data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "(1) Simplify: $\\dfrac {\\tan (3\\pi-\\alpha)\\cos (2\\pi-\\alpha)\\sin (-\\alpha+ \\dfrac {3\\pi}{2})}{\\cos (-\\alpha-\\pi)\\sin (-\\pi+\\alpha)\\cos (\\alpha+ \\dfrac {5\\pi}{2})}$; \n(2) Given $\\tan \\alpha= \\dfrac {1}{4}$, find the value of $\\dfrac {1}{2\\cos ^{2}\\alpha -3\\sin \\alpha \... | {
"ground_truth": "\\dfrac {17}{20}",
"style": "rule"
} | {
"index": "33200",
"split": "train"
} |
math | [
{
"content": "How many ways are there to color the edges of a hexagon orange and black if we assume that two hexagons are indistinguishable if one can be rotated into the other? Note that we are saying the colorings OOBBOB and BOBBOO are distinct; we ignore flips. Please output the final answer within \\boxed{}... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "33201",
"split": "train"
} |
math | [
{
"content": "The leg, the smaller base, and the diagonal of an isosceles trapezoid are 10, 6, and 14 respectively. Find the length of the larger base. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "33202",
"split": "train"
} |
math | [
{
"content": "In a regular triangular prism \\(ABC-A_1B_1C_1\\), all 9 edges are equal in length. Point \\(P\\) is the midpoint of \\(CC_1\\). The dihedral angle \\(B-A_1P-B_1 = \\alpha\\). Find \\(\\sin \\alpha\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{10}}{4}",
"style": "rule"
} | {
"index": "33203",
"split": "train"
} |
math | [
{
"content": "The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer ${n\\ge 4}$, find the least number of edges of a graph that can be obtained by repeated applicati... | {
"ground_truth": "n",
"style": "rule"
} | {
"index": "33204",
"split": "train"
} |
math | [
{
"content": "In a notebook, there is a grid rectangle of size $3 \\times 7$. Igor's robot was asked to trace all the lines with a marker, and it took him 26 minutes (the robot draws lines at a constant speed). How many minutes will it take him to trace all the lines in a $5 \\times 5$ grid square with the mark... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "33205",
"split": "train"
} |
math | [
{
"content": "If a natural number \\(N\\) can be expressed as the sum of 3 consecutive natural numbers, the sum of 11 consecutive natural numbers, and the sum of 12 consecutive natural numbers, then the smallest value of \\(N\\) is \\(\\qquad\\). (Note: The smallest natural number is 0.) Please output the final... | {
"ground_truth": "66",
"style": "rule"
} | {
"index": "33206",
"split": "train"
} |
math | [
{
"content": "In a new diagram showing the miles traveled by bikers Alberto, Bjorn, and Carlos over a period of 6 hours. The straight lines represent their paths on a coordinate plot where the y-axis represents miles and x-axis represents hours. Alberto's line passes through the points (0,0) and (6,90), Bjorn's... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "33207",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $\\sin C+\\cos C=1-\\sin \\frac{C}{2}$.\n\n$(1)$ Find the value of $\\sin C$.\n\n$(2)$ If $a^{2}+b^{2}=4(a+b)-8$, find the value of side $c$. Please output the final ... | {
"ground_truth": "1+ \\sqrt{7}",
"style": "rule"
} | {
"index": "33208",
"split": "train"
} |
math | [
{
"content": "Determine the value of:\n\\[3003 + \\frac{1}{3} \\left( 3002 + \\frac{1}{3} \\left( 3001 + \\dots + \\frac{1}{3} \\left( 4 + \\frac{1}{3} \\cdot 3 \\right) \\right) \\dotsb \\right).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9006.5",
"style": "rule"
} | {
"index": "33209",
"split": "train"
} |
math | [
{
"content": "After lunch, there are dark spots with a total area of $S$ on a transparent square tablecloth. It turns out that if the tablecloth is folded in half along any of the two lines connecting the midpoints of its opposite sides or along one of its two diagonals, the total visible area of the spots beco... | {
"ground_truth": "2/3",
"style": "rule"
} | {
"index": "33210",
"split": "train"
} |
math | [
{
