data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "How many graphs are there on 10 vertices labeled \\(1,2, \\ldots, 10\\) such that there are exactly 23 edges and no triangles? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42840",
"style": "rule"
} | {
"index": "36100",
"split": "train"
} |
math | [
{
"content": "Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is either a 10 or a Jack? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{17}{442}",
"style": "rule"
} | {
"index": "36101",
"split": "train"
} |
math | [
{
"content": "Positive integers $a, b$, and $c$ have the property that $a^{b}, b^{c}$, and $c^{a}$ end in 4, 2, and 9, respectively. Compute the minimum possible value of $a+b+c$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "36102",
"split": "train"
} |
math | [
{
"content": "What is $3^{-1} + 3^{-2} \\pmod{25}$? Express your answer as an integer from $0$ to $24$, inclusive. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "36103",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\tan(\\omega x+\\phi)$ $(\\omega>0, 0<|\\phi|<\\frac{\\pi}{2})$, where two adjacent branches of the graph intersect the coordinate axes at points $A(\\frac{\\pi}{6},0)$ and $B(\\frac{2\\pi}{3},0)$. Find the sum of all solutions of the equation $f(x)=\\sin(2x-\\frac{\\pi}{... | {
"ground_truth": "\\frac{5\\pi}{6}",
"style": "rule"
} | {
"index": "36104",
"split": "train"
} |
math | [
{
"content": "A point is chosen at random within a rectangle in the coordinate plane whose vertices are $(0, 0), (3030, 0), (3030, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\\tfrac{3}{4}$. Find the value of $d$ to the nearest tenth. Please output the fina... | {
"ground_truth": "0.5",
"style": "rule"
} | {
"index": "36105",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer that is both an integer power of 13 and is not a palindrome. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "169",
"style": "rule"
} | {
"index": "36106",
"split": "train"
} |
math | [
{
"content": "Joe will randomly select two letters from the word CAMP, four letters from the word HERBS, and three letters from the word GLOW. What is the probability that he will have all of the letters from the word PROBLEM? Express your answer as a common fraction. Please output the final answer within \\b... | {
"ground_truth": "\\frac{1}{30}",
"style": "rule"
} | {
"index": "36107",
"split": "train"
} |
math | [
{
"content": "If a positive integer is equal to the sum of all its factors (including 1 but excluding the number itself), then this number is called a \"perfect number\". For example, 28 is a \"perfect number\" because $1 + 2 + 4 + 7 + 14 = 28$. If the sum of all factors of a positive integer (including 1 but e... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "36108",
"split": "train"
} |
math | [
{
"content": "An artist arranges 1000 dots evenly around a circle, with each dot being either red or blue. A critic counts faults: each pair of adjacent red dots counts as one fault, and each pair of blue dots exactly two apart (separated by one dot) counts as another fault. What is the smallest number of fault... | {
"ground_truth": "250",
"style": "rule"
} | {
"index": "36109",
"split": "train"
} |
math | [
{
"content": "In the diagram, square PQRS has side length 40. Points J, K, L, and M are on the sides of PQRS, so that JQ = KR = LS = MP = 10. Line segments JZ, KW, LX, and MY are drawn parallel to the diagonals of the square so that W is on JZ, X is on KW, Y is on LX, and Z is on MY. Find the area of quadrilate... | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "36110",
"split": "train"
} |
math | [
{
"content": "Given that the largest four-digit number $M$ has digits with a product of $72$, find the sum of its digits. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "36111",
"split": "train"
} |
math | [
{
"content": "As $n$ ranges over the positive integers, what is the sum of all possible values of the greatest common divisor of $3n+4$ and $n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "36112",
