data_source stringclasses 1 value | prompt listlengths 1 1 | ability stringclasses 1 value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Equilateral $\\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside of the plane of $\\triangle ABC$ and are on the opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\\triangle PAB$ and $\\triangle QAB$ form a $120^{\\circ}$ dihedral angle (The... | math | {
"ground_truth": "450",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Example 8 (2004 National Girls' Mathematical Olympiad) Let $a, b, c$ be positive real numbers. Find the minimum value of $\\frac{a+3 c}{a+2 b+c}+\\frac{4 b}{a+b+2 c}-$ $\\frac{8 c}{a+b+3 c}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-17+12\\sqrt{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "8.29 Find all integers not exceeding 1000 such that the last two digits of their square are the same but not zero.\n(Kyiv Mathematical Olympiad, 1962)\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "40",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Problem 11.4. Given a regular pentagon $A B C D E$. On side $A E$ there is a point $K$, and on side $C D$ there is a point $L$. It is known that $\\angle L A E + \\angle K C D = 108^{\\circ}$, and $A K: K E = 3: 7$. Find $C L: A B$.\n\nA regular pentagon is a pentagon in which all sides are equal ... | math | {
"ground_truth": "0.7",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "6. Point $N$ divides the diagonal $A C$ of trapezoid $A B C D$ in the ratio $C N: N A=4$. The lengths of the bases $B C$ and $A D$ of the trapezoid are in the ratio $2: 3$. A line is drawn through point $N$ and vertex $D$, intersecting the lateral side $A B$ at point $M$. What fraction of the area... | math | {
"ground_truth": "124:325",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Mekkora a kör köré rajzolható egyenlőoldalú háromszög és a körbe rajzolható szabályos hatszög területeinek aránya?\n\nWhat is the ratio of the areas of an equilateral triangle circumscribed around a circle and a regular hexagon inscribed in the same circle?\nPlease reason step by step, and put you... | math | {
"ground_truth": "2:1",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Compute the product of $$ \\left[\\begin{array}{lcc}2&3&4\\\\3&4&5\\\\4&5&6\\end{array}\\right] $$ and\n$$ \\left[\\begin{array}{rcc}1&-3&5\\\\0&2&4\\\\3&0&5\\end{array}\\right] $$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\left[\\begin{array}{rcc}14&0&42\\\\18&-1&56\\\\22&-2&70\\end{array}\\right]",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Example 1. Investigate the convergence of the functional series $\\sum_{n=1}^{\\infty} \\frac{1}{n^{2}+3} \\cdot\\left(\\frac{x+1}{x-1}\\right)^{n}$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "x\\leqslant0",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "In three-dimensional space, let \\( S \\) be the region of points \\( (x, y, z) \\) satisfying \\( -1 \\leq z \\leq 1 \\). Let \\( S_{1}, S_{2}, \\ldots, S_{2022} \\) be 2022 independent random rotations of \\( S \\) about the origin \\( (0,0,0) \\). The expected volume of the region \\( S_{1} \\c... | math | {
"ground_truth": "271619",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "The Pell numbers $P_n$ satisfy $P_0 = 0$, $P_1 = 1$, and $P_n=2P_{n-1}+P_{n-2}$ for $n\\geq 2$. Find $$\\sum \\limits_{n=1}^{\\infty} \\left (\\tan^{-1}\\frac{1}{P_{2n}}+\\tan^{-1}\\frac{1}{P_{2n+2}}\\right )\\tan^{-1}\\frac{2}{P_{2n+1}}$$\nPlease reason step by step, and put your final answer wit... | math | {
"ground_truth": "\\left( \\tan^{-1} \\frac{1}{2} \\right)^2",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "## Task A-1.1.\n\nOn the side $\\overline{A B}$ of triangle $A B C$, there are points $P_{1}, P_{2}$ and $P_{3}$ such that\n\n$$\n\\left|A P_{1}\\right|=\\left|P_{1} P_{2}\\right|=\\left|P_{2} P_{3}\\right|=\\left|P_{3} B\\right|=\\frac{1}{4}|A B|\n$$\n\nParallel lines to side $\\overline{B C}$ ar... | math | {
