data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "How many solutions does the equation\n\n$$\n\\{x\\}^{2}=\\left\\{x^{2}\\right\\}\n$$\n\nhave in the interval $[1, 100]$? ($\\{u\\}$ denotes the fractional part of $u$, which is the difference between $u$ and the largest integer not greater than $u$.) Please output the final answer within \\boxed{}... | {
"ground_truth": "9901",
"style": "rule"
} | {
"index": "800",
"split": "train"
} |
math | [
{
"content": "Find $(100110_2 + 1001_2) \\times 110_2 \\div 11_2$. Express your answer in base 2. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1011110_2",
"style": "rule"
} | {
"index": "801",
"split": "train"
} |
math | [
{
"content": "Let the function \\( f(x) \\) satisfy the following conditions:\n(i) If \\( x > y \\), and \\( f(x) + x \\geq w \\geq f(y) + y \\), then there exists a real number \\( z \\in [y, x] \\), such that \\( f(z) = w - z \\);\n(ii) The equation \\( f(x) = 0 \\) has at least one solution, and among the so... | {
"ground_truth": "2004",
"style": "rule"
} | {
"index": "802",
"split": "train"
} |
math | [
{
"content": "Given that a floor is tiled in a similar pattern with a $4 \\times 4$ unit repeated pattern and each of the four corners looks like the scaled down version of the original, determine the fraction of the tiled floor made up of darker tiles, assuming symmetry and pattern are preserved. Please output... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "803",
"split": "train"
} |
math | [
{
"content": "If $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ are vectors such that $\\mathbf{a} \\cdot \\mathbf{b} = -3,$ $\\mathbf{a} \\cdot \\mathbf{c} = 4,$ and $\\mathbf{b} \\cdot \\mathbf{c} = 6,$ then find\n\\[\\mathbf{b} \\cdot (7 \\mathbf{c} - 2 \\mathbf{a}).\\] Please output the final answer within... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "804",
"split": "train"
} |
math | [
{
"content": "Define\n\\[ A' = \\frac{1}{1^2} + \\frac{1}{7^2} - \\frac{1}{11^2} - \\frac{1}{13^2} + \\frac{1}{19^2} + \\frac{1}{23^2} - \\dotsb, \\]\nwhich omits all terms of the form $\\frac{1}{n^2}$ where $n$ is an odd multiple of 5, and\n\\[ B' = \\frac{1}{5^2} - \\frac{1}{25^2} + \\frac{1}{35^2} - \\frac{1... | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "805",
"split": "train"
} |
math | [
{
"content": "Calculate the limit of the function:\n\n$$\\lim_{x \\rightarrow \\frac{1}{3}} \\frac{\\sqrt[3]{\\frac{x}{9}}-\\frac{1}{3}}{\\sqrt{\\frac{1}{3}+x}-\\sqrt{2x}}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{2 \\sqrt{2}}{3 \\sqrt{3}}",
"style": "rule"
} | {
"index": "806",
"split": "train"
} |
math | [
{
"content": "The set of $x$-values satisfying the equation $x^{\\log_{10} x} = \\frac{x^3}{100}$ consists of: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10 \\text{ or } 100",
"style": "rule"
} | {
"index": "807",
"split": "train"
} |
math | [
{
"content": "What percent of square $EFGH$ is shaded? All angles in the diagram are right angles. [asy]\nimport graph;\ndefaultpen(linewidth(0.7));\n\nxaxis(0,6,Ticks(1.0,NoZero));\nyaxis(0,6,Ticks(1.0,NoZero));\n\nfill((0,0)--(2,0)--(2,2)--(0,2)--cycle);\nfill((3,0)--(4,0)--(4,4)--(0,4)--(0,3)--(3,3)--cycle);... | {
"ground_truth": "61.11\\%",
"style": "rule"
} | {
"index": "808",
"split": "train"
} |
math | [
{
"content": "Given that the vertex of angle $\\alpha$ coincides with the origin $O$, its initial side coincides with the non-negative semi-axis of the $x$-axis, and its terminal side passes through point $P(-\\frac{3}{5}, -\\frac{4}{5})$.\n\n(1) Find the value of $\\sin(\\alpha + \\pi)$;\n\n(2) If angle $\\bet... | {
"ground_truth": "\\frac{16}{65}",
"style": "rule"
} | {
"index": "809",
"split": "train"
} |
math | [
{
"content": "Let $f_0=f_1=1$ and $f_{i+2}=f_{i+1}+f_i$ for all $n\\ge 0$. Find all real solutions to the equation \n\\[x^{2010}=f_{2009}\\cdot x+f_{2008}\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1 + \\sqrt{5}}{2} \\text{ and } \\frac{1 - \\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "810",