"split": "train"
} |
math | [
{
"content": "If the height on the base of an isosceles triangle is $18 \\mathrm{~cm}$ and the median on the leg is $15 \\mathrm{~cm}$, what is the area of this isosceles triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "36113",
"split": "train"
} |
math | [
{
"content": "The real numbers $x, y$ and $z$ satisfy the three equations $x+y=7$, $xz=-180$, and $(x+y+z)^{2}=4$. If $S$ is the sum of the two possible values of $y$, what is $-S$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "36114",
"split": "train"
} |
math | [
{
"content": "A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The yard's remaining area forms a trapezoidal shape, as shown. The lengths of the parallel sides of the trapezoid are $20$ and $30$ meters, respectively. What fraction of the yard is occupied by the fl... | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"index": "36115",
"split": "train"
} |
math | [
{
"content": "For any 2016 complex numbers \\( z_1, z_2, \\ldots, z_{2016} \\), it holds that\n\\[ \n\\sum_{k=1}^{2016} |z_k|^2 \\geq \\lambda \\min_{1 \\leq k \\leq 2016} \\{ |z_{k+1} - z_k|^2 \\}, \n\\]\nwhere \\( z_{2017} = z_1 \\). Find the maximum value of \\( \\lambda \\). Please output the final answer w... | {
"ground_truth": "504",
"style": "rule"
} | {
"index": "36116",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, $A(7,8)$, $B(10,4)$, $C(2,-4)$, then $S_{\\triangle ABC}$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "36117",
"split": "train"
} |
math | [
{
"content": "There exists a polynomial $P$ of degree 5 with the following property: if $z$ is a complex number such that $z^{5}+2004 z=1$, then $P(z^{2})=0$. Calculate the quotient $P(1) / P(-1)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2010012 / 2010013",
"style": "rule"
} | {
"index": "36118",
"split": "train"
} |
math | [
{
"content": "Find the sum of all roots of the equation:\n\n$$\n\\begin{gathered}\n\\sqrt{2 x^{2}-2024 x+1023131} + \\sqrt{3 x^{2}-2025 x+1023132} + \\sqrt{4 x^{2}-2026 x+1023133} = \\\\\n= \\sqrt{x^{2}-x+1} + \\sqrt{2 x^{2}-2 x+2} + \\sqrt{3 x^{2}-3 x+3}\n\\end{gathered}\n$$ Please output the final answer with... | {
"ground_truth": "2023",
"style": "rule"
} | {
"index": "36119",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, given the circle $O: x^2 + y^2 = 1$ and the circle $C: (x-4)^2 + y^2 = 4$, a moving point $P$ is located between two points $E$ and $F$ on the line $x + \\sqrt{3}y - 2 = 0$. Tangents to circles $O$ and $C$ are drawn from point $P$, with the points of tange... | {
"ground_truth": "\\frac{2 \\sqrt{39}}{3}",
"style": "rule"
} | {
"index": "36120",
"split": "train"
} |
math | [
{
"content": "Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The probability that for every positive integer $k<5$, no collection of $k$ pairs made by the child contains ... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "36121",
"split": "train"
} |
math | [
{
"content": "Compute $139+27+23+11$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "36122",
"split": "train"
} |
math | [
{
"content": "Given the sequence $103, 1003, 10003, 100003, \\dots$, find how many of the first 2048 numbers in this sequence are divisible by 101. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "36123",
"split": "train"
} |
math | [
{
"content": "Rectangle $ABCD$ is inscribed in a semicircle with diameter $\\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9.$ What is the area of $ABCD?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "36124",
"split": "train"
} |
math | [
{
"content": "Lauren plays basketball with her friends. She makes 10 baskets. Each of these baskets is worth either 2 or 3 points. Lauren scores a total of 26 points. How many 3 point baskets did she make? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "36125",
"split": "train"
} |
math | [
{