"ground_truth": "16",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "4. Calculate $\\sqrt{5+\\sqrt{24}}-\\sqrt{5-\\sqrt{24}}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{8}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "The sequences of positive integers $1,a_2, a_3,...$ and $1,b_2, b_3,...$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$.\nPlease reason step by step, and put... | math | {
"ground_truth": "262",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "[b]p1A[/b] Positive reals $x$, $y$, and $z$ are such that $x/y +y/x = 7$ and $y/z +z/y = 7$. There are two possible values for $z/x + x/z;$ find the greater value.\n\n\n[b]p1B[/b] Real values $x$ and $y$ are such that $x+y = 2$ and $x^3+y^3 = 3$. Find $x^2+y^2$.\n\n\n[b]p2[/b] Set $A = \\{5, 6, 8,... | math | {
"ground_truth": "\\frac{7}{3}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Example 1 (24th All-Soviet Union Mathematical Olympiad, 1990) Determine for which natural numbers $n$, the number $3^{2 n+1}-2^{2 n+1}-6^{n}$ is composite.\n\nDetermine for which natural numbers $n$, the number $3^{2 n+1}-2^{2 n+1}-6^{n}$ is composite.\nPlease reason step by step, and put your fin... | math | {
"ground_truth": "n\\geqslant2",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Problem 4. Solve the equation:\n\n$$\n(x+1)^{2}+(x+3)^{2}+(x+5)^{2}+\\ldots+(x+2021)^{2}=x^{2}+(x-2)^{2}+(x-4)^{2}+\\ldots+(x-2020)^{2}\n$$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-0.5",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let $k$ be an integer and $k > 1$ . Define a sequence $\\{a_n\\}$ as follows: $a_0 = 0$ , $a_1 = 1$ , and $a_{n+1} = ka_n + a_{n-1}$ for $n = 1,2,...$ .\n\nDetermine, with proof, all possible $k$ for which there exist non-negative integers $l,m (l \\not= m)$ and positive integers $p,q$... | math | {
"ground_truth": "k = 2",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "观察下列各式：\n\n$$1^{2}+(1×2)^{2}+2^{2}=9=3^{2}$$\n$$2^{2}+(2×3)^{2}+3^{2}=49=7^{2}$$\n$$3^{2}+(3×4)^{2}+4^{2}=169=13^{2}$$\n…\n\n你发现了什么规律？请用含有n(n为正整数)的等式表示出来？可以不说理由！\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "n^{2}+[n(n+1)]^{2}+(n+1)^{2}=(n^{2}+n+1)^{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "If $\\log_2(\\log_2(\\log_2(x)))=2$, then how many digits are in the base-ten representation for x?\n$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "5\\quad",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "## Task Condition\n\nFind the derivative.\n\n$y=x^{3^{x}} \\cdot 2^{x}$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "x^{3^{x}}\\cdot2^{x}\\cdot(3^{x}\\cdot\\ln3\\cdot\\ln(x)+\\frac{3^{x}}{x}+\\ln2)",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "For how many positive integral values of $a$ is it true that $x = 2$ is the only positive integer solution of the system of inequalities $$\n\\begin{cases}\n2x>3x-3\\\\\n3x-a>-6\n\\end{cases}\n$$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Three vertices of parallelogram $ABCD$ are $A(-1, 3), B(2, -1), D(7, 6)$ with $A$ and $D$ diagonally opposite. Calculate the product of the coordinates of vertex $C$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "40",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "10.1. Find the smallest of the solutions to the inequality\n\n$$\n\\frac{-\\log _{2}(120-2 x \\sqrt{32-2 x})^{2}+\\left|\\log _{2} \\frac{120-2 x \\sqrt{32-2 x}}{\\left(x^{2}-2 x+8\\right)^{3}}\\right|}{5 \\log _{7}(71-2 x \\sqrt{32-2 x})-2 \\log _{2}(120-2 x \\sqrt{32-2 x})} \\geqslant 0\n$$\nPle... | math | {
"ground_truth": "-13-\\sqrt{57}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "11.177. A cone is formed by rotating a right triangle with area $S$ around one of its legs. Find the volume of the cone if the length of the circumference described by the point of intersection of the triangle's medians during its rotation is $L$.\nPlease reason step by step, and put your final an... | math | {
"ground_truth": "SL",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "There are $2022$ distinct integer points on the plane. Let $I$ be the number of pairs among these points with exactly $1$ unit apart. Find the maximum possible value of $I$. \n([i]Note. An integer point is a point with integer coordinates.[/i])\n\n[i]Proposed by CSJL.[/i]\nPlease reason step by st... | math | {