"split": "train"
} |
math | [
{
"content": "A point is chosen at random on the number line between 0 and 1, and the point is colored green. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored purple. What is the probability that the number of the purple point is greater than the number of th... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "811",
"split": "train"
} |
math | [
{
"content": "Rectangular prism P Q R S W T U V has a square base P Q R S. Point X is on the face T U V W so that P X = 12, Q X = 10, and R X = 8. Determine the maximum possible area of rectangle P Q U T. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "67.82",
"style": "rule"
} | {
"index": "812",
"split": "train"
} |
math | [
{
"content": "Consider the polynomials $P(x) = x^{6} - x^{5} - x^{3} - x^{2} - x$ and $Q(x) = x^{4} - x^{3} - x^{2} - 1.$ Given that $z_{1},z_{2},z_{3},$ and $z_{4}$ are the roots of $Q(x) = 0,$ find $P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4}).$ Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "813",
"split": "train"
} |
math | [
{
"content": "Given the parametric equation of curve \\\\(C_{1}\\\\) as \\\\(\\begin{cases}x=3\\cos \\alpha \\\\ y=\\sin \\alpha\\end{cases} (\\alpha\\\\) is the parameter\\\\()\\\\), and taking the origin \\\\(O\\\\) of the Cartesian coordinate system \\\\(xOy\\\\) as the pole and the positive half-axis of \\\... | {
"ground_truth": "\\dfrac{6 \\sqrt{3}}{5}",
"style": "rule"
} | {
"index": "814",
"split": "train"
} |
math | [
{
"content": "Robert read a book for 10 days. He read an average of 25 pages per day for the first 5 days and an average of 40 pages per day for the next 4 days, and read 30 more pages on the last day. Calculate the total number of pages in the book. Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "315",
"style": "rule"
} | {
"index": "815",
"split": "train"
} |
math | [
{
"content": "The lines $l_1$ and $l_2$ are two tangents to the circle $x^2+y^2=2$. If the intersection point of $l_1$ and $l_2$ is $(1,3)$, then the tangent of the angle between $l_1$ and $l_2$ equals \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "816",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)$ defined on the interval $[-2011, 2011]$ and satisfying $f(x_1+x_2) = f(x_1) + f(x_2) - 2011$ for any $x_1, x_2 \\in [-2011, 2011]$, and $f(x) > 2011$ when $x > 0$, determine the value of $M+N$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4022",
"style": "rule"
} | {
"index": "817",
"split": "train"
} |
math | [
{
"content": "Find the remainder when $9 \\times 99 \\times 999 \\times \\cdots \\times \\underbrace{99\\cdots9}_{\\text{999 9's}}$ is divided by $1000$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "109",
"style": "rule"
} | {
"index": "818",
"split": "train"
} |
math | [
{
"content": "A digital watch displays time in a 24-hour format showing only hours and minutes. Find the largest possible sum of the digits in the display. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "819",
"split": "train"
} |
math | [
{
"content": "Find all positive integers $n>1$ for which $\\frac{n^{2}+7 n+136}{n-1}$ is the square of a positive integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5, 37",
"style": "rule"
} | {
"index": "820",
"split": "train"
} |
math | [
{
"content": "The product of two 2-digit numbers is $4536$. What is the smaller of the two numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "821",
"split": "train"
} |
math | [
{
"content": "If the purchase price of a product is 8 yuan and it is sold for 10 yuan per piece, 200 pieces can be sold per day. If the selling price of each piece increases by 0.5 yuan, the sales volume will decrease by 10 pieces. What should the selling price be set at to maximize the profit? And calculate th... | {
"ground_truth": "720",
"style": "rule"
} | {
"index": "822",
"split": "train"
} |
math | [
{
"content": "A lame king is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the cells in the same column as the current cell. A lame king is placed in the top-left cell of a $7 \\times 7$ grid. Compute the maximum number of cells it can visit without visi... | {
