data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Given $f(x)=5x+2$ and $g(x)=3x^2-4x$, evaluate $f(f(1))-g(g(2))$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let $A$ be a discrete valuation ring with maximal ideal $\\mathfrak{m}$ and fraction field $K$. Suppose $L$ is a finite separable extension of $K$ of degree $n$, unramified with respect to $A$. Let $B$ be the integral closure of $A$ in $L$. Determine whether $B$ can be expressed as $A[x]/(f)$ for ... | math | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Determine if there exists a right triangle with integer side lengths such that the two legs differ in length by exactly 1 unit.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Evaluate the integral \\( \\int \\frac{e^x \\sin x}{\\sinh x + \\cosh x} \\, dx \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-\\cos x + C",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let \\( f \\) be the function defined for all \\((x,y)\\) as follows: \\[ f(x,y)= \\begin{cases} \\frac{xy(x^2-y^2)}{x^2+y^2}, &\\text{if }(x,y)\\ne(0,0)\\\\ 0, &\\text{if }(x,y)=(0,0) \\end{cases} \\] Determine the value of \\( \\frac{\\partial^2 f}{\\partial x \\partial y} \\) at \\((0,0)\\).\nP... | math | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "If $A \\subseteq X \\subseteq \\mathbb{R}^{n}$ and $A$ is closed in $X$ with respect to the subspace topology, does it follow that the closure of $A$ in $X$, denoted $\\overline{A}^{X}$, is equal to $A$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
... | math | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Compute the limit \\(\\lim_{x\\to 0^{+}}(\\cos x-1)e^{\\cot x}\\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-\\infty",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the minimum value of the expression \\( \\cos{a} + \\cos{b} + \\cos(a + b) \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-\\dfrac{3}{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Consider an open subset $G$ of the real numbers that is not bounded above. Determine whether there exists a positive real number $x$ such that the set of all integer multiples of $x$ has infinitely many points in common with $G$. In other words, is it true that there exists $x > 0$ for which the s... | math | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let \\( g(x) = \\int_{\\cos{x}}^{\\sin{x}} \\sqrt{1-t^2} \\,\\mathrm{d}t \\) for \\( x \\in \\left[0,\\dfrac{\\pi}{2}\\right] \\). Find \\( g'(x) \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Consider two sequences of positive numbers, $(x_n)$ and $(y_n)$. Determine whether the following statement is true: If for every $\\epsilon > 0$, the inequality $x_n^{1+\\epsilon} > y_n$ holds for all $n$, then it must be that $x_n \\geq y_n$ for all $n$. Provide a justification for your answer.\n... | math | {
"ground_truth": "True",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, and $DA=12$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?\n$\\textbf{(A)}\\ \\sqrt{15} \\qquad \\textbf{(B)}\\ \\sqrt{21} \\qquad \\textbf{(C)}\\ 2\\sqrt{6} \\qqu... | math | {
"ground_truth": "2\\sqrt{6}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "On a chessboard with 2 rows and 2019 columns, each cell contains a real number. The numbers in the first row are all distinct, and the numbers in the second row are a permutation of the numbers in the first row. In each column, the sum of the two numbers is rational, and the two numbers are differ... | math | {
"ground_truth": "4032",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let $f(x)$ be a real function not identically zero, such that for all $x, y \\in \\mathbb{R}$, the following holds: \\[ f(x+y^{2n+1}) = f(x) + (f(y))^{2n+1} \\] where $n \\in \\mathbb{N}$. If $f'(0) \\ge 0$, find $f'(6)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
... | math | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Given a positive integer \\( A \\) with the prime factorization \\( A = 2^{\\alpha} \\times 3^{\\beta} \\times 5^{\\gamma} \\), where \\(\\alpha, \\beta, \\gamma\\) are natural numbers, find the minimum value of \\(\\alpha + \\beta + \\gamma\\) if one-half of \\( A \\) is a perfect square, one-thi... | math | {