"content": "How many zeroes does the product of \\( 25^{5} \\), \\( 150^{4} \\), and \\( 2008^{3} \\) end with? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "33211",
"split": "train"
} |
math | [
{
"content": "Solve the equation: $(2x+1)^2=3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{-1-\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "33212",
"split": "train"
} |
math | [
{
"content": "Let the altitude of a regular triangular pyramid \\( P-ABC \\) be \\( PO \\). \\( M \\) is the midpoint of \\( PO \\). A plane parallel to edge \\( BC \\) passes through \\( AM \\), dividing the pyramid into two parts, upper and lower. Find the volume ratio of these two parts. Please output the fi... | {
"ground_truth": "4/21",
"style": "rule"
} | {
"index": "33213",
"split": "train"
} |
math | [
{
"content": "Given that $S_{n}$ is the sum of the first $n$ terms of an arithmetic sequence ${a_{n}}$, $S_{1} < 0$, $2S_{21}+S_{25}=0$, find the value of $n$ when $S_{n}$ is minimized. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "33214",
"split": "train"
} |
math | [
{
"content": "Let $n\\ge 3$ be a fixed integer. There are $m\\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\\emph{not}$ possible.\n\n[i]Carl... | {
"ground_truth": "$\\boxed{n^2-n-1}$",
"style": "rule"
} | {
"index": "33215",
"split": "train"
} |
math | [
{
"content": "Monica is trying to solve the following equation by completing the square: $$x^2-8x+8 = 0.$$She successfully rewrites the above equation in the following form: $$(x + b)^2 = c,$$where $b$ and $c$ are integers. What is the value of $b + c$? Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "33216",
"split": "train"
} |
math | [
{
"content": "Given that $2^x+ 2^x+ 2^x+ 2^x= 512$, what is the value of $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "33217",
"split": "train"
} |
math | [
{
"content": "Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen? Ple... | {
"ground_truth": "170",
"style": "rule"
} | {
"index": "33218",
"split": "train"
} |
math | [
{
"content": "The point $(1,1,1)$ is rotated $180^\\circ$ about the $y$-axis, then reflected through the $yz$-plane, reflected through the $xz$-plane, rotated $180^\\circ$ about the $y$-axis, and reflected through the $xz$-plane. Find the coordinates of the point now. Please output the final answer within \\bo... | {
"ground_truth": "(-1,1,1)",
"style": "rule"
} | {
"index": "33219",
"split": "train"
} |
math | [
{
"content": "Given an arithmetic sequence $\\left\\{a_{n}\\right\\}$ with the sum of the first 12 terms being 60, find the minimum value of $\\left|a_{1}\\right| + \\left|a_{2}\\right| + \\cdots + \\left|a_{12}\\right|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "33220",
"split": "train"
} |
math | [
{
"content": "If $\\phi$ is the Golden Ratio, we know that $\\frac1\\phi = \\phi - 1$ . Define a new positive real number, called $\\phi_d$ , where $\\frac1{\\phi_d} = \\phi_d - d$ (so $\\phi = \\phi_1$ ). Given that $\\phi_{2009} = \\frac{a + \\sqrt{b}}{c}$ , $a, b, c$ positive integers, and the gr... | {
"ground_truth": "4038096",
"style": "rule"
} | {
"index": "33221",
"split": "train"
} |
math | [
{
"content": "Let $$\\overset{ .}{a_{n}a_{n-1}a_{n-2}\\cdots a_{1}a_{0}}|_{m}$$ be defined as $a_{0}+a_{1}\\times m+\\ldots+a_{n-1}\\times m^{n-1}+a_{n}\\times m^{n}$, where $n\\leq m$, $m$ and $n$ are positive integers, $a_{k}\\in\\{0,1,2,\\ldots,m-1\\}$ ($k=0,1,2,\\ldots,n$) and $a_{n}\\neq 0$; \n(1) Calcula... | {
"ground_truth": "699",
"style": "rule"
} | {
"index": "33222",
"split": "train"
} |
math | [
{
"content": "Given an ellipse with foci at $(1,0)$ and $(-1,0)$, it intersects with the line $y=x-2$. Determine the maximum value of the eccentricity of this ellipse. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{10}}{5}",
"style": "rule"
} | {
"index": "33223",
"split": "train"
} |
math | [
{
"content": "What is the least positive integer with exactly $12$ positive factors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "33224",
