data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "The value of $\\left(256\\right)^{.16}\\left(256\\right)^{.09}$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "6400",
"split": "train"
} |
math | [
{
"content": "Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(9,3)$, respectively. What is its area? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45 \\sqrt{3}",
"style": "rule"
} | {
"index": "6401",
"split": "train"
} |
math | [
{
"content": "$14 N$ is a 5-digit number composed of 5 different non-zero digits, and $N$ is equal to the sum of all three-digit numbers that can be formed using any 3 of these 5 digits. Find all such 5-digit numbers $N$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "35964",
"style": "rule"
} | {
"index": "6402",
"split": "train"
} |
math | [
{
"content": "Let \\( x \\) and \\( y \\) be real numbers with \\( y > x > 0 \\), satisfying \n\\[ \\frac{x}{y} + \\frac{y}{x} = 8. \\]\nFind the value of \\[ \\frac{x + y}{x - y}. \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{\\frac{5}{3}}",
"style": "rule"
} | {
"index": "6403",
"split": "train"
} |
math | [
{
"content": "If the line $y=ax+b$ is a tangent line of the graph of the function $f(x)=\\ln{x}-\\frac{1}{x}$, then the minimum value of $a+b$ is $\\_\\_\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "6404",
"split": "train"
} |
math | [
{
"content": "The points \\((2,3)\\) and \\((3, 7)\\) lie on a circle whose center is on the \\(x\\)-axis. What is the radius of the circle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{1717}}{2}",
"style": "rule"
} | {
"index": "6405",
"split": "train"
} |
math | [
{
"content": "A bag of fruit contains 10 fruits, including an even number of apples, at most two oranges, a multiple of three bananas, and at most one pear. How many different combinations of these fruits can there be? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "6406",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer that has eight positive odd integer divisors and sixteen positive even integer divisors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3000",
"style": "rule"
} | {
"index": "6407",
"split": "train"
} |
math | [
{
"content": "The measure of angle $ACB$ is 70 degrees. If ray $CA$ is rotated 600 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new obtuse angle $ACB$, in degrees? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "170",
"style": "rule"
} | {
"index": "6408",
"split": "train"
} |
math | [
{
"content": "What is the greatest prime factor of $15! + 18!$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "6409",
"split": "train"
} |
math | [
{
"content": "Out of 100 externally identical marbles, one is radioactive, but I don't know which one it is. A friend of mine would buy only non-radioactive marbles from me, at a price of 1 forint each. Another friend of mine has an instrument that can determine whether or not there is a radioactive marble amon... | {
"ground_truth": "92",
"style": "rule"
} | {
"index": "6410",
"split": "train"
} |
math | [
{
"content": "The sum of the digits of the positive integer $N$ is three times the sum of the digits of $N+1$. What is the smallest possible sum of the digits of $N$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "6411",
"split": "train"
} |
math | [
{
"content": "On a table lie 140 different cards with numbers $3, 6, 9, \\ldots, 417, 420$ (each card has exactly one number, and each number appears exactly once). In how many ways can you choose 2 cards so that the sum of the numbers on the selected cards is divisible by $7?$ Please output the final answer wi... | {
"ground_truth": "1390",
"style": "rule"
} | {
"index": "6412",
"split": "train"
} |
math | [
{
"content": "If $a>0$ and $b>0,$ a new operation $\\nabla$ is defined as follows: $$a \\nabla b = \\frac{a + b}{1 + ab}.$$For example, $$3 \\nabla 6 = \\frac{3 + 6}{1 + 3 \\times 6} = \\frac{9}{19}.$$Calculate $(1 \\nabla 2) \\nabla 3.$ Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "6413",
"split": "train"
} |
math | [
{
"content": "Two different points, $C$ and $D$, lie on the same side of line $AB$ so that $\\triangle ABC$ and $\\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\\tfrac mn$, where $m$ and $n$ are relatively prime positive intege... | {
