data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA projection takes $\\begin{pmatrix} 1 \\\\ -2 \\end{pmatrix}$ to $\\begin{pmatrix} \\frac{3}{2} \\\\ -\\frac{3}{2} \\... | MATH | {
"ground_truth": "\\begin{pmatrix} -5/2 \\\\ 5/2 \\end{pmatrix}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5500"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIf $\\mathbf{a}$ and $\\mathbf{b}$ are vectors such that $\\|\\mathbf{a}\\| = 7$ and $\\|\\mathbf{b}\\| = 11$, then fi... | MATH | {
"ground_truth": "[-77,77]",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5501"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\mathbf{a}$ and $\\mathbf{b}$ be two vectors such that\n\\[\\|\\mathbf{a} + \\mathbf{b}\\| = \\|\\mathbf{b}\\|.\... | MATH | {
"ground_truth": "90^\\circ",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5502"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCompute the least positive value of $t$ such that\n\\[\\arcsin (\\sin \\alpha), \\ \\arcsin (\\sin 2 \\alpha), \\ \\ar... | MATH | {
"ground_truth": "9 - 4 \\sqrt{5}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5503"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $z_1,$ $z_2,$ $\\dots,$ $z_{20}$ be the twenty (complex) roots of the equation\n\\[z^{20} - 4z^{19} + 9z^{18} - 16... | MATH | {
"ground_truth": "\\frac{241}{220}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5504"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $A$ and $B$ be the endpoints of a semicircular arc of radius $2$. The arc is divided into seven congruent arcs by ... | MATH | {
"ground_truth": "28672",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5505"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\mathbf{p}$ be the projection of $\\mathbf{v}$ onto $\\mathbf{w},$ and let $\\mathbf{q}$ be the projection of $\... | MATH | {
"ground_truth": "\\frac{25}{49}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5506"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe volume of the parallelepiped generated by $\\begin{pmatrix} 2 \\\\ 3 \\\\ 4 \\end{pmatrix},$ $\\begin{pmatrix} 1 \... | MATH | {
"ground_truth": "\\frac{9}{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5507"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA sequence $(a_1,b_1)$, $(a_2,b_2)$, $(a_3,b_3)$, $\\ldots$ of points in the coordinate plane satisfies\n\\[(a_{n + 1}... | MATH | {
"ground_truth": "\\frac{1}{2^{98}}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5508"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $P$ be the point on line segment $\\overline{AB}$ such that $AP:PB = 3:2.$ Then\n\\[\\overrightarrow{P} = t \\ove... | MATH | {
"ground_truth": "\\left( \\frac{2}{5}, \\frac{3}{5} \\right)",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5509"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\mathbf{a} = \\begin{pmatrix} 2 \\\\ 1 \\\\ 5 \\end{pmatrix}.$ Find the vector $\\mathbf{b}$ such that $\\mathb... | MATH | {
"ground_truth": "\\begin{pmatrix} -1 \\\\ 3 \\\\ 2 \\end{pmatrix}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5510"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nJoel selected an acute angle $x$ (strictly between 0 and 90 degrees) and wrote the values of $\\sin x$, $\\cos x$, and... | MATH | {
"ground_truth": "\\frac{1 + \\sqrt{5}}{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5511"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe set of points $(x,y,z)$ that are equidistant to $(1,2,-5)$ and point $P$ satisfy an equation of the form\n\\[10x -... | MATH | {
"ground_truth": "(6,0,7)",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5512"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nConvert the point $(0, -3 \\sqrt{3}, 3)$ in rectangular coordinates to spherical coordinates. Enter your answer in th... | MATH | {
"ground_truth": "\\left( 6, \\frac{3 \\pi}{2}, \\frac{\\pi}{3} \\right)",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5513"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe line $y = 2x + 7$ is to be parameterized using vectors. Which of the following options are valid parameterization... | MATH | {
"ground_truth": "\\text{B,E}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5514"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the range of\n\\[f(A)=\\frac{\\sin A(3\\cos^{2}A+\\cos^{4}A+3\\sin^{2}A+\\sin^{2}A\\cos^{2}A)}{\\tan A (\\sec A-\... | MATH | {
"ground_truth": "(3,4)",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5515"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThree of the vertices of parallelogram $ABCD$ are $A = (3,-1,2),$ $B = (1,2,-4),$ and $C = (-1,1,2).$ Find the coordi... | MATH | {