"split": "train"
} |
math | [
{
"content": "The function \\[f(x) = \\left\\{ \\begin{aligned} x-2 & \\quad \\text{ if } x < 4 \\\\ \\sqrt{x} & \\quad \\text{ if } x \\ge 4 \\end{aligned} \\right.\\]has an inverse $f^{-1}.$ Find the value of $f^{-1}(-5) + f^{-1}(-4) + \\dots + f^{-1}(4) + f^{-1}(5).$ Please output the final answer within \\b... | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "33225",
"split": "train"
} |
math | [
{
"content": "A sequence of real numbers $\\{a_n\\}_{n = 1}^\\infty (n=1,2,...)$ has the following property:\n\\begin{align*}\n6a_n+5a_{n-2}=20+11a_{n-1}\\ (\\text{for }n\\geq3).\n\\end{align*}\nThe first two elements are $a_1=0, a_2=1$ . Find the integer closest to $a_{2011}$ . Please output the final ans... | {
"ground_truth": "40086",
"style": "rule"
} | {
"index": "33226",
"split": "train"
} |
math | [
{
"content": "Nine lines parallel to the base of a triangle divide the other sides each into $10$ equal segments and the area into $10$ distinct parts. If the area of the largest of these parts is $38$, calculate the area of the original triangle. Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "33227",
"split": "train"
} |
math | [
{
"content": "Two concentric circles have radii $1$ and $4$ . Six congruent circles form a ring where each of the six circles is tangent to the two circles adjacent to it as shown. The three lightly shaded circles are internally tangent to the circle with radius $4$ while the three darkly shaded circles ... | {
"ground_truth": "126",
"style": "rule"
} | {
"index": "33228",
"split": "train"
} |
math | [
{
"content": "Find the smallest four-digit number whose product of all digits equals 512. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1888",
"style": "rule"
} | {
"index": "33229",
"split": "train"
} |
math | [
{
"content": "The equation $2^{333x-2} + 2^{111x+2} = 2^{222x+1} + 1$ has three real roots. Given that their sum is $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "113",
"style": "rule"
} | {
"index": "33230",
"split": "train"
} |
math | [
{
"content": "Let $A$, $B$, $R$, $M$, and $L$ be positive real numbers such that\n\\begin{align*}\n\\log_{10} (AB) + \\log_{10} (AM) &= 2, \\\\\n\\log_{10} (ML) + \\log_{10} (MR) &= 3, \\\\\n\\log_{10} (RA) + \\log_{10} (RB) &= 5.\n\\end{align*}\nCompute the value of the product $ABRML$. Please output the final... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "33231",
"split": "train"
} |
math | [
{
"content": "Geoff walks on the number line for 40 minutes, starting at the point 0. On the $n$th minute, he flips a fair coin. If it comes up heads he walks $\\frac{1}{n}$ in the positive direction and if it comes up tails he walks $\\frac{1}{n}$ in the negative direction. Let $p$ be the probability that he n... | {
"ground_truth": "8101",
"style": "rule"
} | {
"index": "33232",
"split": "train"
} |
math | [
{
"content": "Given that $\\alpha$ and $\\beta$ are acute angles, and $\\cos(\\alpha+\\beta)=\\frac{\\sin\\alpha}{\\sin\\beta}$, determine the maximum value of $\\tan \\alpha$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "33233",
"split": "train"
} |
math | [
{
"content": "Let \\[f(n) =\n\\begin{cases}\nn^2-1 & \\text{ if }n < 4,\n\\\\ 3n-2 & \\text{ if }n \\geq 4.\n\\end{cases}\n\\]Find $f(f(f(2)))$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "33234",
"split": "train"
} |
math | [
{
"content": "Given that real numbers \\( a \\) and \\( b \\) are such that the equation \\( a x^{3} - x^{2} + b x - 1 = 0 \\) has three positive real roots, find the minimum value of \\( P = \\frac{5 a^{2} - 3 a b + 2}{a^{2}(b - a)} \\) for all such \\( a \\) and \\( b \\). Please output the final answer withi... | {
"ground_truth": "12 \\sqrt{3}",
"style": "rule"
} | {
"index": "33235",
"split": "train"
} |
math | [
{
"content": "A sequence $b_1, b_2, b_3, \\dots,$ is defined recursively by $b_1 = 3,$ $b_2 = 2,$ and for $k \\ge 3,$\n\\[ b_k = \\frac{1}{4} b_{k-1} + \\frac{2}{5} b_{k-2}. \\]\nEvaluate $b_1 + b_2 + b_3 + \\dotsb.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{85}{7}",