"ground_truth": "D",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "What is the smallest possible value of $x$ such that $2x^2+24x-60=x(x+13)$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-15",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Evaluate the integral \\( \\lim_{n\\to \\infty} \\int_0^{\\infty} \\frac{\\cos(nx)}{1+x^2} \\, dx \\) without using complex analysis, specifically the residue theorem.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "The parabolas $y = (x + 1)^2$ and $x + 4 = (y - 3)^2$ intersect at four points $(x_1,y_1),$ $(x_2,y_2),$ $(x_3,y_3),$ and $(x_4,y_4).$ Find\n\\[x_1 + x_2 + x_3 + x_4 + y_1 + y_2 + y_3 + y_4.\\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let $R$ be a commutative ring, and let $P$ be a matrix in $M_n(R)$ such that $\\det(P)$ is a zero divisor in $R$. Must $P$ be a zero divisor in $M_n(R)$? Justify your answer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "A monkey is sitting at a keyboard with only the keys \"a\", \"b\", and \"c\". The monkey presses the keys at random. Let \\( X \\) be the number of keys pressed until the monkey has pressed all the different keys at least once. What is the expected value of \\( X \\)?\nPlease reason step by step, ... | math | {
"ground_truth": "\\dfrac{11}{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "[asy] draw((0,0)--(1,sqrt(3)),black+linewidth(.75),EndArrow); draw((0,0)--(1,-sqrt(3)),black+linewidth(.75),EndArrow); draw((0,0)--(1,0),dashed+black+linewidth(.75)); dot((1,0)); MP(\"P\",(1,0),E); [/asy]\nLet $S$ be the set of points on the rays forming the sides of a $120^{\\circ}$ angle, and le... | math | {
"ground_truth": "\\cos(6^\\circ)\\sin(12^\\circ)\\csc(18^\\circ)",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Determine the order of the group of homomorphisms from the dihedral group $D_{2n}$ to the multiplicative group of complex numbers $\\mathbb{C}^*$, where $n$ is an odd integer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Given the irreducible transition probability matrix (PTM) \\( P = \\begin{bmatrix} 0.25 & 0.25 & 0.25 & 0.25 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\end{bmatrix} \\) and the stationary distribution \\([0.4, 0.3, 0.2, 0.1]\\), find the expected number of steps \\( E[T|X_0=1] \\) ... | math | {
"ground_truth": "\\dfrac{5}{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Evaluate the integral \\( \\int_{0}^{\\infty}\\frac{x\\arctan x}{(1+x^2)^2}dx \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\dfrac{\\pi}{8}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let \\( x, y, z \\) be positive numbers that satisfy the following system of equations:\n\n$$\n\\left\\{\\begin{array}{l}\nx^{2}+x y+y^{2}=12 \\\\\ny^{2}+y z+z^{2}=16 \\\\\nz^{2}+x z+x^{2}=28\n\\end{array}\\right.\n$$\n\nFind the value of the expression \\( x y + y z + x z \\).\nPlease reason step... | math | {
"ground_truth": "16",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Determine the minimum possible real number \\( L \\) such that the area of the intersection of the set \\( I = \\{(x,y) \\mid x > (\\frac{y^4}{9} + 2015)^{1/4}\\} \\) and the disk \\( x^2 + y^2 \\leq r^2 \\) is less than \\( Lr^2 \\) for all \\( r > 0 \\).\nPlease reason step by step, and put your... | math | {
"ground_truth": "\\dfrac{\\pi}{3}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Evaluate the limit: $$\\lim\\limits_{n\\to\\infty}{\\sum\\limits_{k=n}^{5n}{k-1 \\choose n-1}(\\frac{1}{5})^{n}(\\frac{4}{5})^{k-n}}$$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\dfrac{1}{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Evaluate the series:\n$$(x+y)(x-y)+\\frac{1}{2!}(x+y)(x-y)(x^2+y^2)+\\frac{1}{3!}(x+y)(x-y)(x^4+y^4+x^2y^2)+\\ldots$$\nWhich of the following is the value of the series?\n$$(A)\\quad e^{x^2}+e^{y^2}\\quad\\quad(B)\\quad e^{x^2-y^2}\\quad\\quad(C)\\quad e^{x^2}-e^{y^2}\\quad\\quad(D)\\quad \\text{n... | math | {