"ground_truth": "3954",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "19. $A B C$ and $G B D$ are straight lines. $E$ is a point on $C D$ produced, and $A D$ meets $E G$ at $F$. If $\\angle C A D=\\angle E G D, E F=F G, A B: B C=1: 2$ and $C D: D E=3: 2$, find $B D: D F$.\n(3 marks)\n$A B C$ 和 $G B D$ 均為直線, $E$ 是 $C D$ 延線上的一點, 且 $A D$ 交 $E G$ 於 $F$ 。若 $\\angle C A D... | math | {
"ground_truth": "\\sqrt{3}:2",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Consider the $10$-digit number $M=9876543210$. We obtain a new $10$-digit number from $M$ according to the following rule: we can choose one or more disjoint pairs of adjacent digits in $M$ and interchange the digits in these chosen pairs, keeping the remaining digits in their own places. For exa... | math | {
"ground_truth": "88",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "## \n\nSide $A B$ of parallelogram $A B C D$ is equal to $2, \\angle A=45^{\\circ}$. Points $E$ and $F$ are located on diagonal $B D$, such that\n\n$\\angle A E B=\\angle C F D=90^{\\circ}, B F=\\frac{3}{2} B E$.\n\nFind the area of the parallelogram.\nPlease reason step by step, and put your fina... | math | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "If $x,y>0, \\log_y(x)+\\log_x(y)=\\frac{10}{3} \\text{ and } xy=144,\\text{ then }\\frac{x+y}{2}=$\n$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "13\\sqrt{3}\\quad",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "A game publisher, GameBurst, is determined to surpass the executive's company, PlayMax, in terms of market success. Last year, PlayMax sold 1,200,000 units of their top game. GameBurst plans to release a new game and aims to sell 25% more units than PlayMax did last year. \n\nTo achieve this, Game... | math | {
"ground_truth": "1250000",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "9.4. How many three-digit natural numbers $n$ exist for which the number $n^{3}-n^{2}$ is a perfect square\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "22",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "已知数列 ${a_n}$ 的前 $n$ 项和为 $S_n$, $a_1 = 1$, $a_{n+1} = 3S_n$ ($n \\in \\mathbb{N}^*$).\n\n(Ⅰ) 求数列 ${a_n}$ 的通项公式;\n\n(Ⅱ) 若 $b_n = \\log_4 a_n$, 求 $b_1 + b_2 + \\cdots + b_n$ 的值.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(n-1) \\log_4 3 + \\frac{(n-2)(n-1)}{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "1. The sequence is defined by the following conditions: $x_{1}=1, x_{n+1}=\\cos \\left(\\operatorname{arcctg}\\left(x_{n}\\right)\\right)$. Find $x_{4} 000000$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "0.0005",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Given the distance light travels in one year is approximately $9,460,800,000,000$ kilometers, calculate the distance light travels in $70$ years.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "6.62256 \\times 10^{14}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "(given to Juliette Fournier). What is the maximum number of subsequences of the form $n, n+1, n+2$ that a sequence of 2007 natural numbers can have? For example, the sequence $1,2,2,3,3$ has 4 subsequences of this nature.\nPlease reason step by step, and put your final answer within \\boxed{}.",
... | math | {
"ground_truth": "669^3",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the number of odd numbers in the $n$-th row of Pascal's triangle.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2^k",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "[Theorem of the lengths of the tangent and the secant; the product of the entire secant and its external part] [Sine Theorem\n\nA circle passes through vertex $B$ of triangle $A B C$, touches side $A C$ at its midpoint $D$, and intersects sides $A B$ and $B C$ at points $M$ and $N$ respectively, $... | math | {
