data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given $|\\overrightarrow {a}|=\\sqrt {2}$, $|\\overrightarrow {b}|=2$, and $(\\overrightarrow {a}-\\overrightarrow {b})\\bot \\overrightarrow {a}$, determine the angle between $\\overrightarrow {a}$ and $\\overrightarrow {b}$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "39100",
"split": "train"
} |
math | [
{
"content": "Given that $-\\frac{\\pi}{2} < x < 0$ and $\\sin x + \\cos x = \\frac{1}{5}$.\n\n(I) Find the value of $\\sin x - \\cos x$.\n\n(II) Find the value of $\\frac{3\\sin^2{\\frac{x}{2}} - 2\\sin{\\frac{x}{2}}\\cos{\\frac{x}{2}} + \\cos^2{\\frac{x}{2}}}{\\tan x + \\cot x}$. Please output the final answe... | {
"ground_truth": "-\\frac{108}{125}",
"style": "rule"
} | {
"index": "39101",
"split": "train"
} |
math | [
{
"content": "Let $M \\Theta N$ represent the remainder when the larger of $M$ and $N$ is divided by the smaller one. For example, $3 \\Theta 10 = 1$. For non-zero natural numbers $A$ less than 40, given that $20 \\Theta(A \\bigodot 20) = 7$, find $A$. Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "39102",
"split": "train"
} |
math | [
{
"content": "A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the prev... | {
"ground_truth": "585",
"style": "rule"
} | {
"index": "39103",
"split": "train"
} |
math | [
{
"content": "Compute $\\left\\lceil\\displaystyle\\sum_{k=2018}^{\\infty}\\frac{2019!-2018!}{k!}\\right\\rceil$ . (The notation $\\left\\lceil x\\right\\rceil$ denotes the least integer $n$ such that $n\\geq x$ .)\n\n*Proposed by Tristan Shin* Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "2019",
"style": "rule"
} | {
"index": "39104",
"split": "train"
} |
math | [
{
"content": "$AL$ and $BM$ are the angle bisectors of triangle $ABC$. The circumcircles of triangles $ALC$ and $BMC$ intersect again at point $K$, which lies on side $AB$. Find the measure of angle $ACB$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "39105",
"split": "train"
} |
math | [
{
"content": "In acute triangle $\\triangle ABC$, $b=2$, $B=\\frac{\\pi }{3}$, $\\sin 2A+\\sin (A-C)-\\sin B=0$, find the area of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "39106",
"split": "train"
} |
math | [
{
"content": "Find the number of ordered pairs \\((a, b)\\) of positive integers such that \\(a\\) and \\(b\\) both divide \\(20^{19}\\), but \\(ab\\) does not. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "444600",
"style": "rule"
} | {
"index": "39107",
"split": "train"
} |
math | [
{
"content": "Given that $\\dfrac {\\pi}{2} < \\alpha < \\pi$ and $\\sin (\\alpha+ \\dfrac {\\pi}{6})= \\dfrac {3}{5}$, find the value of $\\cos (\\alpha- \\dfrac {\\pi}{6})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac {3\\sqrt {3}-4}{10}",
"style": "rule"
} | {
"index": "39108",
"split": "train"
} |
math | [
{
"content": "Let C be the number of ways to arrange the letters of the word CATALYSIS, T be the number of ways to arrange the letters of the word TRANSPORT, S be the number of ways to arrange the letters of the word STRUCTURE, and M be the number of ways to arrange the letters of the word MOTION. What is $\\f... | {
"ground_truth": "126",
"style": "rule"
} | {
"index": "39109",
"split": "train"
} |
math | [
{
"content": "Determine all functions $f: \\mathbb{Q} \\to \\mathbb{Q}$ such that\n$$f(2xy + \\frac{1}{2}) + f(x-y) = 4f(x)f(y) + \\frac{1}{2}$$\nfor all $x,y \\in \\mathbb{Q}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "f(x) = x^2 + \\frac{1}{2}",
"style": "rule"
} | {
"index": "39110",
"split": "train"
} |
math | [
{
"content": "Let set $A=\\{x|\\left(\\frac{1}{2}\\right)^{x^2-4}>1\\}$, $B=\\{x|2<\\frac{4}{x+3}\\}$ \n(1) Find $A\\cap B$ \n(2) If the solution set of the inequality $2x^2+ax+b<0$ is $B$, find the values of $a$ and $b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-6",
"style": "rule"
} | {
"index": "39111",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $2c=a+\\cos A\\frac{b}{\\cos B}$. \n$(1)$ Find the measure of angle $B$. \n$(2)$ If $b=4$ and $a+c=3\\sqrt{2}$, find the area of $\\triangle ABC$. Please output the fina... | {