"ground_truth": "C",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Evaluate the integral \\( \\int_{\\gamma} \\frac{dz}{(z^2+4)^2} \\) using complex analysis techniques, where \\( \\gamma \\) is a closed path in \\( \\mathbb{C}-\\{\\pm 2i\\} \\) that winds once counterclockwise around \\( 2i \\) and once clockwise around \\( -2i \\).\nPlease reason step by step, ... | math | {
"ground_truth": "\\dfrac{\\pi}{8}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the smallest integer $n$ such that there exist $n$ numbers within the interval $(-1, 1)$ whose sum is 0 and the sum of their squares is 42.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "44",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Given a surface represented by the function $z = f(u, w)$, with a point $p_1(u_1, v_1)$ and a closed boundary $\beta(u, v)$ defined by $(u - u_1)^2 + (v - v_1)^2 = 0.01$, calculate the surface area $s_1$. If there is a tangent plane at $p_1$ and another point $p_2(u_2, v_2)$ lies in the $f_u$ dire... | math | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Evaluate the integral \\( F(y) = \\int_{-\\infty}^{\\infty} \\frac{dx}{(1+x^2)(1+(x+y)^2)} \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\dfrac{2\\pi}{y^2 + 4}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Determine the number of automorphisms of the symmetric group $S_6$. Provide your answer as a single integer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1440",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Determine the sign of the permutation \\( \\sigma \\) given by:\n\\[ \\sigma = \\left[ \\begin{array}{cccccc} 1 & \\cdots & k & k+1 & \\cdots & k+l \\\\ l+1 & \\cdots & l+k & 1 & \\cdots & l \\end{array} \\right] .\\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"... | math | {
"ground_truth": "(-1)^{kl}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Calculate \\( 73^{1567} \\mod 990 \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "127",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Evaluate the double integral \\( I_6 = \\int\\int_{Q}\\frac{x}{\\pi(x^2+y^2)}\\,dxdy \\) where \\( Q = \\{(x,y)|(x-2)^2+y^2\\leq 4, y\\geq0\\} \\) using polar coordinates.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find all the singular points and corresponding residues of the function \\( \\sin \\frac{1}{z} \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "In a circle of 20 people, determine the number of ways to select a committee of 7 people such that no two selected members are adjacent.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2640",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Suppose $C, D \\subseteq \\mathbb{R}$. If $C$ is compact and $D$ is closed, is it true that there exist points $c \\in C$ and $d \\in D$ such that $d(C, D) = |c - d|$? Justify your answer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the value of \\( n \\) such that \\( \\lim_{x\\rightarrow 0}\\frac{x^n\\sin^{n}(x)}{x^{n}-(\\sin x)^{n}} \\) is a finite non-zero number.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the equation of the directrix of the parabola $y = -2x^2 + 4x - 8.$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "y = -\\frac{47}{8}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the value of \\( f(-1) \\) where \\( f(x) = \\lim_{h \\to 0} \\frac{1}{2h} \\int_{x-h}^{x+h} g(y) \\, dy \\) and \\( g(x) = \\begin{cases} 0, & \\text{if } x \\in [-1, 1] \\\\ 1, & \\text{otherwise} \\end{cases} \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
... | math | {
"ground_truth": "\\dfrac{1}{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Evaluate the limit: \\[ \\lim\\limits_{x \\rightarrow 0^+} \\left ( 2 \\sin \\left ( \\frac{1}{x} \\right ) + \\sqrt{x} \\sin \\left ( \\frac{1}{x} \\right ) \\right )^x. \\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "How many four-character license plates consist of a consonant, followed by a vowel, followed by a consonant, and then a digit? (For this problem, consider Y a vowel.)\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "24{,}000",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Determine the order of the entire function \\(f(z) = \\cos\\sqrt{z}\\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\dfrac{1}{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Determine the degree of the splitting field of the polynomial $x^3 - \\pi$ over the field $\\mathbb{R}(\\pi)$. Provide your answer as a single integer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Evaluate the surface integral \\( \\iint_{S} G(x, y, z) \\, dS \\) where \\( G(x, y, z) = (1-x^2-y^2)^{\\frac{3}{2}} \\) and \\( S \\) is the hemisphere defined by \\( z = (1-x^2-y^2)^{\\frac{1}{2}} \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