"ground_truth": "59",
"style": "rule"
} | {
"index": "6414",
"split": "train"
} |
math | [
{
"content": "The numbers $\\sqrt{3v-2}$, $\\sqrt{3v+1}$, and $2\\sqrt{v}$ are the side lengths of a triangle. What is the measure of the largest angle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "6415",
"split": "train"
} |
math | [
{
"content": "A group of children held a grape-eating contest. When the contest was over, the winner had eaten $n$ grapes, and the child in $k$-th place had eaten $n+2-2k$ grapes. The total number of grapes eaten in the contest was $2009$. Find the smallest possible value of $n$. Please output the final answer ... | {
"ground_truth": "89",
"style": "rule"
} | {
"index": "6416",
"split": "train"
} |
math | [
{
"content": "Let $A B C$ be a triangle where $A B=9, B C=10, C A=17$. Let $\\Omega$ be its circumcircle, and let $A_{1}, B_{1}, C_{1}$ be the diametrically opposite points from $A, B, C$, respectively, on $\\Omega$. Find the area of the convex hexagon with the vertices $A, B, C, A_{1}, B_{1}, C_{1}$. Please ou... | {
"ground_truth": "\\frac{1155}{4}",
"style": "rule"
} | {
"index": "6417",
"split": "train"
} |
math | [
{
"content": "Buses travel along a country road, at equal intervals in both directions and at equal speeds. A cyclist, traveling at $16 \\, \\mathrm{km/h}$, begins counting buses from the moment when two buses meet beside him. He counts the 31st bus approaching from the front and the 15th bus from behind, both ... | {
"ground_truth": "46",
"style": "rule"
} | {
"index": "6418",
"split": "train"
} |
math | [
{
"content": "The square \\( STUV \\) is formed by a square bounded by 4 equal rectangles. The perimeter of each rectangle is \\( 40 \\text{ cm} \\). What is the area, in \\( \\text{cm}^2 \\), of the square \\( STUV \\)?\n(a) 400 \n(b) 200 \n(c) 160 \n(d) 100 \n(e) 80 Please output the final answer within \... | {
"ground_truth": "400",
"style": "rule"
} | {
"index": "6419",
"split": "train"
} |
math | [
{
"content": "What is the probability of randomly drawing three different numbers from the set {1, 2, ..., 10} such that their sample variance \\( s^2 \\leqslant 1 \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/15",
"style": "rule"
} | {
"index": "6420",
"split": "train"
} |
math | [
{
"content": "The vectors $\\mathbf{a} = \\begin{pmatrix} 1 \\\\ 1 \\\\ 1 \\end{pmatrix},$ $\\mathbf{b} = \\begin{pmatrix} 2 \\\\ -3 \\\\ 1 \\end{pmatrix},$ and $\\mathbf{c} = \\begin{pmatrix} 4 \\\\ 1 \\\\ -5 \\end{pmatrix}$ are mutually orthogonal. There exist scalars $p,$ $q,$ and $r$ such that\n\\[\\begin{... | {
"ground_truth": "\\left( 2, -\\frac{13}{7}, -\\frac{4}{7} \\right)",
"style": "rule"
} | {
"index": "6421",
"split": "train"
} |
math | [
{
"content": "Given $x, y \\in \\mathbb{N}$, find the maximum value of $y$ such that there exists a unique value of $x$ satisfying the following inequality:\n$$\n\\frac{9}{17}<\\frac{x}{x+y}<\\frac{8}{15}.\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "112",
"style": "rule"
} | {
"index": "6422",
"split": "train"
} |
math | [
{
"content": "Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "6423",
"split": "train"
} |
math | [
{
"content": "The polynomial $1-x+x^2-x^3+\\cdots+x^{16}-x^{17}$ may be written in the form $a_0+a_1y+a_2y^2+\\cdots +a_{16}y^{16}+a_{17}y^{17}$, where $y=x+1$ and the $a_i$'s are constants. Find the value of $a_2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "816",
"style": "rule"
} | {
"index": "6424",
"split": "train"
} |
math | [
{
"content": "Find \\( g(2022) \\) if for any real numbers \\( x, y \\) the following equality holds:\n\n\\[ g(x-y) = g(x) + g(y) - 2021(x+y) \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4086462",
"style": "rule"
} | {
"index": "6425",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer n such that n has exactly 144 positive divisors including 10 consecutive integers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "110880",
"style": "rule"
} | {
"index": "6426",
"split": "train"
} |
math | [
{
"content": "Let $x$ be an angle such that $\\tan x = \\frac{a}{b}$ and $\\tan 2x = \\frac{b}{a + b}.$ Then the least positive value of $x$ equals $\\tan^{-1} k.$ Compute $k.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "6427",