"ground_truth": "\\frac{\\sqrt{3}}{6}",
"style": "rule"
} | {
"index": "39112",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfying $|\\overrightarrow{a}|=2$, $|\\overrightarrow{b}|=\\overrightarrow{a}\\cdot\\overrightarrow{b}=1$, and $(\\overrightarrow{a}-2\\overrightarrow{c}) \\cdot (\\overrightarrow{b}-\\overrightarrow{c})=0$, find the minimum value of... | {
"ground_truth": "\\frac{\\sqrt{7}-\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "39113",
"split": "train"
} |
math | [
{
"content": "The second hand on a clock is 10 cm long. How far in centimeters does the tip of this second hand travel during a period of 15 minutes? Express your answer in terms of $\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "300\\pi",
"style": "rule"
} | {
"index": "39114",
"split": "train"
} |
math | [
{
"content": "An investment of $\\$$10,000 is made in a government bond that will pay 6$\\%$ interest compounded annually. At the end of five years, what is the total number of dollars in this investment? Express your answer to the nearest whole number. Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "13382",
"style": "rule"
} | {
"index": "39115",
"split": "train"
} |
math | [
{
"content": "The total area of all the faces of a rectangular solid is $22\\text{cm}^2$, and the total length of all its edges is $24\\text{cm}$. Then the length in cm of any one of its interior diagonals is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{14}",
"style": "rule"
} | {
"index": "39116",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC^{}_{}$, $A'$, $B'$, and $C'$ are on the sides $BC$, $AC^{}_{}$, and $AB^{}_{}$, respectively. Given that $AA'$, $BB'$, and $CC'$ are concurrent at the point $O^{}_{}$, and that $\\frac{AO^{}_{}}{OA'}+\\frac{BO}{OB'}+\\frac{CO}{OC'}=92$, find $\\frac{AO}{OA'}\\cdot \\frac{BO}{OB'}\... | {
"ground_truth": "94",
"style": "rule"
} | {
"index": "39117",
"split": "train"
} |
math | [
{
"content": "The maximum number that can be formed by the digits 0, 3, 4, 5, 6, 7, 8, 9, given that there are only thirty 2's and twenty-five 1's. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "199",
"style": "rule"
} | {
"index": "39118",
"split": "train"
} |
math | [
{
"content": "Calculate the definite integral:\n\n$$\n\\int_{\\pi / 4}^{\\operatorname{arctg} 3} \\frac{d x}{(3 \\operatorname{tg} x+5) \\sin 2 x}\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{10} \\ln \\frac{12}{7}",
"style": "rule"
} | {
"index": "39119",
"split": "train"
} |
math | [
{
"content": "Twenty cubical blocks are arranged as shown. First, 10 are arranged in a triangular pattern; then a layer of 6, arranged in a triangular pattern, is centered on the 10; then a layer of 3, arranged in a triangular pattern, is centered on the 6; and finally one block is centered on top of the third ... | {
"ground_truth": "114",
"style": "rule"
} | {
"index": "39120",
"split": "train"
} |
math | [
{
"content": "A bag contains $4$ identical small balls, of which there is $1$ red ball, $2$ white balls, and $1$ black ball. Balls are drawn from the bag with replacement, randomly taking one each time.\n\n(1) Find the probability of drawing a white ball two consecutive times;\n\n(2) If drawing a red ball score... | {
"ground_truth": "\\frac{15}{64}",
"style": "rule"
} | {
"index": "39121",
"split": "train"
} |
math | [
{
"content": "Evaluate $x^3 + x^2 + x + 1$ when $x = 3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "39122",
"split": "train"
} |
math | [
{
"content": "In the convex quadrilateral \\(ABCD\\), \\(E\\) is the intersection of the diagonals. The areas of triangles \\(ADE, BCE, CDE\\) are \\(12 \\, \\text{cm}^2, 45 \\, \\text{cm}^2, 18 \\, \\text{cm}^2\\), respectively, and the length of side \\(AB\\) is \\(7 \\, \\text{cm}\\).\n\nDetermine the distan... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "39123",
"split": "train"
} |