"style": "rule"
} | {
"index": "33236",
"split": "train"
} |
math | [
{
"content": "Let $A X B Y$ be a cyclic quadrilateral, and let line $A B$ and line $X Y$ intersect at $C$. Suppose $A X \\cdot A Y=6, B X \\cdot B Y=5$, and $C X \\cdot C Y=4$. Compute $A B^{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{242}{15}",
"style": "rule"
} | {
"index": "33237",
"split": "train"
} |
math | [
{
"content": "Given the sequence $\\left\\{a_{n}\\right\\}$ defined by\n$$\na_{n}=\\left[(2+\\sqrt{5})^{n}+\\frac{1}{2^{n}}\\right] \\quad (n \\in \\mathbf{Z}_{+}),\n$$\nwhere $[x]$ denotes the greatest integer not exceeding the real number $x$, determine the minimum value of the constant $C$ such that for any ... | {
"ground_truth": "1/288",
"style": "rule"
} | {
"index": "33238",
"split": "train"
} |
math | [
{
"content": "For each interger $n\\geq 4$, we consider the $m$ subsets $A_1, A_2,\\dots, A_m$ of $\\{1, 2, 3,\\dots, n\\}$, such that\n$A_1$ has exactly one element, $A_2$ has exactly two elements,...., $A_m$ has exactly $m$ elements and none of these subsets is contained in any other set. Find the maximum val... | {
"ground_truth": "\\left\\lfloor \\frac{-1 + \\sqrt{1 + 8n}}{2} \\right\\rfloor",
"style": "rule"
} | {
"index": "33239",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{M}$ be a matrix such that\n\\[\\mathbf{M} \\begin{pmatrix} 2 \\\\ -1 \\end{pmatrix} = \\begin{pmatrix} 3 \\\\ 0 \\end{pmatrix} \\quad \\text{and} \\quad \\mathbf{M} \\begin{pmatrix} -3 \\\\ 5 \\end{pmatrix} = \\begin{pmatrix} -1 \\\\ -1 \\end{pmatrix}.\\]Compute $\\mathbf{M} \\begin{... | {
"ground_truth": "\\begin{pmatrix} 11 \\\\ -1 \\end{pmatrix}",
"style": "rule"
} | {
"index": "33240",
"split": "train"
} |
math | [
{
"content": "Given that a hyperbola $mx^{2}+ny^{2}=1$ has a focus that is the same as the focus of the parabola $y=\\frac{1}{8}{x^2}$ and an eccentricity of $2$, calculate the distance from the focus of the parabola to one of the asymptotes of the hyperbola. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "33241",
"split": "train"
} |
math | [
{
"content": "Given $\\begin{vmatrix} x & y \\\\ z & w \\end{vmatrix} = 3,$ find $\\begin{vmatrix} 3x & 3y \\\\ 3z & 3w \\end{vmatrix}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "33242",
"split": "train"
} |
math | [
{
"content": "Find the smallest composite number that has no prime factors less than 15. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "289",
"style": "rule"
} | {
"index": "33243",
"split": "train"
} |
math | [
{
"content": "Four steel balls, each with a radius of 1, are completely packed into a container in the shape of a regular tetrahedron. Find the minimum height of this regular tetrahedron. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2+\\frac{2 \\sqrt{6}}{3}",
"style": "rule"
} | {
"index": "33244",
"split": "train"
} |
math | [
{
"content": "How many positive two-digit integers have an odd number of positive factors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "33245",
"split": "train"
} |
math | [
{
"content": "Given that a product originally priced at an unknown value is raised by 20% twice consecutively, and another product originally priced at an unknown value is reduced by 20% twice consecutively, ultimately selling both at 23.04 yuan each, determine the profit or loss situation when one piece of eac... | {
"ground_truth": "5.92",
"style": "rule"
} | {
"index": "33246",
"split": "train"
} |
math | [
{
"content": "The percentage of seventh-grade students participating in the gymnastics section is between 2.9% and 3.1%. Determine the smallest possible number of students in this class. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "33247",
"split": "train"
} |
math | [
{
"content": "The sequence \\(a_{n}\\) is defined as follows:\n\n\\[ a_{1} = 1, \\quad a_{n+1} = a_{n} + \\frac{2a_{n}}{n} \\text{ for } n \\geq 1 \\]\n\nFind \\(a_{100}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5151",