... | math | {
"ground_truth": "\\dfrac{\\pi}{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the constant value $c$ such that $y(t) = c$ is a solution to the differential equation: $$\\frac{d^2y}{dt^2}(t) + (\\sqrt{1+y^2}-2)\\frac{dy}{dt}(t) + y(t) = 0.$$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the value of \\(i^i\\), where \\(i\\) is the imaginary unit.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "e^{-\\pi/2}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Determine whether the equation $a^2 + d^2 + 4 = b^2 + c^2$ has any integer solutions, given that $d < c < b < a$. Provide a justification for your answer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Solve \\[\\frac{3x^2+2x+1}{x-1}=3x+1\\]for $x$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-\\frac{1}2",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Given a linear operator $A: V \\rightarrow V$, where $V$ is a vector space, if $R(A) = R(A^2)$, does this imply that $N(A)$ is a trivial subspace (i.e., $N(A) = \\{0\\}$)? Provide a justification for your answer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role"... | math | {
"ground_truth": "No",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "In a parallelogram \\(ABCD\\), \\(\\angle BAD = 60^\\circ\\). Points \\(K\\) and \\(L\\) are the midpoints of \\(BC\\) and \\(CD\\), respectively. If quadrilateral \\(ABKL\\) is cyclic, determine the measure of \\(\\angle ABD\\).\nPlease reason step by step, and put your final answer within \\boxe... | math | {
"ground_truth": "75^\\circ",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the minimum value of the expression \\( x^{\\ln(y)-\\ln(z)}+y^{\\ln(z)-\\ln(x)}+z^{\\ln(x)-\\ln(y)} \\) for positive real numbers \\( x, y, \\) and \\( z \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the equation of the tangent line to the elliptic curve $E$ defined by $y^2 = x^3 + 2$ over $\\mathbb{F}_{7}$ at the point $P = (3, 6)$. The tangent line should intersect the curve only at $P$ and the point at infinity $\\mathcal{O}$. Provide your answer as an equation.\nPlease reason step by ... | math | {
"ground_truth": "y = 4x + 1",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Can a function with the property $f'(0) = f''(0)$ be a characteristic function of a random variable?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Compute the dimension of the Zariski Tangent space of the affine scheme \\(\\operatorname{Spec} A\\), where \\(A = \\mathbb{Z}[2i] \\cong \\mathbb{Z}[x]/(x^2 + 4)\\), at the point corresponding to the maximal ideal \\((2, x)\\).\nPlease reason step by step, and put your final answer within \\boxed... | math | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Compute the limit \\( \\lim_{a \\to \\infty} \\operatorname{erf}(a+z) \\) for a given \\( z \\in \\overline{\\mathbb{C}} \\). Can you conclude that this limit is equal to 1 by considering the behavior of \\( \\operatorname{erf}(a) \\) as \\( a \\to \\infty \\)?\nPlease reason step by step, and put... | math | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the coefficient of $x^{11}$ in the expansion of $(1+x^2)^4 (1+x^3)^7 (1+x^4)^{12}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1113",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Given a field \\( k \\), determine whether there exists a valuation ring of the rational function field \\( k(x, y) \\) with Krull dimension 2.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Given a map \\( f: T^2 \\rightarrow T^2 \\), where the induced homomorphism on the first homology group \\( f^{**}: H_1(T^2) \\rightarrow H_1(T^2) \\) is given by \\( f(x+y) = mx + ny \\), calculate the degree of the map \\( f \\).\nPlease reason step by step, and put your final answer within \\bo... | math | {