"ground_truth": "(1,-2,8)",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5516"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSolve $\\arcsin x + \\arcsin (1 - x) = \\arccos x.$\n\nRemember to put your answer on its own line after \"Answer:\"."... | MATH | {
"ground_truth": "0, \\frac{1}{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5517"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind $k$ if\n\\[(\\sin \\alpha + \\csc \\alpha)^2 + (\\cos \\alpha + \\sec \\alpha)^2 = k + \\tan^2 \\alpha + \\cot^2 ... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5518"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the matrix $\\mathbf{M}$ such that\n\\[\\mathbf{M} \\mathbf{v} = -5 \\mathbf{v}\\]for all vectors $\\mathbf{v}.$\... | MATH | {
"ground_truth": "\\begin{pmatrix} -5 & 0 \\\\ 0 & -5 \\end{pmatrix}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5519"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor real numbers $a,$ $b,$ and $c,$ the matrix\n\\[\\begin{pmatrix} a & b & c \\\\ b & c & a \\\\ c & a & b \\end{pmat... | MATH | {
"ground_truth": "-3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5520"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA projectile is fired with an initial velocity of $v$ at an angle of $\\theta$ from the ground. Then its trajectory c... | MATH | {
"ground_truth": "\\frac{\\pi}{8}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5521"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the volume of the region in space defined by\n\\[|x + y + z| + |x + y - z| \\le 8\\]and $x,$ $y,$ $z \\ge 0.$\n\n... | MATH | {
"ground_truth": "32",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5522"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind a unit vector that is orthogonal to both $\\begin{pmatrix} 1 \\\\ 1 \\\\ 0 \\end{pmatrix}$ and $\\begin{pmatrix} ... | MATH | {
"ground_truth": "\\begin{pmatrix} -2/3 \\\\ 2/3 \\\\ 1/3 \\end{pmatrix}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5523"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\mathbf{v}$ and $\\mathbf{w}$ be the vectors such that $\\mathbf{v} \\cdot \\mathbf{w} = -3$ and $\\|\\mathbf{w}... | MATH | {
"ground_truth": "\\frac{3}{5}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5524"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many solutions of the equation $\\tan x = \\tan (\\tan x)$ are on the interval $0 \\le x \\le \\tan^{-1} 942$? (H... | MATH | {
"ground_truth": "300",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5525"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA certain point has rectangular coordinates $(10,3)$ and polar coordinates $(r, \\theta).$ What are the rectangular c... | MATH | {
"ground_truth": "(91,60)",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5526"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe matrix\n\\[\\begin{pmatrix} a & \\frac{15}{34} \\\\ c & \\frac{25}{34} \\end{pmatrix}\\]corresponds to a projectio... | MATH | {
"ground_truth": "\\left( \\frac{9}{34}, \\frac{15}{34} \\right)",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5527"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\mathbf{M} = \\begin{pmatrix} 2 & 0 \\\\ 1 & -3 \\end{pmatrix}.$ Find constants $a$ and $b$ so that\n\\[\\mathb... | MATH | {
"ground_truth": "\\left( \\frac{1}{6}, \\frac{1}{6} \\right)",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5528"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn triangle $ABC,$ $\\angle B = 60^\\circ$ and $\\angle C = 45^\\circ.$ The point $D$ divides $\\overline{BC}$ in the... | MATH | {
"ground_truth": "\\frac{\\sqrt{6}}{6}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5529"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA particle is located on the coordinate plane at $(5,0)$. Define a ''move'' for the particle as a counterclockwise rot... | MATH | {
"ground_truth": "(-5 \\sqrt{2}, 5 + 5 \\sqrt{2})",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5530"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe dilation, centered at $2 + 3i,$ with scale factor 3, takes $-1 - i$ to which complex number?\n\nRemember to put yo... | MATH | {
"ground_truth": "-7 - 9i",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5531"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the projection of the vector $\\begin{pmatrix} 4 \\\\ -4 \\\\ -1 \\end{pmatrix}$ onto the line\n\\[2x = -3y = z.\... | MATH | {
"ground_truth": "\\begin{pmatrix} 6/7 \\\\ -4/7 \\\\ 12/7 \\end{pmatrix}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5532"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn triangle $ABC,$ $D,$ $E,$ and $F$ are points on sides $\\overline{BC},$ $\\overline{AC},$ and $\\overline{AB},$ res... | MATH | {