"style": "rule"
} | {
"index": "33248",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has sidelengths $AB=1$ , $BC=\\sqrt{3}$ , and $AC=2$ . Points $D,E$ , and $F$ are chosen on $AB, BC$ , and $AC$ respectively, such that $\\angle EDF = \\angle DFA = 90^{\\circ}$ . Given that the maximum possible value of $[DEF]^2$ can be expressed as $\\frac{a}{b}$ f... | {
"ground_truth": "67",
"style": "rule"
} | {
"index": "33249",
"split": "train"
} |
math | [
{
"content": "Let $\\phi$ be the smallest acute angle for which $\\cos \\phi,$ $\\cos 2\\phi,$ $\\cos 4\\phi$ form an arithmetic progression, in some order. Find $\\sin \\phi.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{\\sqrt{2}}",
"style": "rule"
} | {
"index": "33250",
"split": "train"
} |
math | [
{
"content": "What is the remainder when the sum \\(1^2 + 2^2 + 3^2 + \\dots + 20^2\\) is divided by 13? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "33251",
"split": "train"
} |
math | [
{
"content": "A segment connecting the centers of two intersecting circles is divided by their common chord into segments equal to 5 and 2. Find the length of the common chord, given that the radii of the circles are in the ratio \\(4: 3\\). Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "2\\sqrt{23}",
"style": "rule"
} | {
"index": "33252",
"split": "train"
} |
math | [
{
"content": "For which value of $x$ does the function $f(x) = \\frac{2x^2 - 5x - 7}{x^2 - 4x + 1}$ cross its horizontal asymptote? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "33253",
"split": "train"
} |
math | [
{
"content": "Given $f(x)=1-2x^{2}$ and $g(x)=x^{2}-2x$, let $F(x) = \\begin{cases} f(x), & \\text{if } f(x) \\geq g(x) \\\\ g(x), & \\text{if } f(x) < g(x) \\end{cases}$. Determine the maximum value of $F(x)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{9}",
"style": "rule"
} | {
"index": "33254",
"split": "train"
} |
math | [
{
"content": "Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $7, n,$ and $n+2$ cents, $120$ cents is the greatest postage that cannot be formed. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "33255",
"split": "train"
} |
math | [
{
"content": "How many numbers from $1$ to $150$ are not perfect squares or perfect cubes? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "135",
"style": "rule"
} | {
"index": "33256",
"split": "train"
} |
math | [
{
"content": "In the plane rectangular coordinate system $xOy$, the parametric equations of curve $C$ are $\\left\\{\\begin{array}{l}{x=2+3\\cos\\alpha,}\\\\{y=3\\sin\\alpha}\\end{array}\\right.$ ($\\alpha$ is the parameter). Taking the coordinate origin $O$ as the pole and the non-negative $x$-axis as the pola... | {
"ground_truth": "\\frac{3\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "33257",
"split": "train"
} |
math | [
{
"content": "For some real numbers $a$ and $b$, the equation $9x^3 + 5ax^2 + 4bx + a = 0$ has three distinct positive roots. If the sum of the base-2 logarithms of the roots is 4, what is the value of $a$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-144",
"style": "rule"
} | {
"index": "33258",
"split": "train"
} |
math | [
{
"content": "How many different positive, five-digit integers can be formed using the digits 2, 2, 2, 9 and 9? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "33259",
"split": "train"
} |
math | [
{
"content": "Given the system of equations $\\begin{cases} x - 2y = z - 2u \\\\ 2yz = ux \\end{cases}$, for every set of positive real number solutions $\\{x, y, z, u\\}$ where $z \\geq y$, there exists a positive real number $M$ such that $M \\leq \\frac{z}{y}$. Find the maximum value of $M$. Please output th... | {
"ground_truth": "6 + 4\\sqrt{2}",
"style": "rule"
} | {
"index": "33260",
"split": "train"
} |
math | [
{
"content": "Let $\\lfloor x\\rfloor$ denote the largest integer less than or equal to $x$, and let $\\{x\\}$ denote the fractional part of $x$. For example, $\\lfloor\\pi\\rfloor=3$, and $\\{\\pi\\}=0.14159 \\ldots$, while $\\lfloor 100\\rfloor=100$ and $\\{100\\}=0$. If $n$ is the largest solution to the equ... | {