"ground_truth": "43",
"style": "rule"
} | {
"index": "823",
"split": "train"
} |
math | [
{
"content": "Find all non-negative solutions to the equation $2013^x+2014^y=2015^z$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(0,1,1)",
"style": "rule"
} | {
"index": "824",
"split": "train"
} |
math | [
{
"content": "There is a moving point \\( P \\) on the \\( x \\)-axis. Given the fixed points \\( A(0,2) \\) and \\( B(0,4) \\), what is the maximum value of \\( \\sin \\angle APB \\) when \\( P \\) moves along the entire \\( x \\)-axis? Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "825",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted by $a$, $b$, and $c$ respectively. Given that angle $A = \\frac{\\pi}{4}$, $\\sin A + \\sin (B - C) = 2\\sqrt{2}\\sin 2C$, and the area of triangle $ABC$ is $1$. Find the length of side $BC$. Please output the final answ... | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "826",
"split": "train"
} |
math | [
{
"content": "Let $x = \\frac{\\sum\\limits_{n=1}^{30} \\cos n^\\circ}{\\sum\\limits_{n=1}^{30} \\sin n^\\circ}$. What is the smallest integer that does not fall below $100x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "360",
"style": "rule"
} | {
"index": "827",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and $\\overrightarrow{m}=(\\sqrt{3}b-c,\\cos C)$, $\\overrightarrow{n}=(a,\\cos A)$. Given that $\\overrightarrow{m} \\parallel \\overrightarrow{n}$, determine the value of $\\cos A$. Please ... | {
"ground_truth": "\\dfrac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "828",
"split": "train"
} |
math | [
{
"content": "Consider the infinite series \\(S\\) represented by \\(2 - 1 - \\frac{1}{3} + \\frac{1}{9} - \\frac{1}{27} - \\frac{1}{81} + \\frac{1}{243} - \\cdots\\). Find the sum \\(S\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "829",
"split": "train"
} |
math | [
{
"content": "How many solutions does the system have: $ \\{\\begin{matrix}&(3x+2y) *(\\frac{3}{x}+\\frac{1}{y})=2 & x^2+y^2\\leq 2012 \\end{matrix} $ where $ x,y $ are non-zero integers Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "102",
"style": "rule"
} | {
"index": "830",
"split": "train"
} |
math | [
{
"content": "Charles has two six-sided die. One of the die is fair, and the other die is biased so that it comes up six with probability $\\frac{2}{3}$ and each of the other five sides has probability $\\frac{1}{15}$. Charles chooses one of the two dice at random and rolls it three times. Given that the first ... | {
"ground_truth": "167",
"style": "rule"
} | {
"index": "831",
"split": "train"
} |
math | [
{
"content": "Find the number of sets ${a,b,c}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,61$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "728",
"style": "rule"
} | {
"index": "832",
"split": "train"
} |
math | [
{
"content": "Let $A=\\{m-1,-3\\}$, $B=\\{2m-1,m-3\\}$. If $A\\cap B=\\{-3\\}$, then determine the value of the real number $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "833",
"split": "train"
} |
math | [
{
"content": "Two pedestrians departed simultaneously from point A in the same direction. The first pedestrian met a tourist heading towards point A 20 minutes after leaving point A, and the second pedestrian met the tourist 5 minutes after the first pedestrian. The tourist arrived at point A 10 minutes after t... | {
"ground_truth": "15/8",
"style": "rule"
} | {
"index": "834",
"split": "train"
} |
math | [
{
"content": "Egor, Nikita, and Innokentiy took turns playing chess with each other (two play, one watches). After each game, the loser gave up their place to the spectator (there were no draws). As a result, Egor participated in 13 games, and Nikita participated in 27 games. How many games did Innokentiy play?... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "835",
"split": "train"
} |
math | [
{