"content": "Given an ellipse $C:\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with foci $F_{1}$ and $F_{2}$, point $A$ lies on $C$, point $B$ lies on the $y$-axis, and satisfies $\\overrightarrow{A{F}_{1}}⊥\\overrightarrow{B{F}_{1}}$, $\\overrightarrow{A{F}_{2}}=\\frac{2}{3}\\overrightarrow{{F}_... | {
"ground_truth": "\\frac{\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "36126",
"split": "train"
} |
math | [
{
"content": "The graph of the equation $9x+223y=2007$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant? Please output the final answer within \\boxed{... | {
"ground_truth": "888",
"style": "rule"
} | {
"index": "36127",
"split": "train"
} |
math | [
{
"content": "A bar graph shows the number of hamburgers sold by a fast food chain each season. However, the bar indicating the number sold during the winter is covered by a smudge. If exactly $25\\%$ of the chain's hamburgers are sold in the fall, how many million hamburgers are sold in the winter? Please ou... | {
"ground_truth": "2.5",
"style": "rule"
} | {
"index": "36128",
"split": "train"
} |
math | [
{
"content": "How many sequences of 8 digits $x_1, x_2, \\ldots, x_8$ can we form, given the condition that no two adjacent $x_i$ have the same parity and the sequence starts with an odd digit? Leading zeroes are allowed. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "390625",
"style": "rule"
} | {
"index": "36129",
"split": "train"
} |
math | [
{
"content": "Von Neumann's Poker: The first step in Von Neumann's game is selecting a random number on $[0,1]$. To generate this number, Chebby uses the factorial base: the number $0 . A_{1} A_{2} A_{3} A_{4} \\ldots$ stands for $\\sum_{n=0}^{\\infty} \\frac{A_{n}}{(n+1)!}$, where each $A_{n}$ is an integer be... | {
"ground_truth": "0.57196",
"style": "rule"
} | {
"index": "36130",
"split": "train"
} |
math | [
{
"content": "The figure shows three squares with non-overlapping interiors. The area of the shaded square is 1 square inch. What is the area of rectangle $ABCD$, in square inches?\n\n[asy]size(100);\npair A = (0,0), D = (3,0),C = (3,2),B = (0,2);\ndraw(A--B--C--D--cycle);\ndraw(A--(1,0)--(1,1)--(0,1)--cycle);\... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "36131",
"split": "train"
} |
math | [
{
"content": "Find the residue of $182\\cdot 12 - 15\\cdot 7 + 3\\pmod{14}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "36132",
"split": "train"
} |
math | [
{
"content": "Arrange the positive odd numbers as shown in the pattern below. What is the 5th number from the left in the 21st row?\n\n$$\n1 \\\\\n3 \\quad 5 \\quad 7 \\\\\n9 \\quad 11 \\quad 13 \\quad 15 \\quad 17 \\\\\n19 \\quad 21 \\quad 23 \\quad 25 \\quad 27 \\quad 29 \\quad 31 \\\\\n\\ldots \\quad \\quad ... | {
"ground_truth": "809",
"style": "rule"
} | {
"index": "36133",
"split": "train"
} |
math | [
{
"content": "Parallelogram $ABCD$ has vertices $A(3,4)$, $B(-2,1)$, $C(-5,-2)$, and $D(0,1)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point is left of the $y$-axis? Express your answer as a common fraction. Please output the final answe... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "36134",
"split": "train"
} |
math | [
{
"content": "A convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. For the \\( V \\) vertices, each vertex has \\( T \\) triangular faces and \\( P \\) pentagonal faces intersecting. Find the value of \\( P + T + V \\). Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "36135",
"split": "train"
} |
math | [
{
"content": "Nine people sit down at random seats around a round table. Four of them are math majors, three others are physics majors, and the two remaining people are biology majors. What is the probability that all four math majors sit in consecutive seats? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{1}{14}",
"style": "rule"
} | {
"index": "36136",
"split": "train"
} |
math | [
{
"content": "Find the polynomial $p(x),$ with real coefficients, such that\n\\[p(x^3) - p(x^3 - 2) = [p(x)]^2 + 12\\]for all real numbers $x.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6x^3 - 6",
"style": "rule"
} | {
"index": "36137",
