data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepmath-103k | [
{
"role": "user",
"content": "Our club has 10 members, and wishes to pick a president, secretary, treasurer, and morale officer. In how many ways can we choose the officers, if individual members can only hold at most one office?"
}
] | math | {
"style": "rule",
"ground_truth": "5,\\!040"
} | {
"index": 102978,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "Given that\n\n\\begin{align*}\n\\frac{1}{x}+\\frac{1}{y}&=3,\\\\\nxy+x+y&=4,\n\\end{align*}\n\ncompute $x^2y+xy^2$."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 102979,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "Jonathon is given the following problem: Let $a$ and $b$ be positive real numbers such that $a + b = 1.$ Find the minimum value of\n\\[\\left( a + \\frac{1}{a} \\right)^2 + \\left( b + \\frac{1}{b} \\right)^2.\\]Jonathon reasons as follows: By AM-GM,\n\\[a + \\frac{1}{a} \\ge 2... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{2}"
} | {
"index": 102980,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "If $f(x)=x^3$, what is the value of $f^{-1}(8)\\div (f(8))^{-1}$?"
}
] | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"index": 102981,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "Simplify $\\frac{1}{1+\\sqrt{2}}\\cdot\\frac{1}{1-\\sqrt{2}}$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 102982,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "An infinite geometric series has common ratio $1/8$ and sum 60. What is the first term of the series?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{105}{2}"
} | {
"index": 102983,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "Solve for $x$: $5(3x + 2) - 2 = -2(1 - 7x)$."
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"index": 102984,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "The six faces of a three-inch wooden cube are each painted red. The cube is then cut into one-inch cubes along the lines shown in the diagram. How many of the one-inch cubes have red paint on at least two faces? [asy]\n\npair A,B,C,D,E,F,G;\n\npair a,c,d,f,g,i,j,l,m,o,p,r,s,u,v... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 102985,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "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 $120^{\\circ}$... | math | {
"style": "rule",
"ground_truth": "\\cos(6^\\circ)\\sin(12^\\circ)\\csc(18^\\circ)"
} | {
"index": 102986,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "Let $x,$ $y,$ and $z$ be positive real numbers. Find the minimum value of\n\\[\\frac{\\sqrt{x^2 + y^2} + \\sqrt{x^2 + z^2} + \\sqrt{y^2 + z^2}}{x + y + z}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 102987,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "What is the smallest positive five-digit integer, with all different digits, that is divisible by each of its non-zero digits? Note that one of the digits of the original integer may be a zero."
}
] | math | {
"style": "rule",
"ground_truth": "10,\\!236"
} | {
"index": 102988,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ be three vectors such that $\\|\\mathbf{a}\\| = \\|\\mathbf{b}\\| = \\|\\mathbf{c}\\| = 2.$ Also, the angle between any two of these vectors is $\\arccos \\frac{5}{8}.$ Find the volume of the parallelepiped generated by $\\mat... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 102989,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "Let $f(x)=\\frac{1}{2x-5}$. Find the largest $x$ which is not in the domain of $g(x)=f(f(x))$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{27}{10}"
} | {
"index": 102990,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "The function defined by\n\\[f(x) = \\left\\{\n\\begin{array}{cl}\nx + k & \\text{if $x < 4$}, \\\\\n2x - 3 & \\text{if $x \\ge 4$}\n\\end{array}\n\\right.\\]has an inverse, and the inverse is defined for all real numbers. Enter all possible values of $k,$ separated by commas."
... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 102991,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "One of the roots of\n\\[z^4 + az^3 + 5z^2 - iz - 6 = 0\\]is $2i,$ where $a$ is a complex number. Enter the other three roots, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "1,-1,-3i"
} | {
"index": 102992,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "Consider the six by six grid of unit squares below. How many rectangles of area 3 square units can be formed using only the line segments of the grid as the sides of the rectangles?\n\nunitsize(0.1inch);\ndraw((0,0)--(6,0)--(6,6)--(0,6)--cycle);\ndraw((1,0)--(1,6));\ndraw((2,0)... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 102993,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "Circle $T$ has a circumference of $12\\pi$ inches, and segment $XY$ is a diameter. If the measure of angle $TXZ$ is $60^{\\circ}$, what is the length, in inches, of segment $XZ$?\n\nsize(150);\ndraw(Circle((0,0),13),linewidth(1));\ndraw((-12,-5)--(-5,-12)--(12,5)--cycle,linewid... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 102994,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "Let $n$ be a positive integer and $a$ be an integer such that $a$ is its own inverse modulo $n$. What is the remainder when $a^2$ is divided by $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 102995,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "William runs at a constant speed of 8 miles per hour. Running at this constant speed, it takes William 75 seconds to run from his house to school. How many miles did William run?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 102996,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "In a history class, the probability of earning an A is .7 times the probability of earning a B, and the probability of earning a C is 1.4 times the probability of earning a B. Assuming that all grades are A, B, or C, how many B's will there be in a history class of 31 students?... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 102997,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "When the base-$b$ number $11011_b$ is multiplied by $b-1$, then $1001_b$ is added, what is the result (written in base $b$)?"
}
] | math | {
"style": "rule",
"ground_truth": "100100"
} | {
"index": 102998,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "The coefficients of the polynomial\n\\[a_{10} x^{10} + a_9 x^9 + a_8 x^8 + \\dots + a_2 x^2 + a_1 x + a_0 = 0\\]are all integers, and its roots $r_1,$ $r_2,$ $\\dots,$ $r_{10}$ are all integers. Furthermore, the roots of the polynomial\n\\[a_0 x^{10} + a_1 x^9 + a_2 x^8 + \\dot... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 102999,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "Solve for $x$: $\\left(\\frac{1}{4}\\right)^{2x+8} = (16)^{2x+5}$."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 103000,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "The graph of \\[\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1\\]has its foci at $(0,\\pm 4),$ while the graph of \\[\\frac{x^2}{a^2}-\\frac{y^2}{b^2} = 1\\]has its foci at $(\\pm 6,0).$ Compute the value of $|ab|.$"
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{65}"
} | {
"index": 103001,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "Point $P$ is inside equilateral $\\triangle ABC$. Points $Q$, $R$, and $S$ are the feet of the perpendiculars from $P$ to $\\overline{AB}$, $\\overline{BC}$, and $\\overline{CA}$, respectively. Given that $PQ=1$, $PR=2$, and $PS=3$, what is $AB$ in terms of radicals?"
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"index": 103002,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "Find the integer $n$, $4 \\le n \\le 8$, such that \\[n \\equiv 7882 \\pmod{5}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 103003,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "If $a$, $b$, and $c$ are consecutive integers, find the area of the shaded region in the square below: [asy]\nsize(1.75inch);\npair A, B, C, D, W, X, Y, Z;\nA = (0,0); B = (7,0); C = (7,7); D = (0,7);\nW = (3,0); X = (7,3); Y = (4,7); Z = (0,4);\ndraw(A--B--C--D--cycle); draw(W... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 103004,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "One line is described by\n\\[\\begin{pmatrix} -1 \\\\ -3 \\\\ -5 \\end{pmatrix} + t \\begin{pmatrix} 3 \\\\ k \\\\ 7 \\end{pmatrix}.\\]Another line is described by\n\\[\\begin{pmatrix} 2 \\\\ 4 \\\\ 6 \\end{pmatrix} + u \\begin{pmatrix} 1 \\\\ 4 \\\\ 7 \\end{pmatrix}.\\]Find $k... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 103005,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "There are real numbers $a$ and $b$ such that for every positive number $x$, we have the identity\n\\[\n \\tan^{-1} \\left( \\frac{1}{x} - \\frac{x}{8} \\right) + \\tan^{-1}(ax) + \\tan^{-1}(bx) \n = \\frac{\\pi}{2} \\, .\n\\](Throughout this equation, $\\tan^{-1}$ means the inv... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 103006,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "Compute\n\\[\\frac{1}{2^3 - 2} + \\frac{1}{3^3 - 3} + \\frac{1}{4^3 - 4} + \\dots + \\frac{1}{100^3 - 100}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5049}{20200}"