math | [
{
"content": "For each real number \\( x \\), \\( f(x) \\) is defined to be the minimum of the values of \\( 2x + 3 \\), \\( 3x - 2 \\), and \\( 25 - x \\). What is the maximum value of \\( f(x) \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{53}{3}",
"style": "rule"
} | {
"index": "39124",
"split": "train"
} |
math | [
{
"content": "Petya's bank account contains $500. The bank allows only two types of transactions: withdrawing $300 or adding $198. What is the maximum amount Petya can withdraw from the account, if he has no other money? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "39125",
"split": "train"
} |
math | [
{
"content": "Consider a triangle $DEF$ where the angles of the triangle satisfy\n\\[ \\cos 3D + \\cos 3E + \\cos 3F = 1. \\]\nTwo sides of this triangle have lengths 12 and 14. Find the maximum possible length of the third side. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{127}",
"style": "rule"
} | {
"index": "39126",
"split": "train"
} |
math | [
{
"content": "A sphere passes through two adjacent vertices of a unit cube and touches the planes of the faces that do not contain these vertices. What is the radius of this sphere? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 - \\frac{\\sqrt{7}}{2}",
"style": "rule"
} | {
"index": "39127",
"split": "train"
} |
math | [
{
"content": "The lengths of two sides of a triangle are 33 units and 42 units. The third side also has an integral length. What is the least possible number of units in the perimeter of the triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "85",
"style": "rule"
} | {
"index": "39128",
"split": "train"
} |
math | [
{
"content": "A particle is placed on the curve $y = x^3 - 3x^2 - x + 3$ at a point $P$ whose $y$-coordinate is $5$. It is allowed to roll along the curve until it reaches the nearest point $Q$ whose $y$-coordinate is $-2$. Compute the horizontal distance traveled by the particle.\nA) $|\\sqrt{6} - \\sqrt{3}|$\... | {
"ground_truth": "|\\sqrt{6} - \\sqrt{3}|",
"style": "rule"
} | {
"index": "39129",
"split": "train"
} |
math | [
{
"content": "If $a$ and $b$ are two distinct numbers with $\\frac{a+b}{a-b}=3$, what is the value of $\\frac{a}{b}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "39130",
"split": "train"
} |
math | [
{
"content": "The product of three consecutive positive integers is $8$ times their sum. What is the sum of their squares? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "77",
"style": "rule"
} | {
"index": "39131",
"split": "train"
} |
math | [
{
"content": "Michel starts with the string $H M M T$. An operation consists of either replacing an occurrence of $H$ with $H M$, replacing an occurrence of $M M$ with $M O M$, or replacing an occurrence of $T$ with $M T$. For example, the two strings that can be reached after one operation are $H M M M T$ and ... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "39132",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be real numbers. One of the roots of\n\\[x^3 + ax^2 - x + b = 0\\]is $1 - 2i.$ Enter the ordered pair $(a,b).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(1,15)",
"style": "rule"
} | {
"index": "39133",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, $AB=3$, $AC=4$, and $\\angle BAC=60^{\\circ}$. If $P$ is a point in the plane of $\\triangle ABC$ and $AP=2$, calculate the maximum value of $\\vec{PB} \\cdot \\vec{PC}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10 + 2 \\sqrt{37}",
"style": "rule"
} | {
"index": "39134",
"split": "train"
} |
math | [
{
"content": "Given a defect rate of 3%, products are drawn from the batch without replacement until a non-defective product is found or a maximum of three draws have been made. Let $X$ represent the number of products drawn, and calculate $P(X=3)$. Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "(0.03)^2 \\times 0.97 + (0.03)^3",
"style": "rule"
} | {