"content": "From the sequence of natural numbers $1, 2, 3, 4, \\ldots$, erase every multiple of 3 and 4, but keep every multiple of 5 (for example, 15 and 20 are not erased). After removing the specified numbers, write the remaining numbers in a sequence: $A_{1}=1, A_{2}=2, A_{3}=5, A_{4}=7, \\ldots$. Find th... | {
"ground_truth": "3314",
"style": "rule"
} | {
"index": "836",
"split": "train"
} |
math | [
{
"content": "Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed leaving the remaining six cards in either ascending or descending order. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "837",
"split": "train"
} |
math | [
{
"content": "Given that the area of a cross-section of sphere O is $\\pi$, and the distance from the center O to this cross-section is 1, then the radius of this sphere is __________, and the volume of this sphere is __________. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8\\sqrt{2}}{3}\\pi",
"style": "rule"
} | {
"index": "838",
"split": "train"
} |
math | [
{
"content": "Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Fi... | {
"ground_truth": "929",
"style": "rule"
} | {
"index": "839",
"split": "train"
} |
math | [
{
"content": "The integers -5 and 6 are shown on a number line. What is the distance between them? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "840",
"split": "train"
} |
math | [
{
"content": "Try to divide the set $\\{1,2,\\cdots, 1989\\}$ into 117 mutually disjoint subsets $A_{i}, i = 1,2,\\cdots, 117$, such that\n(1) Each $A_{i}$ contains 17 elements;\n(2) The sum of the elements in each $A_{i}$ is the same. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16915",
"style": "rule"
} | {
"index": "841",
"split": "train"
} |
math | [
{
"content": "Find the point in the plane $3x - 4y + 5z = 30$ that is closest to the point $(1,2,3).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left( \\frac{11}{5}, \\frac{2}{5}, 5 \\right)",
"style": "rule"
} | {
"index": "842",
"split": "train"
} |
math | [
{
"content": "In Mr. Smith's science class, there are 3 boys for every 4 girls. If there are 42 students in total in his class, what percent of them are boys? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42.857\\%",
"style": "rule"
} | {
"index": "843",
"split": "train"
} |
math | [
{
"content": "Define $n!!$ to be $n(n-2)(n-4)\\cdots 3\\cdot 1$ for $n$ odd and $n(n-2)(n-4)\\cdots 4\\cdot 2$ for $n$ even. When $\\sum_{i=1}^{2009} \\frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\\dfrac{ab}{10}$. Please output the final answ... | {
"ground_truth": "401",
"style": "rule"
} | {
"index": "844",
"split": "train"
} |
math | [
{
"content": "Find in explicit form all ordered pairs of positive integers $(m, n)$ such that $mn-1$ divides $m^2 + n^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(2, 1), (3, 1), (1, 2), (1, 3)",
"style": "rule"
} | {
"index": "845",
"split": "train"
} |
math | [
{
"content": "Find the maximum value of\n\\[\\frac{2x + 3y + 4}{\\sqrt{x^2 + 4y^2 + 2}}\\]\nover all real numbers \\( x \\) and \\( y \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{29}",
"style": "rule"
} | {
"index": "846",
"split": "train"
} |
math | [
{
"content": "Given that real numbers $a$ and $b$ satisfy $\\frac{1}{a} + \\frac{2}{b} = \\sqrt{ab}$, calculate the minimum value of $ab$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "847",
"split": "train"
} |
math | [
{
"content": "12 Smurfs are seated around a round table. Each Smurf dislikes the 2 Smurfs next to them, but does not dislike the other 9 Smurfs. Papa Smurf wants to form a team of 5 Smurfs to rescue Smurfette, who was captured by Gargamel. The team must not include any Smurfs who dislike each other. How many wa... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "848",
"split": "train"
} |
math | [
{
"content": "Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form ... | {
"ground_truth": "\\frac{m}{n}",
"style": "rule"
} | {
"index": "849",
"split": "train"
} |
math | [
{