"ground_truth": "\\frac{1}{7}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5533"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere exists a positive real number $x$ such that $ \\cos (\\arctan (x)) = x $. Find the value of $x^2$.\n\nRemember t... | MATH | {
"ground_truth": "\\frac{-1 + \\sqrt{5}}{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5534"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn triangle $ABC,$ $AB = 3,$ $AC = 6,$ $BC = 8,$ and $D$ lies on $\\overline{BC}$ such that $\\overline{AD}$ bisects $... | MATH | {
"ground_truth": "\\frac{\\sqrt{34}}{12}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5535"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nConsider two lines: line $l$ parametrized as\n\\begin{align*} \nx &= 1 + 4t,\\\\\ny &= 4 + 3t\n\\end{align*}and the l... | MATH | {
"ground_truth": "\\begin{pmatrix}-6 \\\\ 8 \\end{pmatrix}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5536"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCube $ABCDEFGH,$ labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the ... | MATH | {
"ground_truth": "\\frac{41}{48}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5537"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe line $y = \\frac{3x - 5}{4}$ is parameterized in the form\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\mathbf{... | MATH | {
"ground_truth": "\\begin{pmatrix} 4/5 \\\\ 3/5 \\end{pmatrix}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5538"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nExpress $\\sin 4x + \\sin 6x$ as a product of trigonometric functions.\n\nRemember to put your answer on its own line ... | MATH | {
"ground_truth": "2 \\sin 5x \\cos x",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5539"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn polar coordinates, the point $\\left( -2, \\frac{3 \\pi}{8} \\right)$ is equivalent to what other point, in the sta... | MATH | {
"ground_truth": "\\left( 2, \\frac{11 \\pi}{8} \\right)",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5540"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nOne angle of a triangle is twice another, and the sides opposite these angles have lengths 15 and 9. Compute the leng... | MATH | {
"ground_truth": "16",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5541"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSimplify\n\\[\\cos ^2 x + \\cos^2 (x + y) - 2 \\cos x \\cos y \\cos (x + y).\\]\n\nRemember to put your answer on its ... | MATH | {
"ground_truth": "\\sin^2 y",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5542"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the number of solutions to\n\\[\\sin x = \\left( \\frac{1}{2} \\right)^x\\]on the interval $(0,100 \\pi).$\n\nRem... | MATH | {
"ground_truth": "100",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5543"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe complex number $(3 \\operatorname{cis} 18^\\circ)(-2\\operatorname{cis} 37^\\circ)$ is expressed in polar form as ... | MATH | {
"ground_truth": "(6,235^\\circ)",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5544"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\mathbf{M}$ be a matrix such that\n\\[\\mathbf{M} \\begin{pmatrix} 2 \\\\ -1 \\end{pmatrix} = \\begin{pmatrix} 3... | MATH | {
"ground_truth": "\\begin{pmatrix} 11 \\\\ -1 \\end{pmatrix}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5545"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\mathbf{w} = \\begin{pmatrix} 2 \\\\ -1 \\\\ 2 \\end{pmatrix}.$ The set of vectors $\\mathbf{v}$ such that\n\\[... | MATH | {
"ground_truth": "2x - y + 2z - 18 = 0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5546"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the number of ordered quadruples $(a,b,c,d)$ of real numbers such that\n\\[\\begin{pmatrix} a & b \\\\ c & d \\en... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5547"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\overline{AD},$ $\\overline{BE},$ $\\overline{CF}$ be the altitudes of acute triangle $ABC.$ If\n\\[9 \\overrig... | MATH | {
"ground_truth": "60^\\circ",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5548"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSimplify\n\\[\\cos \\frac{2 \\pi}{13} + \\cos \\frac{6 \\pi}{13} + \\cos \\frac{8 \\pi}{13}.\\]\n\nRemember to put you... | MATH | {
"ground_truth": "\\frac{\\sqrt{13} - 1}{4}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5549"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind\n\\[\\sin \\left( \\sin^{-1} \\frac{3}{5} + \\tan^{-1} 2 \\right).\\]\n\nRemember to put your answer on its own l... | MATH | {