"ground_truth": "4:9",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let $\\theta$ be the angle between the planes $3x - 2y + z - 4 = 0$ and $9x - 6y - 4z + 2 = 0.$ Find $\\cos \\theta.$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\cos \\theta = \\frac{35}{\\sqrt{1862}}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the maximum number of different integers that can be selected from the set $ \\{1,2,...,2013\\}$ so that no two exist that their difference equals to $17$ .\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1010",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "If $\\frac{1}{8}$ of $2^{36}$ is $4^x$, then what is the value of \\( x \\) ?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "16.5",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "327. Find $\\left(\\cos ^{3} x\\right)^{\\prime}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-3\\cos^{2}x\\sinx",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "10. [60] Let $n$ be a fixed positive integer, and choose $n$ positive integers $a_{1}, \\ldots, a_{n}$. Given a permutation $\\pi$ on the first $n$ positive integers, let $S_{\\pi}=\\left\\{i \\left\\lvert\\, \\frac{a_{i}}{\\pi(i)}\\right.\\right.$ is an integer $\\}$. Let $N$ denote the number of... | math | {
"ground_truth": "2^{n}-n",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "10. Define the \"coordinate distance\" between two points $P\\left(x_{1}, y_{1}\\right)$ and $Q\\left(x_{2}, y_{2}\\right)$ as:\n$$\nd(P, Q)=\\left|x_{1}-x_{2}\\right|+\\left|y_{1}-y_{2}\\right| .\n$$\n\nIf $C(x, y)$ has equal coordinate distances to points $A(1,3)$ and $B(6,9)$, where the real nu... | math | {
"ground_truth": "5(\\sqrt{2}+1)",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "4. Find the smallest positive number $c$ such that for any integer $n \\geqslant 4$, and any set $A \\subseteq\\{1,2, \\cdots, n\\}$, if $|A|>c n$, then there exists a function $f: A \\rightarrow\\{1,-1\\}$, satisfying $\\left|\\sum_{a \\in A} f(a) \\cdot a\\right| \\leqslant 1$.\nPlease reason st... | math | {
"ground_truth": "\\frac{2}{3}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "1. Kolya went to the store in the neighboring village on an electric scooter at a speed of 10 km/h. Having traveled exactly one third of the entire distance, he realized that with the previous speed, he would arrive exactly when the store closes, and he doubled his speed. But when he had traveled ... | math | {
"ground_truth": "6\\frac{2}{3}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "4. Kuzya the flea can make jumps from each vertex of a regular tetrahedron \\(ABCD\\) to the three adjacent vertices, with the choice of these vertices being random and equally probable. Kuzya started jumping from vertex \\(A\\) and, after 2018 jumps, ended up at vertex \\(C\\). What is the probab... | math | {
"ground_truth": "\\frac{3^{2018}-1}{4\\cdot3^{2018}}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "8. [5] The incircle $\\omega$ of equilateral triangle $A B C$ has radius 1. Three smaller circles are inscribed tangent to $\\omega$ and the sides of $A B C$, as shown. Three smaller circles are then inscribed tangent to the previous circles and to each of two sides of $A B C$. This process is rep... | math | {
"ground_truth": "5\\pi",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the maximum value of real number $k$ such that \n\\[\\frac{a}{1+9bc+k(b-c)^2}+\\frac{b}{1+9ca+k(c-a)^2}+\\frac{c}{1+9ab+k(a-b)^2}\\geq \\frac{1}{2}\\]\nholds for all non-negative real numbers $a,\\ b,\\ c$ satisfying $a+b+c=1$ .\nPlease reason step by step, and put your final answer with... | math | {
"ground_truth": "k = 4",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "在平面直角坐标系内，已知点 $(1-2a, a-2)$ 在第三象限的角平分线上，求 $a$ 的值及点的坐标。\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "a = 1, (-1, -1)",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "4. Consider the cards $A, 2, \\cdots, J, Q, K$ as the numbers $1,2, \\cdots, 11,12,13$. Now, place the 13 spades and 13 hearts together, and randomly draw 2 cards. The probability that the two cards have the same suit and their product is a perfect square is $\\qquad$\nPlease reason step by step, ... | math | {