"split": "train"
} |
math | [
{
"content": "Given that $a$ is a positive real number and $b$ is an integer between $1$ and $201$, inclusive, find the number of ordered pairs $(a,b)$ such that $(\\log_b a)^{2023}=\\log_b(a^{2023})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "603",
"style": "rule"
} | {
"index": "36138",
"split": "train"
} |
math | [
{
"content": "Given $\\vec{a}=(\\cos \\alpha, \\sin \\alpha), \\vec{b}=(\\cos \\beta, \\sin \\beta)$, and $|\\vec{a}-\\vec{b}|=\\frac{2 \\sqrt{5}}{5}$. If $0 < \\alpha < \\frac{\\pi}{2}$, $-\\frac{\\pi}{2} < \\beta < 0$, and $\\sin \\beta=-\\frac{5}{13}$, then $\\sin \\alpha=$ Please output the final answer wit... | {
"ground_truth": "$\\frac{33}{65}$",
"style": "rule"
} | {
"index": "36139",
"split": "train"
} |
math | [
{
"content": "Given Professor Lee has ten different mathematics books on a shelf, consisting of three calculus books, four algebra books, and three statistics books, determine the number of ways to arrange the ten books on the shelf keeping all calculus books together and all statistics books together. Please o... | {
"ground_truth": "25920",
"style": "rule"
} | {
"index": "36140",
"split": "train"
} |
math | [
{
"content": "In a plane, a right angle is given. A circle with a center located outside of this angle is tangent to the bisector of the right angle. It intersects one side of the right angle at points \\(A\\) and \\(B\\) and the extension of the other side at points \\(C\\) and \\(D\\). Given that \\(AB = \\sq... | {
"ground_truth": "1.5",
"style": "rule"
} | {
"index": "36141",
"split": "train"
} |
math | [
{
"content": "Given $$\\overrightarrow {m} = (\\sin x, \\frac {3}{4})$$, $$\\overrightarrow {n} = (\\cos x, -1)$$, in triangle ABC, $\\sin A + \\cos A = \\sqrt {2}$. \n(1) When $$\\overrightarrow {m} \\parallel \\overrightarrow {n}$$, find the value of $\\sin^2 x + \\sin 2x$; \n(2) Let the function $f(x) = 2(... | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "36142",
"split": "train"
} |
math | [
{
"content": "Consider this histogram of the scores for $81$ students taking a test:\n\nThe median is in the interval labeled Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "36143",
"split": "train"
} |
math | [
{
"content": "Given that $m$ is a positive integer, and given that $\\mathop{\\text{lcm}}[40,m]=120$ and $\\mathop{\\text{lcm}}[m,45]=180$, what is $m$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "36144",
"split": "train"
} |
math | [
{
"content": "Solve the equation $x^2 + 14x = 72$. The positive solution has the form $\\sqrt{c} - d$ for positive natural numbers $c$ and $d$. What is $c + d$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "128",
"style": "rule"
} | {
"index": "36145",
"split": "train"
} |
math | [
{
"content": "If $x$ and $y$ are positive integers with $xy = 6$, what is the sum of all possible values of $\\frac{2^{x+y}}{2^{x-y}}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4180",
"style": "rule"
} | {
"index": "36146",
"split": "train"
} |
math | [
{
"content": "In a jar of red, green, and blue marbles, all but 6 are red marbles, all but 8 are green, and all but 4 are blue. How many marbles are in the jar? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "36147",
"split": "train"
} |
math | [
{
"content": "A rectangle has an area of $A$. The midpoints of each side of the rectangle are connected to form a new, smaller rectangle inside the original. What is the ratio of the area of the smaller rectangle to the area of the original rectangle? Express your answer as a common fraction. Please output the ... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "36148",
"split": "train"
} |
math | [
{
"content": "Given the plane vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$, where $|\\overrightarrow{a}| = 1$, $|\\overrightarrow{b}| = \\sqrt{2}$, and $\\overrightarrow{a} \\cdot \\overrightarrow{b} = 1$, calculate the angle between vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Please out... | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "36149",