"content": "Four people, A, B, C, and D, stand on a staircase with 7 steps. If each step can accommodate up to 3 people, and the positions of people on the same step are not distinguished, then the number of different ways they can stand is (answer in digits). Please output the final answer within \\boxed... | {
"ground_truth": "2394",
"style": "rule"
} | {
"index": "850",
"split": "train"
} |
math | [
{
"content": "Find the least real number $K$ such that for all real numbers $x$ and $y$ , we have $(1 + 20 x^2)(1 + 19 y^2) \\ge K xy$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8\\sqrt{95}",
"style": "rule"
} | {
"index": "851",
"split": "train"
} |
math | [
{
"content": "The teacher plans to give children a problem of the following type. He will tell them that he has thought of a polynomial \\( P(x) \\) of degree 2017 with integer coefficients, whose leading coefficient is 1. Then he will tell them \\( k \\) integers \\( n_{1}, n_{2}, \\ldots, n_{k} \\), and separ... | {
"ground_truth": "2017",
"style": "rule"
} | {
"index": "852",
"split": "train"
} |
math | [
{
"content": "When $n$ standard 6-sided dice are rolled, find the smallest possible value of $S$ such that the probability of obtaining a sum of 2000 is greater than zero and is the same as the probability of obtaining a sum of $S$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "338",
"style": "rule"
} | {
"index": "853",
"split": "train"
} |
math | [
{
"content": "In triangle \\( \\triangle ABC \\), the lengths of sides opposite to angles A, B, C are denoted by a, b, c respectively. If \\( c = \\sqrt{3} \\), \\( b = 1 \\), and \\( B = 30^\\circ \\),\n(1) find angles A and C;\n(2) find the area of \\( \\triangle ABC \\). Please output the final answer within... | {
"ground_truth": "\\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "854",
"split": "train"
} |
math | [
{
"content": "For positive integers $n$, let $c_{n}$ be the smallest positive integer for which $n^{c_{n}}-1$ is divisible by 210, if such a positive integer exists, and $c_{n}=0$ otherwise. What is $c_{1}+c_{2}+\\cdots+c_{210}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "329",
"style": "rule"
} | {
"index": "855",
"split": "train"
} |
math | [
{
"content": "In 2019, our county built 4 million square meters of new housing, of which 2.5 million square meters are mid-to-low-priced houses. It is expected that in the coming years, the average annual increase in the area of new housing in our county will be 8% higher than the previous year. In addition, th... | {
"ground_truth": "2024",
"style": "rule"
} | {
"index": "856",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer $n$ which cannot be written in any of the following forms? - $n=1+2+\\cdots+k$ for a positive integer $k$. - $n=p^{k}$ for a prime number $p$ and integer $k$ - $n=p+1$ for a prime number $p$. - $n=p q$ for some distinct prime numbers $p$ and $q$ Please output ... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "857",
"split": "train"
} |
math | [
{
"content": "Triangle $\\triangle DEF$ has a right angle at $F$, $\\angle D = 60^\\circ$, and $DF = 6$. Find the radius of the incircle and the circumference of the circumscribed circle around $\\triangle DEF$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12\\pi",
"style": "rule"
} | {
"index": "858",
"split": "train"
} |
math | [
{
"content": "Alice's favorite number is between $90$ and $150$. It is a multiple of $13$, but not a multiple of $4$. The sum of its digits should be a multiple of $4$. What is Alice's favorite number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "143",
"style": "rule"
} | {
"index": "859",
"split": "train"
} |
math | [
{
"content": "A transgalactic ship encountered an astonishing meteor stream. Some meteors fly along a straight line at the same speed, equally spaced from each other. Another group of meteors flies in the exact same manner along another straight line, parallel to the first, but in the opposite direction, also e... | {
"ground_truth": "4.6",
"style": "rule"
} | {
"index": "860",
"split": "train"
} |
math | [
{