"ground_truth": "mn",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Evaluate the limit: $$\\lim_{\\theta\\to 0} \\dfrac {\\sin\\left ( \\sin a \\theta \\right ) }{\\theta}$$ and determine if $$\\lim_{\\theta\\to 0} \\dfrac {\\sin\\left ( \\sin \\left ( \\sin \\left ( \\sin \\cdots a \\theta \\right ) \\right )\\cdots \\right ) }{\\theta} =a.$$\nPlease reason step ... | math | {
"ground_truth": "a",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Calculate the unconditional variance of the random variable $N$, where $N|\\Lambda$ follows a binomial distribution with parameters $\\Lambda$ and $q = 0.4$, and $\\Lambda$ has a probability function defined by $p(1) = p(2) = p(3) = p(4) = 0.25$. Provide your answer as a numerical value.\nPlease r... | math | {
"ground_truth": "0.8",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Calculate the joint conditional distribution of $Y$ and $Z$ given $X = 3$ for the probability distribution $f(x,y,z) = \\frac{xyz}{108}$, where $x = 1, 2, 3$, $y = 1, 2, 3$, and $z = 1, 2$. Verify if the conditional distribution is $\\frac{zy}{18}$.\nPlease reason step by step, and put your final ... | math | {
"ground_truth": "\\dfrac{zy}{18}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "For what positive integer $A$ is the sequence $(a_k)_{k=0}^\\infty$, defined by $a_0 = A$ and $a_{k+1} = a_k + S(a_k)$ for $k \\ge 0$, eventually constant? Here, $S(n) = n - m^2$, where $m$ is the greatest integer such that $m^2 \\le n$. Determine the positive integer $A$ for which the sequence is... | math | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Calculate the limit: \\[ \\lim_{n \\to \\infty} \\frac{1+\\sqrt[n]{e}+\\sqrt[n]{e^2}+\\dots+\\sqrt[n]{e^{n-1}}}{n} \\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "e - 1",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "I have 10 distinguishable socks in my drawer: 4 white, 4 brown, and 2 blue. In how many ways can I choose a pair of socks, provided that I get two socks of the same color?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "In \\( \\triangle ABC \\), the circumcenter is denoted as \\( O \\). Given that \\( AB = 2a \\), \\( AC = \\frac{2}{a} \\), and \\( \\angle BAC = 120^\\circ \\), express \\( \\overrightarrow{AO} \\) as \\( \\alpha \\overrightarrow{AB} + \\beta \\overrightarrow{AC} \\). Determine the minimum value ... | math | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the derivative of the function at $x = 0$ given by:\n\\[ f(x) = \\begin{cases} e^{-\\frac{1}{x^2}} \\sin \\frac{1}{x}, & \\text{if } x \\neq 0 \\\\ 0, & \\text{if } x = 0 \\end{cases} \\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Suppose $f, u, u_n \\in L^p(\\Omega)$ where $\\Omega \\subset \\mathbb{R}^N$ is a bounded domain and $f, u, u_n \\geq 0$. Given that \\( \\|u+u_n\\|_p \\rightarrow \\|u+f\\|_p \\), \\( u_n \\rightarrow f \\) almost everywhere in \\( \\Omega \\), and \\( u_n > f \\), does it follow that \\( \\|u_n-... | math | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "For a tame quiver $Q$ with an underlying connected undirected graph that is one of the extended, simply laced Dynkin diagrams (i.e., $\\tilde A_n$ for $n \\geq 1$, $\\tilde D_n$ for $n \\geq 4$, or $\\tilde E_6$, $\\tilde E_7$, or $\\tilde E_8$), and a vector $d$ of nonnegative integers, if $d$ sa... | math | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Evaluate the double integral \\( \\int \\int_D \\frac{1}{\\sqrt{x^2+y^2}} \\, dx \\, dy \\), where \\( D \\) is the domain defined by \\( 1 \\le x^2+y^2 \\le 4 \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2\\pi",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Determine whether there exists a distributive lattice \\( L \\) such that for no set \\( X \\), there is a surjective lattice homomorphism \\( s: \\text{Part}(X) \\to L \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Given that \\( x, y, z \\) are positive real numbers such that \\( x + y + z = 1 \\), find the minimum value of the function \\( f(x, y, z) = \\frac{3x^{2} - x}{1 + x^{2}} + \\frac{3y^{2} - y}{1 + y^{2}} + \\frac{3z^{2} - z}{1 + z^{2}} \\).