"split": "train"
} |
math | [
{
"content": "A hotel consists of a $2 \\times 8$ square grid of rooms, each occupied by one guest. All the guests are uncomfortable, so each guest would like to move to one of the adjoining rooms (horizontally or vertically). Of course, they should do this simultaneously, in such a way that each room will agai... | {
"ground_truth": "1156",
"style": "rule"
} | {
"index": "36150",
"split": "train"
} |
math | [
{
"content": "At Euclid High School, the number of students taking the AMC 10 was $60$ in 2002, $66$ in 2003, $70$ in 2004, $76$ in 2005, $78$ in 2006, and is $85$ in 2007. Between what two consecutive years was there the largest percentage increase? Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "2002 and 2003",
"style": "rule"
} | {
"index": "36151",
"split": "train"
} |
math | [
{
"content": "The coefficient of $\\frac{1}{x}$ in the expansion of $(1-x^2)^4\\left(\\frac{x+1}{x}\\right)^5$ needs to be determined. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-29",
"style": "rule"
} | {
"index": "36152",
"split": "train"
} |
math | [
{
"content": "In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is , $X$ in centimeters? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "36153",
"split": "train"
} |
math | [
{
"content": "What is the greatest product obtainable from two integers whose sum is 1998? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "998001",
"style": "rule"
} | {
"index": "36154",
"split": "train"
} |
math | [
{
"content": "After reading the following solution, answer the question: Xiaofang found in the simplification of $\\sqrt{7+4\\sqrt{3}}$ that first, $\\sqrt{7+4\\sqrt{3}}$ can be simplified to $\\sqrt{7+2\\sqrt{12}}$. Since $4+3=7$ and $4\\times 3=12$, that is, ${(\\sqrt{4})^2}+{(\\sqrt{3})^2}=7$, $\\sqrt{4}×\\s... | {
"ground_truth": "\\frac{\\sqrt{10}}{2}-\\frac{\\sqrt{6}}{2}",
"style": "rule"
} | {
"index": "36155",
"split": "train"
} |
math | [
{
"content": "A graph $G(V,E)$ is triangle-free, but adding any edges to the graph will form a triangle. It's given that $|V|=2019$, $|E|>2018$, find the minimum of $|E|$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4033",
"style": "rule"
} | {
"index": "36156",
"split": "train"
} |
math | [
{
"content": "What is the number of terms with rational coefficients among the $1001$ terms in the expansion of $\\left(x\\sqrt[3]{2}+y\\sqrt{3}\\right)^{1000}?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "167",
"style": "rule"
} | {
"index": "36157",
"split": "train"
} |
math | [
{
"content": "The area of a circle whose circumference is $24\\pi$ is $k\\pi$. What is the value of $k$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "36158",
"split": "train"
} |
math | [
{
"content": "Professor Lee has eleven different language books lined up on a bookshelf: three Arabic, four German, and four Spanish. Calculate the number of ways to arrange the eleven books on the shelf while keeping the Arabic books together. Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "2,177,280",
"style": "rule"
} | {
"index": "36159",
"split": "train"
} |
math | [
{
"content": "Seven frogs are sitting in a row. They come in four colors: two green, two red, two yellow, and one blue. Green frogs refuse to sit next to red frogs, and yellow frogs refuse to sit next to blue frogs. In how many ways can the frogs be positioned respecting these restrictions? Please output the fi... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "36160",
"split": "train"
} |
math | [
{
"content": "What is the largest number of digits that can be erased from the 1000-digit number 201820182018....2018 so that the sum of the remaining digits is 2018? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "741",
"style": "rule"
} | {
"index": "36161",
"split": "train"
} |
math | [
{
"content": "Xiaopang and Xiaoya both have their birthdays in May, and both fall on a Wednesday. Xiaopang's birthday is later, and the sum of their birth dates is 38. What is Xiaopang's birthday in May? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "36162",