"content": "In a different factor tree, each value is also the product of the two values below it, unless the value is a prime number. Determine the value of $X$ for this factor tree:\n\n[asy]\ndraw((-1,-.3)--(0,0)--(1,-.3),linewidth(1));\ndraw((-2,-1.3)--(-1.5,-.8)--(-1,-1.3),linewidth(1));\ndraw((1,-1.3)--(... | {
"ground_truth": "4620",
"style": "rule"
} | {
"index": "861",
"split": "train"
} |
math | [
{
"content": "Let\n\\[f(x)=\\int_0^1 |t-x|t \\, dt\\]\nfor all real $x$ . Sketch the graph of $f(x)$ . What is the minimum value of $f(x)$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2 - \\sqrt{2}}{6}",
"style": "rule"
} | {
"index": "862",
"split": "train"
} |
math | [
{
"content": "Find all functions $f$ defined on all real numbers and taking real values such that \\[f(f(y)) + f(x - y) = f(xf(y) - x),\\] for all real numbers $x, y.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "f(x) = 0",
"style": "rule"
} | {
"index": "863",
"split": "train"
} |
math | [
{
"content": "Determine the value of \\(a\\) if \\(a\\) and \\(b\\) are integers such that \\(x^3 - x - 1\\) is a factor of \\(ax^{19} + bx^{18} + 1\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2584",
"style": "rule"
} | {
"index": "864",
"split": "train"
} |
math | [
{
"content": "Calculate the value of \n\\[\\left(\\left(\\left((3+2)^{-1}+1\\right)^{-1}+2\\right)^{-1}+1\\right)^{-1}+1.\\]\nA) $\\frac{40}{23}$\nB) $\\frac{17}{23}$\nC) $\\frac{23}{17}$\nD) $\\frac{23}{40}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{40}{23}",
"style": "rule"
} | {
"index": "865",
"split": "train"
} |
math | [
{
"content": "Let $T$ be a subset of $\\{1,2,3,...,100\\}$ such that no pair of distinct elements in $T$ has a sum divisible by $5$. What is the maximum number of elements in $T$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "41",
"style": "rule"
} | {
"index": "866",
"split": "train"
} |
math | [
{
"content": "The circle $2x^2 = -2y^2 + 12x - 4y + 20$ is inscribed inside a square which has a pair of sides parallel to the x-axis. What is the area of the square? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "867",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\left\\{ a_n \\right\\}$ satisfying $a_1=4$ and $a_1+a_2+\\cdots +a_n=a_{n+1}$, and $b_n=\\log_{2}a_n$, calculate the value of $\\frac{1}{b_1b_2}+\\frac{1}{b_2b_3}+\\cdots +\\frac{1}{b_{2017}b_{2018}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3025}{4036}",
"style": "rule"
} | {
"index": "868",
"split": "train"
} |
math | [
{
"content": "In triangle \\(ABC\\), the angles \\(A\\) and \\(B\\) are \\(45^{\\circ}\\) and \\(30^{\\circ}\\) respectively, and \\(CM\\) is the median. The circles inscribed in triangles \\(ACM\\) and \\(BCM\\) touch segment \\(CM\\) at points \\(D\\) and \\(E\\). Find the area of triangle \\(ABC\\) if the le... | {
"ground_truth": "16(\\sqrt{3}+1)",
"style": "rule"
} | {
"index": "869",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = \\cos^4x + 2\\sin x\\cos x - \\sin^4x$ \n(1) Determine the parity, the smallest positive period, and the intervals of monotonic increase for the function $f(x)$. \n(2) When $x \\in [0, \\frac{\\pi}{2}]$, find the maximum and minimum values of the function $f(x)$. Pleas... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "870",
"split": "train"
} |
math | [
{
"content": "In the polar coordinate system, let curve \\(C_1\\) be defined by \\(\\rho\\sin^2\\theta = 4\\cos\\theta\\). Establish a Cartesian coordinate system \\(xOy\\) with the pole as the origin and the polar axis as the positive \\(x\\)-axis. The curve \\(C_2\\) is described by the parametric equations: ... | {
"ground_truth": "\\frac{32}{3}",
"style": "rule"
} | {
"index": "871",
"split": "train"
} |
math | [
{
"content": "A pentagon is formed by placing an equilateral triangle atop a square. Each side of the square is equal to the height of the equilateral triangle. What percent of the area of the pentagon is the area of the equilateral triangle? Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "\\frac{3(\\sqrt{3} - 1)}{6} \\times 100\\%",