\nPlease reason step by step, and put your final answer wi... | math | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let $f(x)$ be a Lipschitz real-valued function defined on a closed interval $I$. Is the derivative $f'(x)$ necessarily square integrable, i.e., in $L^2(I)$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let $f$ be an entire function stable on $\\mathbb{R}$, and let $E_{\\mathbb{R}}$ be its real escaping set defined as: \\[E_{\\mathbb{R}} = \\{ x \\in \\mathbb{R} : f^{(k)}(x) \\rightarrow_{k \\to \\infty} \\infty \\} \\] Given that $E_{\\mathbb{R}}$ is a measurable null set with respect to the Leb... | math | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Determine whether the quotient module $V/I$ is Jacobson semisimple for any $A$-module $V$ and ideal $I$ of $A$. Provide a true or false answer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "False",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the limit of the sequence \\( \\left\\{ \\left(\\frac{n^2+2}{2n^2+1}\\right)^{n^2}\\right\\}_{n\\geq 1} \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Given the parity check matrix \\(H=\\begin{pmatrix} 2 & 0 & 1 & 1 & 1 \\\\ 0 & 2 & 2 & 0 & 0 \\\\ 0 & 0 & 2 & 2 & 1 \\\\ 1 & 1 & 2 & 2 & 0 \\end{pmatrix}\\), determine the minimum distance \\(d(C)\\) of the ternary linear code \\(C\\).\nPlease reason step by step, and put your final answer within ... | math | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let $S$ be a set and $T$ be a family of subsets of $S$. Consider a function $f: S \\to T$. Suppose $S_1 \\in T$ with $S_1 \\neq \\emptyset$ and there exists $s_1 \\in S_1$ such that $f(s_1) \\neq \\emptyset$. Define $S_2 = f(s_1)$. Assume that for every set $S_i$ obtained in this manner, there exi... | math | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Determine the number of distinct terms in the expansion of \\((1 + a^3 + a^{-3})^{100}\\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "201",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let $M_R$ be a right $R$-module. A homogeneous component $H$ of $M_R$ is defined as the sum $\\sum_{i\\in I}B_i$, where $\\{ B_i \\}_{i\\in I}$ is a family of mutually isomorphic simple submodules $B_i \\subseteq M$. Determine if it is true that if the homogeneous component $H$ is non-simple, then... | math | {
"ground_truth": "True",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let \\(X\\) and \\(Y\\) be independent random variables, each following a standard normal distribution \\(\\mathcal{N}(0,1)\\). Define \\(D_r\\) as the disk centered at the origin with radius \\(r\\). Determine the value of \\(r\\) such that the probability \\(\\mathbb{P}[(X,Y) \\in D_r] = 0.3\\).... | math | {
"ground_truth": "\\sqrt{-2 \\ln 0.7}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the integer solution to the congruence equation \\( y^{31} \\equiv 3 \\pmod{100} \\), given that \\( \\phi(100) = 40 \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "47",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let\n\\[\\mathbf{a} = \\begin{pmatrix} 5 \\\\ -3 \\\\ -4 \\end{pmatrix} \\quad \\text{and} \\quad \\mathbf{b} = \\begin{pmatrix} -11 \\\\ 1 \\\\ 28 \\end{pmatrix}.\\]There exist vectors $\\mathbf{p}$ and $\\mathbf{d}$ such that the line containing $\\mathbf{a}$ and $\\mathbf{b}$ can be expressed i... | math | {
"ground_truth": "\\begin{pmatrix} -4/9 \\\\ 1/9 \\\\ 8/9 \\end{pmatrix}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Determine the number of roots of the polynomial \\( g(z) = z^2 + 5z^5 + 2z^8 + z^{12} \\) within the annulus defined by \\( 2 < |z| < 3 \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Assuming $P \\neq NP$, determine whether the language $L_G = \\{H \\mid G \\text{ contains a copy of } H\\}$, for some fixed graph $G$, is in $P$ or $NPC$. Here, $G$ contains a copy of $H = (V_H, E_H)$ if there exists a subset $S \\subseteq V_G$ of size $|V_H|$ such that the subgraph induced by $S... | math | {