"split": "train"
} |
math | [
{
"content": "Given the ratio of the legs of a right triangle is $3: 4$, determine the ratio of the corresponding segments of the hypotenuse created by dropping a perpendicular from the opposite vertex of the right angle onto the hypotenuse. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac{16}{9}",
"style": "rule"
} | {
"index": "36163",
"split": "train"
} |
math | [
{
"content": "We defined an operation denoted by $*$ on the integers, which satisfies the following conditions: \n\n1) $x * 0 = x$ for every integer $x$;\n2) $0 * y = -y$ for every integer $y$;\n3) $((x+1) * y) + (x * (y+1)) = 3(x * y) - x y + 2 y$ for every integer $x$ and $y$.\n\nDetermine the result of the o... | {
"ground_truth": "1639",
"style": "rule"
} | {
"index": "36164",
"split": "train"
} |
math | [
{
"content": "The mean of one set of four numbers is 15, and the mean of a separate set of eight numbers is 20. What is the mean of the set of all twelve numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{55}{3}",
"style": "rule"
} | {
"index": "36165",
"split": "train"
} |
math | [
{
"content": "For what value of $k$ does the line represented by the equation $-\\frac{1}{2}-2kx = 5y$ contain the point $\\left(\\frac{1}{4},-6\\right)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "59",
"style": "rule"
} | {
"index": "36166",
"split": "train"
} |
math | [
{
"content": "Given $sinα-cosα=\\frac{1}{5},0≤α≤π$, calculate $sin(2α-\\frac{π}{4})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{31\\sqrt{2}}{50}",
"style": "rule"
} | {
"index": "36167",
"split": "train"
} |
math | [
{
"content": "Rationalize the denominator of $\\frac{2}{3\\sqrt{5} + 2\\sqrt{11}}$ and write your answer in the form $\\displaystyle \\frac{A\\sqrt{B} + C\\sqrt{D}}{E}$, where $B < D$, the fraction is in lowest terms and all radicals are in simplest radical form. What is $A+B+C+D+E$? Please output the final ans... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "36168",
"split": "train"
} |
math | [
{
"content": "A target consisting of five zones is hanging on the wall: a central circle (bullseye) and four colored rings. The width of each ring equals the radius of the bullseye. It is known that the number of points for hitting each zone is inversely proportional to the probability of hitting that zone, and... | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "36169",
"split": "train"
} |
math | [
{
"content": "A corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, then what fraction of the tiled floor is made of darker tiles? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{9}",
"style": "rule"
} | {
"index": "36170",
"split": "train"
} |
math | [
{
"content": "The quadratic $x^2-20x+36$ can be written in the form $(x+b)^2+c$, where $b$ and $c$ are constants. What is $b+c$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-74",
"style": "rule"
} | {
"index": "36171",
"split": "train"
} |
math | [
{
"content": "A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer two layers of unit cubes are removed from the block, more than half the original unit cubes will still remain? Please out... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "36172",
"split": "train"
} |
math | [
{
"content": "Given the line $l: \\sqrt{3}x-y-4=0$, calculate the slope angle of line $l$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": "36173",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be a cyclic quadrilateral where the side lengths are distinct primes less than $20$ with $AB\\cdot BC + CD\\cdot DA = 280$. What is the largest possible value of $BD$?\nA) 17 \nB) $\\sqrt{310}$ \nC) $\\sqrt{302.76}$ \nD) $\\sqrt{285}$ Please output the final answer within \\boxed{}."... | {
"ground_truth": "\\sqrt{302.76}",
"style": "rule"
} | {
"index": "36174",
"split": "train"
} |
math | [
{
"content": "For what real value of $v$ is $\\frac{-21-\\sqrt{301}}{10}$ a root of $5x^2+21x+v$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "36175",