"ground_truth": "\\frac{2}{65}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "At the base of a quadrilateral pyramid lies a rhombus $A B C D$, in which $\\angle B A D=60^{\\circ}$. It is known that $S A=S C$, $S D=S B=A B$. A point $E$ is taken on the edge $D C$ such that the area of triangle $B S E$ is the smallest among the areas of all sections of the pyramid containing ... | math | {
"ground_truth": "\\frac{2}{3}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Completely factor the expression: $$x^8-256$$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(x^4+16)(x^2+4)(x+2)(x-2)",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find all positive integers $m$ for which there exist three positive integers $a,b,c$ such that $abcm=1+a^2+b^2+c^2$ .\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "m = 4",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let \\(ABCD\\) be a parallelogram. Let \\(M\\) be the midpoint of segment \\([BC]\\) and \\(N\\) be the midpoint of segment \\([CD]\\). The lines \\((AN)\\) and \\((BD)\\) intersect at \\(Q\\), and the lines \\((AM)\\) and \\((BD)\\) intersect at \\(P\\). Show that \\(BP = PQ = QD\\).\nPlease reas... | math | {
"ground_truth": "PQ = QD",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let $\\mathcal E$ be an ellipse with foci $F_1$ and $F_2$. Parabola $\\mathcal P$, having vertex $F_1$ and focus $F_2$, intersects $\\mathcal E$ at two points $X$ and $Y$. Suppose the tangents to $\\mathcal E$ at $X$ and $Y$ intersect on the directrix of $\\mathcal P$. Compute the eccentricity ... | math | {
"ground_truth": "\\frac{2 + \\sqrt{13}}{9}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Solve the equation $\\sqrt{3 + 4x} = 7$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "11.5",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Example 7.1.1 If $x$ is a positive number, find the minimum value of the expression $x^{x}$.\n\n\n\nThe above text is translated into English, preserving the original text's line breaks and format.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{1}{e^{\\frac{1}{e}}}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Moving only south and east along the line segments, how many paths are there from $A$ to $B$? [asy]\nimport olympiad; size(250); defaultpen(linewidth(0.8)); dotfactor=4;\nfor(int i = 0; i <= 9; ++i)\n\nif (i!=4 && i !=5)\n\ndraw((2i,0)--(2i,3));\nfor(int j = 0; j <= 3; ++j)\n\ndraw((0,j)--(18,j));... | math | {
"ground_truth": "160",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Given a regular 2007-gon. Find the minimal number $k$ such that: Among every $k$ vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the polygon.\nPlease reason step by step, and put your final answer withi... | math | {
"ground_truth": "1506",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "## Task Condition\n\nFind the derivative of the specified order.\n\n$y=\\frac{\\ln x}{x^{3}}, y^{IV}=?$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{-342+360\\lnx}{x^{7}}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "## Subject III. (10 points)\n\nCalculate: $I=\\int_{-\\frac{1}{2}}^{\\frac{1}{2}} \\frac{(\\arcsin x)^{2}}{\\sqrt{1-x^{2}}+1+x} d x$\n\nProf. Ilie Diaconu, \"Avram Iancu\" Theoretical High School, Cluj-Napoca\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "u... | math | {
"ground_truth": "\\frac{\\pi^{3}}{648}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Given that $\\sqrt{5}+3$ is a root of the equation \\[x^3 + ax^2 + bx + 16 = 0\\]and that $a$ and $b$ are rational numbers, compute $b.$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-20",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Peter and Basil play the following game on a horizontal table $1\\times{2019}$ . Initially Peter chooses $n$ positive integers and writes them on a board. After that Basil puts a coin in one of the cells. Then at each move, Peter announces a number s among the numbers written on the board, and ... | math | {
"ground_truth": "n = 2",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, which of the following number of words would be an appropriate length for her speech?\n\n$\nP... | math | {