"index": "39135",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $\\sqrt {2}csinAcosB=asinC$.\n(I) Find the measure of $\\angle B$;\n(II) If the area of $\\triangle ABC$ is $a^2$, find the value of $cosA$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {3 \\sqrt {10}}{10}",
"style": "rule"
} | {
"index": "39136",
"split": "train"
} |
math | [
{
"content": "Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S,$ the probability that it is divisible by $15$ is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$ Please output the final ans... | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "39137",
"split": "train"
} |
math | [
{
"content": "A traffic light runs repeatedly through the following cycle: green for 45 seconds, then yellow for 5 seconds, and then red for 50 seconds. Mark picks a random five-second time interval to watch the light. What is the probability that the color changes while he is watching? Please output the final ... | {
"ground_truth": "\\frac{3}{20}",
"style": "rule"
} | {
"index": "39138",
"split": "train"
} |
math | [
{
"content": "In a given triangle, for $\\angle P$ to be the largest angle of the triangle, it must be that $a < y < b$. The side lengths are given by $y+6$, $2y+1$, and $5y-10$. What is the least possible value of $b-a$, expressed as a common fraction? Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "4.5",
"style": "rule"
} | {
"index": "39139",
"split": "train"
} |
math | [
{
"content": "The probability of an event occurring in each of 900 independent trials is 0.5. Find the probability that the relative frequency of the event will deviate from its probability by no more than 0.02. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.7698",
"style": "rule"
} | {
"index": "39140",
"split": "train"
} |
math | [
{
"content": "In a Martian civilization, all logarithms whose bases are not specified as assumed to be base $b$, for some fixed $b\\ge2$. A Martian student writes down \\[3\\log(\\sqrt{x}\\log x)=56\\] \\[\\log_{\\log x}(x)=54\\] and finds that this system of equations has a single real number solution $x>1$. F... | {
"ground_truth": "216",
"style": "rule"
} | {
"index": "39141",
"split": "train"
} |
math | [
{
"content": "Cinderella and her fairy godmother released a collection of seven new models of crystal slippers. The storybook heroines held a presentation of the collection for some guests: the audience had to state which slippers they liked. The guests wrote in a survey which models they considered the best. I... | {
"ground_truth": "128",
"style": "rule"
} | {
"index": "39142",
"split": "train"
} |
math | [
{
"content": "If $\\cos \\theta = \\frac{1}{4},$ then find $\\cos 3 \\theta.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{11}{16}",
"style": "rule"
} | {
"index": "39143",
"split": "train"
} |
math | [
{
"content": "Lara is trying to solve the following equation by completing the square: $$100x^2 + 60x - 49 = 0.$$ She seeks to rewrite the above equation in the form: $$(ax + b)^2 = c,$$ where $a$, $b$, and $c$ are integers and $a > 0$. What is the value of $a + b + c$? Please output the final answer within \\b... | {
"ground_truth": "71",
"style": "rule"
} | {
"index": "39144",
"split": "train"
} |
math | [
{
"content": "Let $r_{k}$ denote the remainder when $\\binom{127}{k}$ is divided by 8. Compute $r_{1}+2 r_{2}+3 r_{3}+\\cdots+63 r_{63}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8096",
"style": "rule"
} | {
"index": "39145",
"split": "train"
} |
math | [
{
"content": "On a blackboard a stranger writes the values of $s_{7}(n)^{2}$ for $n=0,1, \\ldots, 7^{20}-1$, where $s_{7}(n)$ denotes the sum of digits of $n$ in base 7 . Compute the average value of all the numbers on the board. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3680",
"style": "rule"
} | {
"index": "39146",
"split": "train"
} |
math | [
{
"content": "Mary chose an even $4$-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,...,\\dfrac{n}{2},n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$?\n$\\textbf{... | {
"ground_truth": "340",
"style": "rule"