"split": "train"
} |
math | [
{
"content": "There are 8 lily pads in a pond numbered $1,2, \\ldots, 8$. A frog starts on lily pad 1. During the $i$-th second, the frog jumps from lily pad $i$ to $i+1$, falling into the water with probability $\\frac{1}{i+1}$. The probability that the frog lands safely on lily pad 8 without having fallen int... | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "36176",
"split": "train"
} |
math | [
{
"content": "A watch loses $2\\frac{1}{2}$ minutes per day. It is set right at $1$ P.M. on March 15. Let $n$ be the positive correction, in minutes, to be added to the time shown by the watch at a given time. When the watch shows $9$ A.M. on March 21, $n$ equals: Please output the final answer within \\boxed{}... | {
"ground_truth": "14\\frac{14}{23}",
"style": "rule"
} | {
"index": "36177",
"split": "train"
} |
math | [
{
"content": "A certain school organized a Chinese traditional culture activity week to promote traditional Chinese culture. During the activity period, a Chinese traditional culture knowledge competition was held, with classes participating in the competition. Each class selected 5 representatives through a Ch... | {
"ground_truth": "\\frac{537}{125}",
"style": "rule"
} | {
"index": "36178",
"split": "train"
} |
math | [
{
"content": "Ann and Barbara were comparing their ages and found that Barbara is as old as Ann was when Barbara was as old as Ann had been when Barbara was half as old as Ann is. If the sum of their present ages is $44$ years, then Ann's age is Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "36179",
"split": "train"
} |
math | [
{
"content": "What is the smallest number that could be the date of the first Saturday after the second Monday following the second Thursday of a month? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "36180",
"split": "train"
} |
math | [
{
"content": "An ancient human tribe had a hierarchical system where there existed one chief with 2 supporting chiefs (supporting chief A and supporting chief B), each of whom had 2 equal, inferior officers. If the tribe at one point had 10 members, what is the number of different ways to choose the leadership ... | {
"ground_truth": "151200",
"style": "rule"
} | {
"index": "36181",
"split": "train"
} |
math | [
{
"content": "Diagonals \\( AC \\) and \\( BD \\) of the cyclic quadrilateral \\( ABCD \\) are perpendicular and intersect at point \\( M \\). It is known that \\( AM = 3 \\), \\( BM = 4 \\), and \\( CM = 6 \\). Find \\( CD \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10.5",
"style": "rule"
} | {
"index": "36182",
"split": "train"
} |
math | [
{
"content": "If $∀x∈(0,+\\infty)$, $ln2x-\\frac{ae^{x}}{2}≤lna$, then find the minimum value of $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{e}",
"style": "rule"
} | {
"index": "36183",
"split": "train"
} |
math | [
{
"content": "Evaluate the expression $\\frac {1}{3 - \\frac {1}{3 - \\frac {1}{3 - \\frac{1}{4}}}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{11}{29}",
"style": "rule"
} | {
"index": "36184",
"split": "train"
} |
math | [
{
"content": "Given a rectangular storage with length 20 feet, width 15 feet, and height 10 feet, and with the floor and each of the four walls being two feet thick, calculate the total number of one-foot cubical blocks needed for the construction. Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "1592",
"style": "rule"
} | {
"index": "36185",
"split": "train"
} |
math | [
{
"content": "Find all positive integers that can be represented in decimal form as $\\overline{13 x y 45 z}$ and are divisible by 792, where $x, y, z$ are unknown digits. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1380456",
"style": "rule"
} | {
"index": "36186",
"split": "train"
} |
math | [
{
"content": "There is a question: \"If the value of the algebraic expression $5a+3b$ is $-4$, then what is the value of the algebraic expression $2\\left(a+b\\right)+4\\left(2a+b\\right)$?\" Tang, who loves to use his brain, solved the problem as follows: \nOriginal expression $=2a+2b+8a+4b=10a+6b=2\\left(5a+... | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "36187",