"ground_truth": "\\frac{2014}{2015}",
"style": "rule"
} | {
"index": "33261",
"split": "train"
} |
math | [
{
"content": "Evaluate $2000^3-1999\\cdot 2000^2-1999^2\\cdot 2000+1999^3$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3999",
"style": "rule"
} | {
"index": "33262",
"split": "train"
} |
math | [
{
"content": "Given that $α$ is an angle in the second quadrant and $\\cos (α+π)= \\frac {3}{13}$.\n(1) Find the value of $\\tan α$;\n(2) Find the value of $\\sin (α- \\frac {π}{2}) \\cdot \\sin (-α-π)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{12\\sqrt{10}}{169}",
"style": "rule"
} | {
"index": "33263",
"split": "train"
} |
math | [
{
"content": "In how many ways can $10001$ be written as the sum of two primes? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "33264",
"split": "train"
} |
math | [
{
"content": "In the two concentric circles shown, the radius of the outer circle is twice the radius of the inner circle. What is the area of the gray region, in square feet, if the width of the gray region is 2 feet? Express your answer in terms of $\\pi$.\n\n[asy]\nfilldraw(circle((0,0),4),gray);\nfilldraw(c... | {
"ground_truth": "12\\pi",
"style": "rule"
} | {
"index": "33265",
"split": "train"
} |
math | [
{
"content": "A unicorn is tethered by a $20$-foot silver rope to the base of a magician's cylindrical tower whose radius is $8$ feet. The rope is attached to the tower at ground level and to the unicorn at a height of $4$ feet. The unicorn has pulled the rope taut, the end of the rope is $4$ feet from the near... | {
"ground_truth": "813",
"style": "rule"
} | {
"index": "33266",
"split": "train"
} |
math | [
{
"content": "In convex quadrilateral $ ABCD$, $ AB\\equal{}a$, $ BC\\equal{}b$, $ CD\\equal{}c$, $ DA\\equal{}d$, $ AC\\equal{}e$, $ BD\\equal{}f$. If $ \\max \\{a,b,c,d,e,f \\}\\equal{}1$, then find the maximum value of $ abcd$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 - \\sqrt{3}",
"style": "rule"
} | {
"index": "33267",
"split": "train"
} |
math | [
{
"content": "Find the sum $$\\frac{3^1}{9^1 - 1} + \\frac{3^2}{9^2 - 1} + \\frac{3^3}{9^3 - 1} + \\frac{3^4}{9^4 - 1} + \\cdots.$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "33268",
"split": "train"
} |
math | [
{
"content": "Express $367_{8}+4CD_{13}$ as a base 10 integer, where $C$ and $D$ denote the digits whose values are 12 and 13, respectively, in base 13. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1079",
"style": "rule"
} | {
"index": "33269",
"split": "train"
} |
math | [
{
"content": "In a jar, there are 5 salted duck eggs of the same size and shape, among which 3 have green shells, and 2 have white shells. If two eggs are taken out one after another without replacement, calculate:\n(1) The probability of the first egg taken out being green-shelled;\n(2) The probability that bo... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "33270",
"split": "train"
} |
math | [
{
"content": "The graph shows the total distance Sam drove from 6 a.m to 11 a.m. How many miles per hour is the car's average speed for the period from 6 a.m. to 11 a.m.?\n\n[asy]\nunitsize(0.2inch);\ndraw((0,0)--(5.5,0));\ndraw((0,0)--(0,8.5));\ndraw((1,0)--(1,8.5));\ndraw((2,0)--(2,8.5));\ndraw((3,0)-... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "33271",
"split": "train"
} |
math | [
{
"content": "Given that $\\alpha$ is an angle in the second quadrant, and $\\cos\\left( \\frac {\\pi}{2}-\\alpha\\right)= \\frac {4}{5}$, then $\\tan\\alpha= \\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac {4}{3}",
"style": "rule"
} | {
"index": "33272",
"split": "train"
} |
math | [
{
"content": "Given an arithmetic sequence $\\{a_n\\}$ with the first term $a$ ($a \\in \\mathbb{R}, a \\neq 0$). Let the sum of the first $n$ terms of the sequence be $S_n$, and for any positive integer $n$, it holds that $$\\frac {a_{2n}}{a_{n}}= \\frac {4n-1}{2n-1}$$.\n(1) Find the general formula for $\\{a_... | {