} | {
"index": "39147",
"split": "train"
} |
math | [
{
"content": "If line $l_1: (2m+1)x - 4y + 3m = 0$ is parallel to line $l_2: x + (m+5)y - 3m = 0$, determine the value of $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{9}{2}",
"style": "rule"
} | {
"index": "39148",
"split": "train"
} |
math | [
{
"content": "Find the number of odd digits in the base-4 representation of $233_{10}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "39149",
"split": "train"
} |
math | [
{
"content": "In how many ways can a President and a Vice-President be chosen from a group of 5 people (assuming that the President and the Vice-President cannot be the same person)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "39150",
"split": "train"
} |
math | [
{
"content": "Given the piecewise function $f(x)= \\begin{cases} x+2 & (x\\leq-1) \\\\ x^{2} & (-1<x<2) \\\\ 2x & (x\\geq2)\\end{cases}$, if $f(x)=3$, determine the value of $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "39151",
"split": "train"
} |
math | [
{
"content": "The equation of the asymptotes of the hyperbola \\\\(x^{2}- \\frac {y^{2}}{2}=1\\\\) is \\_\\_\\_\\_\\_\\_; the eccentricity equals \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt {3}",
"style": "rule"
} | {
"index": "39152",
"split": "train"
} |
math | [
{
"content": "Compute: $\\displaystyle \\frac{66,\\!666^4}{22,\\!222^4}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "81",
"style": "rule"
} | {
"index": "39153",
"split": "train"
} |
math | [
{
"content": "Let $z_1$ and $z_2$ be the complex roots of $z^2 + az + b = 0,$ where $a$ and $b$ are complex numbers. In the complex plane, 0, $z_1,$ and $z_2$ form the vertices of an equilateral triangle. Find $\\frac{a^2}{b}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "39154",
"split": "train"
} |
math | [
{
"content": "The calculation result of the expression \\(143 \\times 21 \\times 4 \\times 37 \\times 2\\) is $\\qquad$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "888888",
"style": "rule"
} | {
"index": "39155",
"split": "train"
} |
math | [
{
"content": "How many non-empty subsets $S$ of $\\{1, 2, 3, \\ldots, 10\\}$ satisfy the following two conditions?\n1. No two consecutive integers belong to $S$.\n2. If $S$ contains $k$ elements, then $S$ contains no number less than $k$. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "59",
"style": "rule"
} | {
"index": "39156",
"split": "train"
} |
math | [
{
"content": "Mady has an infinite number of balls and boxes available to her. The empty boxes, each capable of holding sixteen balls, are arranged in a row from left to right. At the first step, she places a ball in the first box (the leftmost box) of the row. At each subsequent step, she places a ball in the ... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "39157",
"split": "train"
} |
math | [
{
"content": "Convert $199_{10}$ to base 2. Let $x$ be the number of zeros and $y$ be the number of ones in base 2. What is the value of $y-x?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "39158",
"split": "train"
} |
math | [
{
"content": "In the cells of an $8 \\times 8$ chessboard, there are 8 white and 8 black pieces arranged such that no two pieces are in the same cell. Additionally, no pieces of the same color are in the same row or column. For each white piece, the distance to the black piece in the same column is calculated. ... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "39159",
"split": "train"
} |
math | [
{
"content": "Given $\\cos x + \\cos y = \\frac{1}{2}$ and $\\sin x + \\sin y = \\frac{1}{3}$, find the value of $\\cos (x - y)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$-\\frac{59}{72}$",
"style": "rule"
} | {
"index": "39160",
"split": "train"
} |
math | [
{
"content": "Let $AB$ be a segment of length $2$ . The locus of points $P$ such that the $P$ -median of triangle $ABP$ and its reflection over the $P$ -angle bisector of triangle $ABP$ are perpendicular determines some region $R$ . Find the area of $R$ . Please output the final answer within \\b... | {
