055f"
}
|
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": "When Trilisa takes pictures, they turn out with probability $\\frac{1}{5}$. She wants to take enough pictures so that the probability of at least one turning out is at least $\\frac{3}{4}$. How few pictures can she take to accomplish this?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "6edf8bc0-2a4f-4941-a62e-aec69f6ff030"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "In $\\triangle ABC$, we have $AC=BC=7$ and $AB=2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD=8$. What is the length of $BD$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "9729b267-36cc-4a2f-81a5-515ef883d81e"
}
|
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 $A=(a_{ij})$ be the $n \\times n$ matrix, where $a_{ij}$ is the remainder of the division of $i^j + j^i$ by $3$ for $i,j=1,2,\\ldots,n$. Find the greatest $n$ for which $\\det A \\neq 0$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "e4b23ef1-698b-42c4-acfc-b019829c5073"
}
|
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 value of $\\frac{2013^3-2 \\cdot 2013^2 \\cdot 2014+3 \\cdot 2013 \\cdot 2014^2-2014^3+1}{2013 \\cdot 2014}$?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "2013",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "02b88408-e0c4-4b6d-8122-68a91f25f5ea"
}
|
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 a complex number constant $c$, and real number constants $p$ and $q$, there exist three distinct complex values of $x$ that satisfy the equation:\n\\[\nx^3 + cx + p(1 + qi) = 0\n\\]\nSuppose $c$, $p$, and $q$ were chosen such that all three complex roots $x$ satisfy the inequality:\n\\[\n\\frac{5}{6} \\leq \\frac{\\mathrm{Im}(x)}{\\mathrm{Re}(x)} \\leq \\frac{6}{5}\n\\]\nwhere $\\mathrm{Im}(x)$ and $\\mathrm{Re}(x)$ are the imaginary and real parts of $x$, respectively. The largest possible value of $|q|$ can be expressed as a common fraction \\( \\frac{m}{n} \\), where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "8a352aae-7ecd-4c9d-969e-921865994ecd"
}
|
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": "Starting with a positive integer $M$ written on the board, Alice plays the following game: in each move, if $x$ is the number on the board, she replaces it with $3x + 2$. Similarly, starting with a positive integer $N$ written on the board, Bob plays the following game: in each move, if $x$ is the number on the board, he replaces it with $2x + 27$. Given that Alice and Bob reach the same number after playing $4$ moves each, find the smallest value of $M + N$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "a6e97f34-7e3f-45d4-bb94-cf12900f87e6"
}
|
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 smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "2c9b111d-e5b7-4a31-b149-e4ebaab70223"
}
|
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 regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form $a + b \\sqrt{2} + c \\sqrt{3} + d \\sqrt{6},$ where $a^{}_{}$, $b^{}_{}$, $c^{}_{}$, and $d^{}_{}$ are positive integers. Find $a + b + c + d^{}_{}$.\n\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "720",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "ee45cab4-dfb0-41be-8500-1341fe232f5e"
}
|
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 any integer $k \\geq 1$, let $p(k)$ be the smallest prime which does not divide $k.$ Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k) > 2$, and $X(k) = 1$ if $p(k) = 2.$ Let $\\{x_n\\}$ be the sequence defined by $x_0 = 1$, and $x_{n+1}X(x_n) = x_n p(x_n)$ for $n \\geq 0.$ Find the smallest positive integer $t$ such that $x_t = 2090.$ \n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "149",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "43fd9cad-39d9-496c-a178-1e3ba2e2eb47"
}
|
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 values of $k$ is $12^{12}$ the least common multiple of the positive integers $6^6$ , $8^8$ , and $k$ ?\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "25",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "c91159e1-d9f5-4929-a5fb-3a6148448b66"
}
|
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": "1cf9206d-9df6-45ba-aa08-3026350d8337"
}
|
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": "Alexis noticed her brother Joshua measuring something on the floor. He was designing a sandcastle and needed to measure the base of a spiral staircase. The castle walls form a rectangle, and the center of the staircase is: \n- 9 inches from one corner, \n- 15 inches from another corner, \n- 16 inches from a third corner, \n- and some whole number of inches from the furthest corner. \n\nFind the distance from the center of the staircase to the furthest corner of the rectangular castle, assuming all four distances are measured on the same plane (the ground).\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "7e43a05f-7997-4efc-968a-343048eb91b1"
}
|
math_dapo
|
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section.",
"role": "system"
},
{
"content": "Let $\\triangle ABC$ be a triangle with side lengths $AB = 9$, $BC = 10$, $CA = 11$. Let $O$ be the circumcenter of $\\triangle ABC$. Denote $D = AO \\cap BC$, $E = BO \\cap CA$, $F = CO \\cap AB$. If $\\frac{1}{AD} + \\frac{1}{BE} + \\frac{1}{FC}$ can be written in simplest form as $\\frac{a \\sqrt{b}}{c}$, find $a + b + c$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "43",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "91613e3c-b19a-4002-8efb-0751065c8e8a"
}
|
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 cube with side length $10$ is suspended above a plane. The vertex closest to the plane is labeled $A$ . The three vertices adjacent to vertex $A$ are at heights $10, 11$, and $12$ above the plane. The distance from vertex $A$ to the plane can be expressed as $\\frac{r-\\sqrt{s}}{t}$ , where $r$ , $s$ , and $t$ are positive integers. Find $r+s+t$ .\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "330",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "3a6ed64f-7411-4313-9aba-6173ccddd6ee"
}
|
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 $m$ and $n$ be positive integers such that\n$$(m^3 -27)(n^3- 27) = 27(m^2n^2 + 27):$$\nFind the maximum possible value of $m^3 + n^3$.\n\nPresent the answer in LaTeX format: \\boxed{Your answer}.",
"role": "user"
}
] |
MATH
|
{
"ground_truth": "1792",
"style": "rule-lighteval/MATH_v2"
}
|
{
"index": "97911af1-f5e3-4c72-963b-0956876801ca"
}
|
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