"ground_truth": "k=3",
"style": "rule"
} | {
"index": "33273",
"split": "train"
} |
math | [
{
"content": "Find the area of a triangle if it is known that its medians \\(CM\\) and \\(BN\\) are 6 and 4.5 respectively, and \\(\\angle BKM = 45^\\circ\\), where \\(K\\) is the point of intersection of the medians. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9\\sqrt{2}",
"style": "rule"
} | {
"index": "33274",
"split": "train"
} |
math | [
{
"content": "If 15 bahs are equal to 24 rahs, and 9 rahs are equal in value to 15 yahs, how many bahs are equal in value to 1000 yahs? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "375",
"style": "rule"
} | {
"index": "33275",
"split": "train"
} |
math | [
{
"content": "In what ratio does the line $TH$ divide the side $BC$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1:1",
"style": "rule"
} | {
"index": "33276",
"split": "train"
} |
math | [
{
"content": "What fraction of the volume of a parallelepiped is the volume of a tetrahedron whose vertices are the centroids of the tetrahedra cut off by the planes of a tetrahedron inscribed in the parallelepiped? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/24",
"style": "rule"
} | {
"index": "33277",
"split": "train"
} |
math | [
{
"content": "Find the area of triangle \\(ABC\\), if \\(AC = 3\\), \\(BC = 4\\), and the medians \\(AK\\) and \\(BL\\) are mutually perpendicular. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{11}",
"style": "rule"
} | {
"index": "33278",
"split": "train"
} |
math | [
{
"content": "Eight points are spaced around at intervals of one unit around a $2 \\times 2$ square, as shown. Two of the $8$ points are chosen at random. What is the probability that the two points are one unit apart?\n[asy]\nsize((50));\ndot((5,0));\ndot((5,5));\ndot((0,5));\ndot((-5,5));\ndot((-5,0));\ndot((... | {
"ground_truth": "\\frac{2}{7}",
"style": "rule"
} | {
"index": "33279",
"split": "train"
} |
math | [
{
"content": "Given the base of a triangle is $24$ inches, two lines are drawn parallel to the base, with one of the lines dividing the triangle exactly in half by area, and the other line dividing one of the resulting triangles further into equal areas. If the total number of areas the triangle is divided into... | {
"ground_truth": "12\\sqrt{2}",
"style": "rule"
} | {
"index": "33280",
"split": "train"
} |
math | [
{
"content": "Find the slope of the line that is perpendicular to the line $2x + 3y = 6$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "33281",
"split": "train"
} |
math | [
{
"content": "Given $b-a=-6$ and $ab=7$, find the value of $a^2b-ab^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-42",
"style": "rule"
} | {
"index": "33282",
"split": "train"
} |
math | [
{
"content": "Let $A B C$ be a triangle with $A B=A C=5$ and $B C=6$. Denote by $\\omega$ the circumcircle of $A B C$. We draw a circle $\\Omega$ which is externally tangent to $\\omega$ as well as to the lines $A B$ and $A C$ (such a circle is called an $A$-mixtilinear excircle). Find the radius of $\\Omega$. ... | {
"ground_truth": "\\frac{75}{8}",
"style": "rule"
} | {
"index": "33283",
"split": "train"
} |
math | [
{
"content": "For some positive integer $n$, the number $120n^3$ has $120$ positive integer divisors, including $1$ and the number $120n^3$. How many positive integer divisors does the number $64n^4$ have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "375",
"style": "rule"
} | {
"index": "33284",
"split": "train"
} |
math | [
{
"content": "The café \"Buratino\" operates 6 days a week with Mondays off. Kolya said that from April 1 to April 20, the café was open for 17 days, and from April 10 to April 30, it was open for 18 days. It is known that he made a mistake once. What was the date of the last Tuesday in April? Please output the... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "33285",
"split": "train"
} |
math | [
{
"content": "Given that the lines $ax+2y+6=0$ and $x+\\left(a-1\\right)y+a^{2}-1=0$ are parallel to each other, find the value(s) of the real number $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "33286",