"ground_truth": "2\\pi",
"style": "rule"
} | {
"index": "39161",
"split": "train"
} |
math | [
{
"content": "A positive integer $n$ is picante if $n$ ! ends in the same number of zeroes whether written in base 7 or in base 8 . How many of the numbers $1,2, \\ldots, 2004$ are picante? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "39162",
"split": "train"
} |
math | [
{
"content": "A sphere is inscribed in the tetrahedron whose vertices are $A = (6,0,0), B = (0,4,0), C = (0,0,2),$ and $D = (0,0,0).$ The radius of the sphere 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": "5",
"style": "rule"
} | {
"index": "39163",
"split": "train"
} |
math | [
{
"content": "(1) Find the constant term in the expansion of ${\\left( \\frac{1}{x}- \\sqrt{\\frac{x}{2}}\\right)}^{9}$;\n\n(2) Given that ${x}^{10}={a}_{0}+{a}_{1}\\left( x+2 \\right)+{a}_{2}{\\left( x+2 \\right)}^{2}+... +{a}_{10}{\\left( x+2 \\right)}^{10}$, find the value of ${{a}_{1}+{a}_{2}+{a}_{3}+... +{... | {
"ground_truth": "-1023",
"style": "rule"
} | {
"index": "39164",
"split": "train"
} |
math | [
{
"content": "Let $min|a, b|$ denote the minimum value between $a$ and $b$. When positive numbers $x$ and $y$ vary, let $t = min|2x+y, \\frac{2y}{x^2+2y^2}|$, then the maximum value of $t$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "39165",
"split": "train"
} |
math | [
{
"content": "The sum other than $11$ which occurs with the same probability when all 8 dice are rolled is equal to what value. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "39166",
"split": "train"
} |
math | [
{
"content": "From the numbers 0, 1, 2, 3, 4, select three different digits to form a three-digit number. What is the sum of the units digit of all these three-digit numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "39167",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $2\\sin^2 \\frac{A+B}{2}-\\cos 2C=1$, and the radius of the circumcircle $R=2$. \n$(1)$ Find $C$; \n$(2)$ Find the maximum value of $S_{\\triangle ABC}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "39168",
"split": "train"
} |
math | [
{
"content": "Let $S=\\{1,2, \\ldots, 9\\}$. Compute the number of functions $f: S \\rightarrow S$ such that, for all $s \\in S, f(f(f(s)))=s$ and $f(s)-s$ is not divisible by 3. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "288",
"style": "rule"
} | {
"index": "39169",
"split": "train"
} |
math | [
{
"content": "The sum of the digits of the integer equal to \\( 777777777777777^2 - 222222222222223^2 \\) is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "74",
"style": "rule"
} | {
"index": "39170",
"split": "train"
} |
math | [
{
"content": "Evaluate the expression $\\frac{2020^3 - 3 \\cdot 2020^2 \\cdot 2021 + 5 \\cdot 2020 \\cdot 2021^2 - 2021^3 + 4}{2020 \\cdot 2021}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4042 + \\frac{3}{4080420}",
"style": "rule"
} | {
"index": "39171",
"split": "train"
} |
math | [
{
"content": "What is \\( \\frac{1}{4} \\) more than 32.5? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32.75",
"style": "rule"
} | {
"index": "39172",
"split": "train"
} |
math | [
{
"content": "For positive integers $n$, define $S_n$ to be the minimum value of the sum\n\\[\\sum_{k=1}^n \\sqrt{(2k-1)^2+a_k^2},\\]where $a_1,a_2,\\ldots,a_n$ are positive real numbers whose sum is $17$. Find the unique positive integer $n$ for which $S_n$ is also an integer. Please output the final answer wi... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "39173",
"split": "train"
} |
math | [
{
"content": "Compute the sum of the squares of cosine for the angles progressing by 10 degrees starting from 0 degrees up to 180 degrees:\n\\[\\cos^2 0^\\circ + \\cos^2 10^\\circ + \\cos^2 20^\\circ + \\dots + \\cos^2 180^\\circ.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{19}{2}",
"style": "rule"
} | {
"index": "39174",
"split": "train"
} |
math | [
{