"ground_truth": "5650",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let $f(x) = 4x - x^{2}$. Give $x_{0}$, consider the sequence defined by $x_{n} = f(x_{n-1})$ for all $n \\ge 1$. \nFor how many real numbers $x_{0}$ will the sequence $x_{0}, x_{1}, x_{2}, \\ldots$ take on only a finite number of different values?\n$\nPlease reason step by step, and put your final... | math | {
"ground_truth": "infty",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "6. Through the midpoints of sides $A B$ and $A D$ of the base of a regular quadrilateral pyramid $S A B C D$, a plane is drawn parallel to the median of the lateral face $S D C$, drawn from vertex $D$. Find the area of the section of the pyramid by this plane, if the side of the base of the pyrami... | math | {
"ground_truth": "\\frac{5}{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Solve for $y$: $$3^y + 9 = 4 \\cdot 3^y - 34$$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\log_3 \\left(\\frac{43}{3}\\right)",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "In the city where Absent-Minded Scientist lives, phone numbers consist of 7 digits. The Scientist easily remembers a phone number if it is a palindrome, which means it reads the same forwards and backwards. For example, the number 4435344 is easily remembered by the Scientist because it is a palin... | math | {
"ground_truth": "0.001",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "4. On the coordinate plane, consider squares all of whose vertices have natural coordinates, and the center is located at the point $(55 ; 40)$. Find the number of such squares.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1560",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "5. Let $P$ be the vertex of a cone, and $A, B, C$ be three points on the circumference of the base circle, satisfying $\\angle A B C=90^{\\circ}, M$ is the midpoint of $A P$. If $A B=1, A C=2, A P=\\sqrt{2}$, then the size of the dihedral angle $M-B C-A$ is\nPlease reason step by step, and put you... | math | {
"ground_truth": "\\arctan\\frac{2}{3}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "10.3. In trapezoid $A B C D$, the bases $B C=3$ and $A D=9$, angle $\\angle B A D=30$ degrees, and $\\angle A D C=60$ degrees. A line through point $D$ divides the trapezoid into two equal areas. Find the length of the segment of this line that is inside the trapezoid.\nPlease reason step by step,... | math | {
"ground_truth": "\\sqrt{39}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "1.52 In coin tossing, if $Z$ represents heads and $F$ represents tails, then the sequence of coin tosses is represented by a sequence of $Z$ and $F$. We can count the occurrences of heads followed by tails $(F Z)$, heads followed by heads $(Z Z)$, etc. For example, the sequence\nZZFFZZZZFZZFFFF\n\... | math | {
"ground_truth": "560",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Initially given $31$ tuplets\n$$(1,0,0,\\dots,0),(0,1,0,\\dots,0),\\dots, (0,0,0,\\dots,1)$$ \nwere written on the blackboard. At every move we choose two written $31$ tuplets as $(a_1,a_2,a_3,\\dots, a_{31})$ and $(b_1,b_2,b_3,\\dots,b_{31})$, then write the $31$ tuplet $(a_1+b_1,a_2+b_2,a_3+b_3... | math | {
"ground_truth": "87",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Suppose you forgot one digit of the phone number you need and you are dialing it at random. What is the probability that you will need to make no more than two calls?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "0.2",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "What is the greatest number of positive integers lesser than or equal to 2016 we can choose such that it doesn't have two of them differing by 1,2, or 6?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "576",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "The expression\n\\[a^3 (b^2 - c^2) + b^3 (c^2 - a^2) + c^3 (a^2 - b^2)\\]can be factored into the form $(a - b)(b - c)(c - a) p(a,b,c),$ for some polynomial $p(a,b,c).$ Find $p(a,b,c).$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-(ab + ac + bc)",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "2. How many six-digit numbers of the form $\\overline{a b c a c b}$ are divisible by 23?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "42",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k,\\thinspace 1 \\le k \\le 8.$ A tower is to be built using all 8 cubes according to the rules:\r\n\r\n$\\bullet$ Any cube may be the bottom cube in the tower.\r\n$\\bullet$ The cube immediately on top of a cube ... | math | {