} | {
"index": 103007,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "A sheet of 8-inch by 10-inch paper is placed on top of a sheet of $8 \\frac{1}{2}$-inch by 11-inch paper, as shown. What is the area of the region of overlap in square inches?\n\ndraw((0,0)--(10,0)--(10,8)--(0,8)--(0,0)--cycle,linewidth(2));\ndraw((0,8)--(8.5,8)--(8.5,11.5)--(0... | math | {
"style": "rule",
"ground_truth": "68"
} | {
"index": 103008,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "In trapezoid $ABCD$, leg $\\overline{BC}$ is perpendicular to bases $\\overline{AB}$ and $\\overline{CD}$, and diagonals $\\overline{AC}$ and $\\overline{BD}$ are perpendicular. Given that $AB=\\sqrt{11}$ and $AD=\\sqrt{1001}$, find $BC^2$."
}
] | math | {
"style": "rule",
"ground_truth": "110"
} | {
"index": 103009,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c$ be positive real numbers such that $a + b + c = 1.$ Find the minimum value of $a^2 + 2b^2 + c^2.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 103010,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "What is the coefficient of $x^3$ when $$24x^4 + 6x^3 + 4x^2-7x - 5$$is multiplied by $$6x^3 + 3x^2 + 3x + 4$$and the like terms are combined?"
}
] | math | {
"style": "rule",
"ground_truth": "-15"
} | {
"index": 103011,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "If $\\left\\lfloor n^2/4 \\right\\rfloor - \\lfloor n/2 \\rfloor^2 = 2$, then find all integer values of $n$."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 103012,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "The diagonals of rectangle $PQRS$ intersect at point $X$. If $PS = 6$ and $RS=8$, then what is $\\sin \\angle PXS$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{24}{25}"
} | {
"index": 103013,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "There are two concentric spheres of radii 3 units and 6 units. What is the volume, in cubic units, of the region within the larger sphere and not within the smaller sphere? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "252\\pi"
} | {
"index": 103014,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "If three coins are tossed at the same time, what is the probability of getting two tails and one head? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"index": 103015,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "If the odds for pulling a prize out of the box are $3:4$, what is the probability of not pulling the prize out of the box? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{7}"
} | {
"index": 103016,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "Let $S$ denote the value of the sum\\[\\sum_{n=0}^{668} (-1)^{n} {2004 \\choose 3n}\\]Determine the remainder obtained when $S$ is divided by $1000$."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 103017,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "How many numbers in the list $43$, $4343$, $434343$, $\\dots$, are prime?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 103018,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB=120,BC=220$, and $AC=180$. Lines $\\ell_A,\\ell_B$, and $\\ell_C$ are drawn parallel to $\\overline{BC},\\overline{AC}$, and $\\overline{AB}$, respectively, such that the intersections of $\\ell_A,\\ell_B$, and $\\ell_C$ with the interior of ... | math | {
"style": "rule",
"ground_truth": "715"
} | {
"index": 103019,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "A car travels 40 kph for 20 kilometers, 50 kph for 25 kilometers, 60 kph for 45 minutes and 48 kph for 15 minutes. What is the average speed of the car, in kph?"
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 103020,
"original_dataset": "deepmath-103k",
"split": "train"
} |
deepmath-103k | [
{
"role": "user",
"content": "What is the largest $n$ such that $a = 2^{306} \\cdot 3^{340}$ is a perfect $n$th power?"
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 103021,
"original_dataset": "deepmath-103k",
"split": "train"
} |
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