"content": "(1) Calculate $\\dfrac{2A_{8}^{5}+7A_{8}^{4}}{A_{8}^{8}-A_{9}^{5}}$,\n\n(2) Calculate $C_{200}^{198}+C_{200}^{196}+2C_{200}^{197}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "67331650",
"style": "rule"
} | {
"index": "39175",
"split": "train"
} |
math | [
{
"content": "An equilateral triangle $ABC$ is inscribed in the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ such that vertex $B$ is at $(0,b)$, and side $\\overline{AC}$ is parallel to the $x$-axis. The foci $F_1$ and $F_2$ of the ellipse lie on sides $\\overline{BC}$ and $\\overline{AB}$, respectively. D... | {
"ground_truth": "\\frac{8}{5}",
"style": "rule"
} | {
"index": "39176",
"split": "train"
} |
math | [
{
"content": "Find the maximum value of $S$ such that any finite number of small squares with a total area of $S$ can be placed inside a unit square $T$ with side length 1, in such a way that no two squares overlap. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "39177",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be an acute-angled triangle and $P$ be a point in its interior. Let $P_A,P_B$ and $P_c$ be the images of $P$ under reflection in the sides $BC,CA$ , and $AB$ , respectively. If $P$ is the orthocentre of the triangle $P_AP_BP_C$ and if the largest angle of the triangle tha... | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "39178",
"split": "train"
} |
math | [
{
"content": "In a convex 1950-sided polygon, all the diagonals are drawn, dividing it into smaller polygons. Consider the polygon with the greatest number of sides among these smaller polygons. What is the maximum number of sides it can have? Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "1949",
"style": "rule"
} | {
"index": "39179",
"split": "train"
} |
math | [
{
"content": "Let \\( a \\) and \\( b \\) be real numbers such that \\( a + b = 1 \\). Then, the minimum value of \n\\[ \nf(a, b) = 3 \\sqrt{1 + 2a^2} + 2 \\sqrt{40 + 9b^2} \n\\] \nis ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5 \\sqrt{11}",
"style": "rule"
} | {
"index": "39180",
"split": "train"
} |
math | [
{
"content": "Let $a,$ $b,$ $c,$ $d$ be real numbers such that\n\\[\\frac{(a - b)(c - d)}{(b - c)(d - a)} = \\frac{2}{5}.\\]Find the sum of all possible values of\n\\[\\frac{(a - c)(b - d)}{(a - b)(c - d)}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{2}",
"style": "rule"
} | {
"index": "39181",
"split": "train"
} |
math | [
{
"content": "Given $m$ points on a plane, where no three points are collinear, and their convex hull is an $n$-gon. Connecting the points appropriately can form a mesh region composed of triangles. Let $f(m, n)$ represent the number of non-overlapping triangles in this region. Find $f(2016, 30)$. Please output... | {
"ground_truth": "4000",
"style": "rule"
} | {
"index": "39182",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be positive integers for which $45a+b=2021$. What is the minimum possible value of $a+b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "85",
"style": "rule"
} | {
"index": "39183",
"split": "train"
} |
math | [
{
"content": "Find the sum of the distinct prime factors of $7^7 - 7^4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "39184",
"split": "train"
} |
math | [
{
"content": "Consider the set $\\{45, 52, 87, 90, 112, 143, 154\\}$. How many subsets containing three different numbers can be selected from this set so that the sum of the three numbers is odd? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "39185",
"split": "train"
} |
math | [
{
"content": "Let $$ A=\\lim _{n \\rightarrow \\infty} \\sum_{i=0}^{2016}(-1)^{i} \\cdot \\frac{\\binom{n}{i}\\binom{n}{i+2}}{\\binom{n}{i+1}^{2}} $$ Find the largest integer less than or equal to $\\frac{1}{A}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "39186",
"split": "train"
} |
math | [
{
"content": "Let $\\triangle ABC$ be an acute scalene triangle with circumcircle $\\omega$. The tangents to $\\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppose $BT = CT = 16$, $BC = 22$, and $TX^2 + TY^2 + XY^2 = 1143$. Find $XY^2... | {