"ground_truth": "\\frac{11 \\sqrt{5}}{25}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5550"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the matrix that corresponds to rotating about the origin by an angle of $120^\\circ$ counter-clockwise.\n\nRememb... | MATH | {
"ground_truth": "\\begin{pmatrix} -1/2 & -\\sqrt{3}/2 \\\\ \\sqrt{3}/2 & -1/2 \\end{pmatrix}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5551"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair ... | MATH | {
"ground_truth": "3 \\sqrt{3} + 2 \\pi",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5552"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the maximum $y$-coordinate of a point on the graph of $r = \\sin 2 \\theta.$\n\nRemember to put your answer on it... | MATH | {
"ground_truth": "\\frac{4 \\sqrt{3}}{9}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5553"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the minimum value of\n\\[\\frac{\\sin^6 x + \\cos^6 x + 1}{\\sin^4 x + \\cos^4 x + 1}\\]over all real values $x.$... | MATH | {
"ground_truth": "\\frac{5}{6}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5554"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe vectors $\\mathbf{a} = \\begin{pmatrix} 3 \\\\ 1 \\\\ -2 \\end{pmatrix}$ and $\\mathbf{b} = \\begin{pmatrix} 0 \\\... | MATH | {
"ground_truth": "-\\frac{1}{6}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5555"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $P$ be the plane passing through the origin with normal vector $\\begin{pmatrix} 1 \\\\ -2 \\\\ 1 \\end{pmatrix}.$... | MATH | {
"ground_truth": "\\begin{pmatrix} \\frac{5}{6} & \\frac{1}{3} & -\\frac{1}{6} \\\\ \\frac{1}{3} & \\frac{1}{3} & \\frac{1}{3} \\\\ -\\frac{1}{6} & \\frac{1}{3} & \\frac{5}{6} \\end{pmatrix}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5556"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the range of the function $y=\\log_2 (\\sqrt{\\cos x})$ for $-90^\\circ< x < 90^\\circ$?\n\nRemember to put yo... | MATH | {
"ground_truth": "(-\\infty,0]",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5557"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn spherical coordinates, the point $\\left( 3, \\frac{2 \\pi}{7}, \\frac{8 \\pi}{5} \\right)$ is equivalent to what o... | MATH | {
"ground_truth": "\\left( 3, \\frac{9 \\pi}{7}, \\frac{2 \\pi}{5} \\right)",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5558"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIf $e^{i \\alpha} = \\frac{3}{5} +\\frac{4}{5} i$ and $e^{i \\beta} = -\\frac{12}{13} + \\frac{5}{13} i,$ then find $... | MATH | {
"ground_truth": "-\\frac{33}{65}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5559"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\mathbf{R}$ be the matrix for reflecting over the vector $\\begin{pmatrix} 3 \\\\ 1 \\end{pmatrix}.$ Find $\\ma... | MATH | {
"ground_truth": "\\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5560"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe sides of an isosceles triangle are $\\cos x,$ $\\cos x,$ and $\\cos 7x,$ and its vertex angle is $2x.$ (All angle... | MATH | {
"ground_truth": "10^\\circ, 50^\\circ, 54^\\circ",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5561"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIf $\\sin x + \\sin y = \\frac{96}{65}$ and $\\cos x + \\cos y = \\frac{72}{65}$, then what is the value of $\\tan x +... | MATH | {
"ground_truth": "\\frac{507}{112}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5562"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor each integer $n$ greater than 1, let $F(n)$ be the number of solutions of the equation $\\sin x = \\sin nx$ on the... | MATH | {
"ground_truth": "2{,}016{,}532",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5563"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn triangle $ABC,$ the midpoint of $\\overline{BC}$ is $(1,5,-1),$ the midpoint of $\\overline{AC}$ is $(0,4,-2),$ and... | MATH | {
"ground_truth": "(1, 2, 3)",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5564"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\mathbf{D}$ be a matrix representing a dilation with scale factor $k > 0,$ and let $\\mathbf{R}$ be a matrix rep... | MATH | {
"ground_truth": "\\frac{1}{2}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5565"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor a certain value of $k,$ the system\n\\begin{align*}\nx + ky + 3z &= 0, \\\\\n3x + ky - 2z &= 0, \\\\\n2x + 4y - 3z... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5566"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all $a,$ $0^\\circ < a < 360^\\circ,$ such that $\\cos a,$ $\\cos 2a,$ and $\\cos 3a$ form an arithmetic sequence... | MATH | {