"style": "rule"
} | {
"index": "872",
"split": "train"
} |
math | [
{
"content": "The population of a small town is $480$. The graph indicates the number of females and males in the town, but the vertical scale-values are omitted. How many males live in the town? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "160",
"style": "rule"
} | {
"index": "873",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\sin(2x- \\frac{\\pi}{6})$, determine the horizontal shift required to obtain the graph of the function $g(x)=\\sin(2x)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{12}",
"style": "rule"
} | {
"index": "874",
"split": "train"
} |
math | [
{
"content": "Let \\( M \\) be a set composed of a finite number of positive integers,\n\\[ \nM = \\bigcup_{i=1}^{20} A_i = \\bigcup_{i=1}^{20} B_i, \\text{ where}\n\\]\n\\[ \nA_i \\neq \\varnothing, B_i \\neq \\varnothing \\ (i=1,2, \\cdots, 20)\n\\]\nsatisfying the following conditions:\n1. For any \\( 1 \\le... | {
"ground_truth": "180",
"style": "rule"
} | {
"index": "875",
"split": "train"
} |
math | [
{
"content": "Determine with proof the number of positive integers $n$ such that a convex regular polygon with $n$ sides has interior angles whose measures, in degrees, are integers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "876",
"split": "train"
} |
math | [
{
"content": "How many solutions does the equation $\\tan x = \\tan (\\tan x + \\frac{\\pi}{4})$ have in the interval $0 \\leq x \\leq \\tan^{-1} 1884$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "600",
"style": "rule"
} | {
"index": "877",
"split": "train"
} |
math | [
{
"content": "Rectangle $EFGH$ has area $4032.$ An ellipse with area $4032\\pi$ passes through points $E$ and $G$ and has its foci at points $F$ and $H$. Determine the perimeter of the rectangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8\\sqrt{2016}",
"style": "rule"
} | {
"index": "878",
"split": "train"
} |
math | [
{
"content": "If $\\frac{x-y}{x+y}=5$, what is the value of $\\frac{2x+3y}{3x-2y}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "879",
"split": "train"
} |
math | [
{
"content": "Given $\\binom{18}{11}=31824$, $\\binom{18}{12}=18564$, and $\\binom{20}{13}=77520$, find the value of $\\binom{19}{13}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27132",
"style": "rule"
} | {
"index": "880",
"split": "train"
} |
math | [
{
"content": "Any six points are taken inside or on a rectangle with dimensions $2 \\times 1$. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Determine the value of... | {
"ground_truth": "\\frac{\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "881",
"split": "train"
} |
math | [
{
"content": "Given points $A$, $B$, $C$ with coordinates $(4,0)$, $(0,4)$, $(3\\cos \\alpha,3\\sin \\alpha)$ respectively, and $\\alpha\\in\\left( \\frac {\\pi}{2}, \\frac {3\\pi}{4}\\right)$. If $\\overrightarrow{AC} \\perp \\overrightarrow{BC}$, find the value of $\\frac {2\\sin ^{2}\\alpha-\\sin 2\\alpha}{1... | {
"ground_truth": "- \\frac {7 \\sqrt {23}}{48}",
"style": "rule"
} | {
"index": "882",
"split": "train"
} |
math | [
{
"content": "Let $S$ be the set of numbers of the form $n^5 - 5n^3 + 4n$ , where $n$ is an integer that is not a multiple of $3$ . What is the largest integer that is a divisor of every number in $S$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "360",
"style": "rule"
} | {
"index": "883",
"split": "train"
} |
math | [
{
"content": "Given that point $(x, y)$ moves on the circle $x^{2}+(y-1)^{2}=1$.\n(1) Find the maximum and minimum values of $\\frac{y-1}{x-2}$;\n(2) Find the maximum and minimum values of $2x+y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1 - \\sqrt{5}",
"style": "rule"
} | {
"index": "884",
"split": "train"
} |
math | [
{
"content": "Find the smallest integer \\( n \\) such that the expanded form of \\( (xy - 7x - 3y + 21)^n \\) has 2012 terms. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "44",
"style": "rule"
} | {
"index": "885",
"split": "train"
} |
math | [
{