"ground_truth": "458",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "11.016. The base of a quadrilateral pyramid is a rectangle with a diagonal equal to $b$, and the angle between the diagonals is $60^{\\circ}$. Each of the lateral edges forms an angle of $45^{\\circ}$ with the base plane. Find the volume of the pyramid.\nPlease reason step by step, and put your fi... | math | {
"ground_truth": "\\frac{b^{3}\\sqrt{3}}{24}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "9.5 What is the largest number of non-overlapping groups into which all integers from 1 to 20 can be divided so that the sum of the numbers in each group is a perfect square?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "11.5. On the coordinate plane, the graph of $y=\\frac{2020}{x}$ is constructed. How many points on the graph have a tangent that intersects both coordinate axes at points with integer coordinates?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "40",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the largest value of $c$ such that $4$ is in the range of $f(x) = x^2 + 5x + c$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "10.25",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Example 1.12 A and B are playing table tennis, and the final score is $20: 20$. How many scenarios are there in which A has been leading B throughout the game?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{38!}{20!19!}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "[Theorem of Three Perpendiculars] $[\\quad$ Dihedral Angle $\\quad]$\n\nIn the base of the pyramid $A B C D$, there lies a right triangle $A B C$ with hypotenuse $A C$, and $D C$ is the height of the pyramid, $A B=1, B C=2, C D=3$. Find the dihedral angle between the planes $A D B$ and $A D C$.\nP... | math | {
"ground_truth": "\\arcsin\\frac{2\\sqrt{14}}{\\sqrt{65}}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "3. Find the smallest distance from the point with coordinates $(10 ; 5 ; 10)$ to a point whose coordinates are positive and satisfy the inequality $(x+y+z)\\left(\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}\\right) \\geq 9 \\sqrt{1-(2 x+y)^{2}}$. In your answer, write the square of the found distance.\n... | math | {
"ground_truth": "115.2",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let $\\overline{MN}$ be a diameter of a circle with diameter 1. Let $A$ and $B$ be points on one of the semicircular arcs determined by $\\overline{MN}$ such that $A$ is the midpoint of the semicircle and $MB=\\frac{3}5$. Point $C$ lies on the other semicircular arc. Let $d$ be the length of the l... | math | {
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find all functions $f:\\mathbb{N}_0\\to\\mathbb{N}_0$ for which $f(0)=0$ and\n\\[f(x^2-y^2)=f(x)f(y) \\]\nfor all $x,y\\in\\mathbb{N}_0$ with $x>y$ .\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "f(x) = 0",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Compute the number of subsets $S$ of $\\{0,1,\\dots,14\\}$ with the property that for each $n=0,1,\\dots,\n6$, either $n$ is in $S$ or both of $2n+1$ and $2n+2$ are in $S$.\n\n[i]Proposed by Evan Chen[/i]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"... | math | {
"ground_truth": "2306",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Determine the positive value for the slope of an asymptote of the hyperbola described by the equation:\n\\[\n\\sqrt{(x-2)^2 + (y-3)^2} - \\sqrt{(x-2)^2 + (y+1)^2} = 4\n\\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "2. In tetrahedron $ABCD$, $\\triangle ABC$ is an equilateral triangle, $AD=BD=2, AD \\perp BD, AD \\perp CD$, then the distance from $D$ to plane $ABC$ is $\\qquad$ .\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{2\\sqrt{3}}{3}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the matrix $\\mathbf{M}$ that doubles the first row and triples the second row of any given 2x2 matrix $\\mathbf{A}= \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}$. That is, find $\\mathbf{M}$ such that\n\\[\n\\mathbf{M} \\mathbf{A} = \\begin{pmatrix} 2a & 2b \\\\ 3c & 3d \\end{pmatrix}.\n... | math | {