"ground_truth": "717",
"style": "rule"
} | {
"index": "39187",
"split": "train"
} |
math | [
{
"content": "A positive integer is *happy* if:\n\n1. All its digits are different and not $0$ ,\n2. One of its digits is equal to the sum of the other digits.\n\nFor example, 253 is a *happy* number. How many *happy* numbers are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "39188",
"split": "train"
} |
math | [
{
"content": "The second hand on the clock pictured below is 6 cm long. How far in centimeters does the tip of this second hand travel during a period of 30 minutes? Express your answer in terms of $\\pi$.\n\n[asy]\ndraw(Circle((0,0),20));\nlabel(\"12\",(0,20),S);\nlabel(\"9\",(-20,0),E);\nlabel(\"6\",(0,-20),... | {
"ground_truth": "360\\pi",
"style": "rule"
} | {
"index": "39189",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ and $BCFG$ be two faces of a cube with $AB=12$. A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P$, which is 7 units from $\\overline{BG}$ and 5 units from $\\overline{BC}$. The beam continues to be reflected off the faces of the cube. The length of the l... | {
"ground_truth": "230",
"style": "rule"
} | {
"index": "39190",
"split": "train"
} |
math | [
{
"content": "There are two colors Jessica can use to color this 2 by 2 grid. If non-overlapping regions that share a side cannot be the same color, how many possible ways are there for Jessica to color the grid?\n\n[asy]\nsize(101);\ndraw(unitsquare);\ndraw((0,.5)--(1,.5));\ndraw((.5,0)--(.5,1));\nlabel(\"$A$\... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "39191",
"split": "train"
} |
math | [
{
"content": "How many nonzero complex numbers $z$ have the property that $0, z,$ and $z^3,$ when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "39192",
"split": "train"
} |
math | [
{
"content": "Given that $α∈(0, \\dfrac {π}{2})$ and $β∈(0, \\dfrac {π}{2})$, with $cosα= \\dfrac {1}{7}$ and $cos(α+β)=- \\dfrac {11}{14}$, find the value of $sinβ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "39193",
"split": "train"
} |
math | [
{
"content": "$ABCDEFGH$ is a cube. Find $\\sin \\angle HAD$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "39194",
"split": "train"
} |
math | [
{
"content": "The vertices of a regular hexagon are labeled $\\cos (\\theta), \\cos (2 \\theta), \\ldots, \\cos (6 \\theta)$. For every pair of vertices, Bob draws a blue line through the vertices if one of these functions can be expressed as a polynomial function of the other (that holds for all real $\\theta$... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "39195",
"split": "train"
} |
math | [
{
"content": "A box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. What is the probability that $3$ draws are required? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{5}",
"style": "rule"
} | {
"index": "39196",
"split": "train"
} |
math | [
{
"content": "Given that $P$ is a moving point on the parabola $y^{2}=4x$, and $Q$ is a moving point on the circle $x^{2}+(y-4)^{2}=1$, find the minimum value of the sum of the distance between points $P$ and $Q$ and the distance between point $P$ and the axis of symmetry of the parabola. Please output the fina... | {
"ground_truth": "\\sqrt{17}-1",
"style": "rule"
} | {
"index": "39197",
"split": "train"
} |
math | [
{
"content": "In the right triangle $ABC$, where $\\angle B = \\angle C$, the length of $AC$ is $8\\sqrt{2}$. Calculate the area of triangle $ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "39198",
"split": "train"
} |
math | [
{
"content": "Suppose that $a_1, a_2, a_3, \\ldots$ is an infinite geometric sequence such that for all $i \\ge 1$ , $a_i$ is a positive integer. Suppose furthermore that $a_{20} + a_{21} = 20^{21}$ . If the minimum possible value of $a_1$ can be expressed as $2^a 5^b$ for positive integers $a$ and ... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "39199",
"split": "train"
} |
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