"ground_truth": "P",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Suppose $m$ and $n$ are positive integers such that the greatest common divisor of $m$ and $n$ is divisible by exactly 5 distinct primes, and the least common multiple of $m$ and $n$ is divisible by exactly 20 distinct primes. If $m$ has more distinct prime factors than $n$, what is the maximum nu... | math | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let $X$ be a subset of $[0,1]$ with Lebesgue measure 1. For each $x \\in X$, there is some $\\epsilon_x > 0$. Define $I_x$ as the open interval $(x, x+\\epsilon_x)$. Consider the union of all these intervals: $$U := \\bigcup_{x\\in X} I_x$$ Is the length of $U$ always at least 1?\nPlease reason st... | math | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Determine all functions \\( f: \\mathbb{N} \\to \\mathbb{N} \\) such that \\( f(f(n)) < f(n+1) \\) for all natural numbers \\( n \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "f(n) = n",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Find the limit of the expression as $x$ approaches 0: $$\\lim_{x\\to0}\\left(\\frac{\\sin(x)}{x}\\right)^{\\frac{1}{x^2}}$$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "e^{-\\frac{1}{6}}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Determine whether it is possible to construct a vector space structure on a countable set such that the underlying group is non-cyclic.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Let $\\mathrm{O}$ be the intersection point of the diagonals of a convex quadrilateral $A B C D$. Let $P, Q, R$, and $S$ be the centroids of triangles $A O B$, $B O C$, $C O D$, and $D O A$, respectively. Determine the ratio of the area of quadrilateral $P Q R S$ to the area of quadrilateral $A B ... | math | {
"ground_truth": "\\dfrac{2}{9}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Is it possible for a Blaschke product $B(z)$ with non-negative coefficients to have a zero at a complex number $z_0$ such that $\\overline{z_0}$ is not a zero of $B(z)$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "No",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^{\\infty} \\frac{x\\sin(kx)}{x^2+a^2} \\, \\mathrm{d}x \\) using real analysis techniques, where \\( k > 0 \\) and \\( a \\) is a positive constant.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\pi e^{-a k}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Determine the fundamental group \\( \\pi_1(\\mathbb{C}^*/G) \\), where \\( G \\) is the group of homeomorphisms \\( \\{\\phi^n ; n\\in \\mathbb{Z}\\} \\) with \\( \\phi(z)=2z \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\mathbb{Z} \\times \\mathbb{Z}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Evaluate the integral \\( \\int_{0}^{a}\\frac{1}{1+f(x)}\\,dx \\) given that \\( f(x)f(a-x)=1 \\), \\( a>0 \\), and \\( f(x) \\) is continuous and positive on \\([0,a]\\). Use the fact that the integral is independent of \\( f(x) \\) to find the answer.\nPlease reason step by step, and put your fi... | math | {
"ground_truth": "\\dfrac{a}{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Determine the number of elements in the set \\( \\{z \\in \\mathbb{C} : z^{60} = -1 , z^k \\neq -1, 0<k<60 \\} \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "32",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Determine whether there exists a sequence \\( \\{S'_n\\} \\) that diverges to infinity but has a relative rate of divergence to zero compared to another divergent sequence \\( \\{S_n\\} \\), such that \\( \\lim_{n \\to \\infty} \\frac{S'_n}{S_n} = 0 \\).\nPlease reason step by step, and put your f... | math | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
DigitalLearningGmbH/MATH-lighteval | [
{
"content": "Calculate the sectional curvature of the Poincaré half-plane model using the basis vectors $\\partial_1=\\partial/ \\partial x$ and $\\partial_2=y\\partial/ \\partial y$. Provide your answer as a single numerical value.\nPlease reason step by step, and put your final answer within \\boxed{}.",
... | math | {
"ground_truth": "-1",
"style": "rule-lighteval/MATH_v2"
} | {
"dataset": "deepmath103k",
"membership": null
} |
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