"ground_truth": "\\mathbf{M} = \\begin{pmatrix} 2 & 0 \\\\ 0 & 3 \\end{pmatrix}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let $S$ be the set of all lattice points $(x, y)$ in the plane satisfying $|x|+|y|\\le 10$ . Let $P_1,P_2,\\ldots,P_{2013}$ be a sequence of 2013 (not necessarily distinct) points such that for every point $Q$ in $S$ , there exists at least one index $i$ such that $1\\le i\\le 2013$ ... | math | {
"ground_truth": "222",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "## C3\n\nFor a given positive integer n, two players $A$ and B play the following game: Given is pile of $\\boldsymbol{\\Omega}$ stones. The players take turn alternatively with A going first. On each turn the player is allowed to take one stone, a prime number of stones, or a multiple of $n$ ston... | math | {
"ground_truth": "n-1",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Question 104, Given complex numbers $\\mathrm{z}_{1}, \\mathrm{z}_{2}$ satisfy the following conditions: $\\left|\\mathrm{z}_{1}\\right|=2,\\left|\\mathrm{z}_{2}\\right|=3,3 \\mathrm{z}_{1}-2 \\mathrm{z}_{2}=2-\\mathrm{i}$, then $\\mathrm{z}_{1} \\mathrm{z}_{2}=$ $\\qquad$ -\nPlease reason step by... | math | {
"ground_truth": "-\\frac{18}{5}+\\frac{24}{5}i",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "A circle with radius $r$ is contained within the region bounded by a circle with radius $R$. The area bounded by the larger circle is $\\frac{a}{b}$ times the area of the region outside the smaller circle and inside the larger circle. Then $R:r$ equals:\n$\nPlease reason step by step, and put your... | math | {
"ground_truth": "sqrt{a}:\\sqrt{a-b}\\quad",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "5. Given 9 points in space, where no four points are coplanar, a line segment is drawn between every pair of points. These line segments can be colored blue or red, or left uncolored. Find the minimum value of $n$ such that if any $n$ of these line segments are arbitrarily colored red or blue, the... | math | {
"ground_truth": "33",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "12.309. The base of the prism is a rectangle. The lateral edge forms equal angles with the sides of the base and is inclined to the plane of the base at an angle $\\alpha$. Find the angle between the lateral edge and the side of the base.\nPlease reason step by step, and put your final answer with... | math | {
"ground_truth": "\\arccos\\frac{\\sqrt{2}\\cos\\alpha}{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "2. In the ellipse $\\Gamma$, $A$ is one endpoint of the major axis, $B$ is one endpoint of the minor axis, and $F_{1}, F_{2}$ are the two foci. If $\\overrightarrow{A F_{1}} \\cdot \\overrightarrow{A F_{2}}+\\overrightarrow{B F_{1}} \\cdot \\overrightarrow{B F_{2}}=0$, then the value of $\\frac{|A... | math | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "4. On each of the lines $x=0$ and $x=2$, there are 62 points with ordinates $1, 2, 3, \\ldots, 62$. In how many ways can three points be chosen from the marked 124 points so that they form the vertices of a right triangle?\nPlease reason step by step, and put your final answer within \\boxed{}.",
... | math | {
"ground_truth": "7908",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "3. (physics) In a vessel, there is a mixture of equal masses of nitrogen $N_{2}$ and helium He at a pressure $p$. The absolute temperature of the gas is doubled, during which all nitrogen molecules dissociate into atoms. Find the pressure of the gas mixture at this temperature. The molar masses of... | math | {
"ground_truth": "\\frac{9}{4}p",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "orz_math_13k_hard"
} |
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