"ground_truth": "45^\\circ, 135^\\circ, 225^\\circ, 315^\\circ",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5567"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe set of vectors $\\mathbf{v}$ such that\n\\[\\operatorname{proj}_{\\begin{pmatrix} 5 \\\\ 2 \\end{pmatrix}} \\mathb... | MATH | {
"ground_truth": "y = -\\frac{5}{2} x - \\frac{29}{4}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5568"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nPoints $A,$ $B,$ $C,$ and $D$ are equally spaced along a line such that $AB = BC = CD.$ A point $P$ is located so tha... | MATH | {
"ground_truth": "\\frac{18}{25}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5569"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn triangle $ABC,$ $AB = 9,$ $BC = 10,$ and $AC = 11.$ If $D$ and $E$ are chosen on $\\overline{AB}$ and $\\overline{... | MATH | {
"ground_truth": "\\frac{280 \\sqrt{2}}{33}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5570"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the smallest positive integer $n$ such that\n\\[\\begin{pmatrix} \\cos 170^\\circ & -\\sin 170^\\circ \\\\ \\sin ... | MATH | {
"ground_truth": "36",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5571"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the point where the line passing through $(3,4,1)$ and $(5,1,6)$ intersects the $xy$-plane.\n\nRemember to put yo... | MATH | {
"ground_truth": "\\left( \\frac{13}{5}, \\frac{23}{5}, 0 \\right)",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5572"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c}$ be unit vectors such that\n\\[\\mathbf{a} \\times (\\mathbf{b} \\times... | MATH | {
"ground_truth": "135^\\circ",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5573"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDefine $\\mathbf{A} = \\begin{pmatrix} 0 & 1 \\\\ 3 & 0 \\end{pmatrix}.$ Find the vector $\\mathbf{v}$ such that\n\\[... | MATH | {
"ground_truth": "\\begin{pmatrix} 0 \\\\ 1/11 \\end{pmatrix}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5574"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the ordered pair $(a,b)$ of integers such that\n\\[\\sqrt{9 - 8 \\sin 50^\\circ} = a + b \\csc 50^\\circ.\\]\n\nR... | MATH | {
"ground_truth": "(3,-1)",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5575"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEvaluate\n\\[\\begin{vmatrix} 1 & x & y \\\\ 1 & x + y & y \\\\ 1 & x & x + y \\end{vmatrix}.\\]\n\nRemember to put yo... | MATH | {
"ground_truth": "xy",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5576"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe matrix $\\mathbf{A} = \\begin{pmatrix} 2 & 3 \\\\ 5 & d \\end{pmatrix}$ satisfies\n\\[\\mathbf{A}^{-1} = k \\mathb... | MATH | {
"ground_truth": "\\left( -2, \\frac{1}{19} \\right)",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5577"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe projection of $\\begin{pmatrix} 0 \\\\ 3 \\end{pmatrix}$ onto a certain vector $\\mathbf{w}$ is $\\begin{pmatrix} ... | MATH | {
"ground_truth": "\\begin{pmatrix} 33/10 \\\\ -11/10 \\end{pmatrix}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5578"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA plane is expressed parametrically by\n\\[\\mathbf{v} = \\begin{pmatrix} 1 + s - t \\\\ 2 - s \\\\ 3 - 2s + 2t \\end{... | MATH | {
"ground_truth": "2x + z - 5 = 0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5579"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the smallest positive integer $k$ such that $\nz^{10} + z^9 + z^6+z^5+z^4+z+1\n$ divides $z^k-1$.\n\nRemember to ... | MATH | {
"ground_truth": "84",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5580"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIf\n\\[\\sin x + \\cos x + \\tan x + \\cot x + \\sec x + \\csc x = 7,\\]then find $\\sin 2x.$\n\nRemember to put your ... | MATH | {
"ground_truth": "22 - 8 \\sqrt{7}",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5581"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDefine the sequence $a_1, a_2, a_3, \\ldots$ by $a_n = \\sum\\limits_{k=1}^n \\sin{k}$, where $k$ represents radian me... | MATH | {
"ground_truth": "628",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5582"
} |
dapo_math_train | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the number of real solutions of the equation\n\\[\\frac{x}{100} = \\sin x.\\]\n\nRemember to put your answer on i... | MATH | {
"ground_truth": "63",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5583"
} |
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