"content": "The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules:\n\n 1. The largest isn't first, but it is in one of the first three places. \n 2. The smallest isn't last, but it is in one of the last three places. \n 3. The median isn't first or last.\n\nWhat is th... | {
"ground_truth": "6.5",
"style": "rule"
} | {
"index": "886",
"split": "train"
} |
math | [
{
"content": "What is the maximum value that the expression \\(\\frac{1}{a+\\frac{2010}{b+\\frac{1}{c}}}\\) can take, where \\(a, b, c\\) are distinct non-zero digits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/203",
"style": "rule"
} | {
"index": "887",
"split": "train"
} |
math | [
{
"content": "Compute \\(\\arccos(\\cos 8.5)\\). All functions are in radians. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2.217",
"style": "rule"
} | {
"index": "888",
"split": "train"
} |
math | [
{
"content": "Find the largest prime factor of $15^3+10^4-5^5$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "41",
"style": "rule"
} | {
"index": "889",
"split": "train"
} |
math | [
{
"content": "Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "890",
"split": "train"
} |
math | [
{
"content": "Given that \\( f(x) \\) is an odd function with a period of 4, and for \\( x \\in (0,2) \\), \\( f(x) = x^2 - 16x + 60 \\). Find the value of \\( f(2 \\sqrt{10}) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-36",
"style": "rule"
} | {
"index": "891",
"split": "train"
} |
math | [
{
"content": "A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find $n$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "324",
"style": "rule"
} | {
"index": "892",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)={x^3}+\\frac{{{{2023}^x}-1}}{{{{2023}^x}+1}}+5$, if real numbers $a$ and $b$ satisfy $f(2a^{2})+f(b^{2}-2)=10$, then the maximum value of $a\\sqrt{1+{b^2}}$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3\\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "893",
"split": "train"
} |
math | [
{
"content": "Find $A^2$, where $A$ is the sum of the absolute values of all roots of the following equation:\n\\[x = \\sqrt{19} + \\frac{91}{{\\sqrt{19}+\\frac{91}{{\\sqrt{19}+\\frac{91}{{\\sqrt{19}+\\frac{91}{{\\sqrt{19}+\\frac{91}{x}}}}}}}}}.\\] Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "383",
"style": "rule"
} | {
"index": "894",
"split": "train"
} |
math | [
{
"content": "Given that $A$, $B$, and $C$ are the three interior angles of $\\triangle ABC$, $a$, $b$, and $c$ are the three sides, $a=2$, and $\\cos C=-\\frac{1}{4}$. \n$(1)$ If $\\sin A=2\\sin B$, find $b$ and $c$; \n$(2)$ If $\\cos (A-\\frac{π}{4})=\\frac{4}{5}$, find $c$. Please output the final answer w... | {
"ground_truth": "\\frac{5\\sqrt{30}}{2}",
"style": "rule"
} | {
"index": "895",
"split": "train"
} |
math | [
{
"content": "Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, how many of the four quantities listed below change?\n(a) the length of the segment $MN$\n(b) the perimeter of $\\triangle PAB$\n(c) the area of $\\triangle PAB... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "896",
"split": "train"
} |
math | [
{
"content": "What is the remainder when $3001 \\cdot 3002 \\cdot 3003 \\cdot 3004 \\cdot 3005$ is divided by 17? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "897",
"split": "train"
} |
math | [
{
"content": "How many positive integers less than 1998 are relatively prime to 1547 ? (Two integers are relatively prime if they have no common factors besides 1.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1487",
"style": "rule"
} | {
"index": "898",
"split": "train"
} |
math | [
{
"content": "A portion of the graph of a quadratic function $f(x)$ is shown below.\n\nLet $g(x)=-f(x)$ and $h(x)=f(-x)$. If $a$ is the number of points where the graphs of $y=f(x)$ and $y=g(x)$ intersect, and $b$ is the number of points where the graphs of $y=f(x)$ and $y=h(x)$ intersect, then what is $10a+b$?... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "899",
"split": "train"
} |
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