"split": "train"
} |
math | [
{
"content": "In the game of *Winners Make Zeros*, a pair of positive integers $(m,n)$ is written on a sheet of paper. Then the game begins, as the players make the following legal moves:\n\n\n- If $m\\geq n$ , the player choose a positive integer $c$ such that $m-cn\\geq 0$ , and replaces $(m,n)$ with... | {
"ground_truth": "999",
"style": "rule"
} | {
"index": "33287",
"split": "train"
} |
math | [
{
"content": "Define $\\#N$ by the formula $\\#N = .5(N) + 1$. Calculate $\\#(\\#(\\#50))$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "33288",
"split": "train"
} |
math | [
{
"content": "Find $a_{2012}$ if $a_{n} \\equiv a_{n-1}+n(\\bmod 2012)$ and $a_{1}=1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1006",
"style": "rule"
} | {
"index": "33289",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system, the line $l$ passes through the origin with an inclination angle of $\\frac{\\pi}{4}$; the parametric equations of the curve $C\\_1$ are $\\begin{cases} x = \\frac{\\sqrt{3}}{3} \\cos \\alpha \\\\ y = \\sin \\alpha \\end{cases}$ (where $\\alpha$ is the paramet... | {
"ground_truth": "\\frac{9\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "33290",
"split": "train"
} |
math | [
{
"content": "In a regular hexagon $ABCDEF$, points $P$, $Q$, $R$, and $S$ are chosen on sides $\\overline{AB}$, $\\overline{CD}$, $\\overline{DE}$, and $\\overline{FA}$ respectively, so that lines $PC$ and $RA$, as well as $QS$ and $EB$, are parallel. Moreover, the distances between these parallel lines are eq... | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "33291",
"split": "train"
} |
math | [
{
"content": "The cost of 1 piece of gum is 2 cents. What is the cost of 500 pieces of gum, in cents and in dollars? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10.00",
"style": "rule"
} | {
"index": "33292",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$, $ADE$, and $EFG$ are all equilateral. Points $D$ and $G$ are midpoints of $\\overline{AC}$ and $\\overline{AE}$, respectively. If $AB=4$, what is the perimeter of figure $ABCDEFG$? [asy]\n/* AMC8 2000 #15 Problem */\ndraw((0,0)--(4,0)--(5,2)--(5.5,1)--(4.5,1));\ndraw((0,0)--(2,4)--... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "33293",
"split": "train"
} |
math | [
{
"content": "Determine which of the following expressions has the largest value: $4^2$, $4 \\times 2$, $4 - 2$, $\\frac{4}{2}$, or $4 + 2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "33294",
"split": "train"
} |
math | [
{
"content": "Find the units digit of the following within the indicated number base: $52_7 + 62_7$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "33295",
"split": "train"
} |
math | [
{
"content": "Suppose that $(u_n)$ is a sequence of real numbers satisfying\n\\[u_{n+2}=2u_{n+1}+u_n\\]and that $u_3=9$ and $u_6=128$. What is $u_5$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "53",
"style": "rule"
} | {
"index": "33296",
"split": "train"
} |
math | [
{
"content": "How many positive integer divisors of $2004^{2004}$ are divisible by exactly 2004 positive integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "33297",
"split": "train"
} |
math | [
{
"content": "A parallelogram $ABCD$ is inscribed in an ellipse $\\frac{x^2}{4}+\\frac{y^2}{2}=1$. The slope of line $AB$ is $k_1=1$. Calculate the slope of line $AD$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "33298",
"split": "train"
} |
math | [
{
"content": "In a chess tournament, each pair of players plays exactly one game. The winner of each game receives 2 points, the loser receives 0 points, and in case of a draw, both players receive 1 point each. Four scorers have recorded the total score of the tournament, but due to negligence, the scores reco... | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "33299",
"split": "train"
} |
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