"split": "train"
} |
math | [
{
"content": "For lines $l_{1}$: $(3+a)x+4y=5-3a$ and $l_{2}$: $2x+(5+a)y=8$ to be parallel, $a=$ ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-7",
"style": "rule"
} | {
"index": "36188",
"split": "train"
} |
math | [
{
"content": "Every Sunday, a married couple has breakfast with their mothers. Unfortunately, the relationships each spouse has with the other's mother are quite strained: both know that there is a two-thirds chance of getting into an argument with the mother-in-law. In the event of a conflict, the other spouse... | {
"ground_truth": "4/9",
"style": "rule"
} | {
"index": "36189",
"split": "train"
} |
math | [
{
"content": "If $\\frac{4^x}{2^{x+y}}=8$ and $\\frac{9^{x+y}}{3^{5y}}=243$, $x$ and $y$ real numbers, then $xy$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "36190",
"split": "train"
} |
math | [
{
"content": "Convert the following radians to degrees: convert degrees to radians:\n(1) $\\frac{\\pi}{12} =$ \\_\\_\\_\\_\\_\\_ ; (2) $\\frac{13\\pi}{6} =$ \\_\\_\\_\\_\\_\\_ ; (3) $- \\frac{5}{12}\\pi =$ \\_\\_\\_\\_\\_\\_ .\n(4) $36^{\\circ} =$ \\_\\_\\_\\_\\_\\_ $rad$ ; (5) $-105^{\\circ} =$ \\_\\_\\_\\_\\_... | {
"ground_truth": "-\\frac{7\\pi}{12}",
"style": "rule"
} | {
"index": "36191",
"split": "train"
} |
math | [
{
"content": "A sequence $s_{0}, s_{1}, s_{2}, s_{3}, \\ldots$ is defined by $s_{0}=s_{1}=1$ and, for every positive integer $n, s_{2 n}=s_{n}, s_{4 n+1}=s_{2 n+1}, s_{4 n-1}=s_{2 n-1}+s_{2 n-1}^{2} / s_{n-1}$. What is the value of $s_{1000}$? Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "720",
"style": "rule"
} | {
"index": "36192",
"split": "train"
} |
math | [
{
"content": "Given fixed points \\( A(3,0) \\), \\( B(0,4) \\), and point \\( P \\) on the incircle of triangle \\( \\triangle AOB \\) (where \\( O \\) is the origin), find the maximum value of \\( |PA|^2 + |PB|^2 + |PO|^2 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "36193",
"split": "train"
} |
math | [
{
"content": "In coordinate space, $A = (1,2,3),$ $B = (5,3,1),$ and $C = (3,4,5).$ Find the orthocenter of triangle $ABC.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left( \\frac{5}{2}, 3, \\frac{7}{2} \\right)",
"style": "rule"
} | {
"index": "36194",
"split": "train"
} |
math | [
{
"content": "**p22.** Consider the series $\\{A_n\\}^{\\infty}_{n=0}$ , where $A_0 = 1$ and for every $n > 0$ , $$ A_n = A_{\\left[ \\frac{n}{2023}\\right]} + A_{\\left[ \\frac{n}{2023^2}\\right]}+A_{\\left[ \\frac{n}{2023^3}\\right]}, $$ where $[x]$ denotes the largest integer value smaller than or... | {
"ground_truth": "653",
"style": "rule"
} | {
"index": "36195",
"split": "train"
} |
math | [
{
"content": "A $7\\times 1$ board is completely covered by $m\\times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7\\times 1$ board in which all thre... | {
"ground_truth": "106",
"style": "rule"
} | {
"index": "36196",
"split": "train"
} |
math | [
{
"content": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl} x^2-4 & \\text{ if }x < 7, \\\\\nx-13 & \\text{ if } x \\geq 7.\n\\end{array}\n\\right.\\] What is $f(f(f(17)))$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "36197",
"split": "train"
} |
math | [
{
"content": "Find all functions $f: (0, \\infty) \\to (0, \\infty)$ such that\n\\begin{align*}\nf(y(f(x))^3 + x) = x^3f(y) + f(x) \n\\end{align*}\nfor all $x, y>0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "f(x) = x",
"style": "rule"
} | {
"index": "36198",
"split": "train"
} |
math | [
{
"content": "Let \\( S_{n} = 1 + \\frac{1}{2} + \\cdots + \\frac{1}{n} \\) for \\( n = 1, 2, \\cdots \\). Find the smallest positive integer \\( n \\) such that \\( S_{n} > 10 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12367",
"style": "rule"
} | {
"index": "36199",
"split": "train"
} |
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