"split": "train"
} |
math | [
{
"content": "Ten points are selected on the positive $x$-axis, $X^+$, and five points are selected on the positive $y$-axis, $Y^+$. The fifty segments connecting the ten points on $X^+$ to the five points on $Y^+$ are drawn. What is the maximum possible number of points of intersection of these fifty segments ... | {
"ground_truth": "450",
"style": "rule"
} | {
"index": "6428",
"split": "train"
} |
math | [
{
"content": "A triangle is divided into 1000 smaller triangles. What is the minimum number of distinct points that can be the vertices of these triangles? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "503",
"style": "rule"
} | {
"index": "6429",
"split": "train"
} |
math | [
{
"content": "Calculate the value of the expression \\( \\sqrt{\\frac{16^{12} + 8^{15}}{16^5 + 8^{16}}} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3\\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "6430",
"split": "train"
} |
math | [
{
"content": "A polygon is said to be friendly if it is regular and it also has angles that, when measured in degrees, are either integers or half-integers (i.e., have a decimal part of exactly 0.5). How many different friendly polygons are there? Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "6431",
"split": "train"
} |
math | [
{
"content": "Calculate the limit of the numerical sequence:\n\n$$\\lim _{n \\rightarrow \\infty} \\frac{\\sqrt{n+6}-\\sqrt{n^{2}-5}}{\\sqrt[3]{n^{3}+3}+\\sqrt[4]{n^{3}+1}}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "6432",
"split": "train"
} |
math | [
{
"content": "What is the result of adding 12.8 to a number that is three times more than 608? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2444.8",
"style": "rule"
} | {
"index": "6433",
"split": "train"
} |
math | [
{
"content": "Let $a, b$ be integers chosen independently and uniformly at random from the set $\\{0,1,2, \\ldots, 80\\}$. Compute the expected value of the remainder when the binomial coefficient $\\binom{a}{b}=\\frac{a!}{b!(a-b)!}$ is divided by 3. Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "\\frac{1816}{6561}",
"style": "rule"
} | {
"index": "6434",
"split": "train"
} |
math | [
{
"content": "Find the number of different recommendation plans for the high school given that 3 male and 2 female students are selected as candidates, where both Russian and Japanese exams must include male participants, and 2 spots are available for Russian, 2 for Japanese, and 1 for Spanish. Please output th... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "6435",
"split": "train"
} |
math | [
{
"content": "The pairwise greatest common divisors of five positive integers are $2,3,4,5,6,7,8, p, q, r$ in some order, for some positive integers $p, q, r$. Compute the minimum possible value of $p+q+r$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "6436",
"split": "train"
} |
math | [
{
"content": "Two-thirds of a number plus two times another number equals 20. If one-fourth of the first number minus the second number equals 2, find these two numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{22}{7}",
"style": "rule"
} | {
"index": "6437",
"split": "train"
} |
math | [
{
"content": "The expression $x^2 + 13x + 30$ can be written as $(x + a)(x + b),$ and the expression $x^2 + 5x - 50$ written as $(x + b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "6438",
"split": "train"
} |
math | [
{
"content": "Given the function f(x) = $\\frac{1}{3}$x^3^ + $\\frac{1−a}{2}$x^2^ - ax - a, x ∈ R, where a > 0.\n(1) Find the monotonic intervals of the function f(x);\n(2) If the function f(x) has exactly two zeros in the interval (-3, 0), find the range of values for a;\n(3) When a = 1, let the maximum value ... | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "6439",
"split": "train"
} |
math | [
{
"content": "Let $ n$ be a positive integer and $ [ \\ n ] = a.$ Find the largest integer $ n$ such that the following two conditions are satisfied:\r\n\r $ (1)$ $ n$ is not a perfect square;\r\n\r $ (2)$ $ a^{3}$ divides $ n^{2}$ . Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "6440",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{a} = \\begin{pmatrix} 3 \\\\ 4 \\\\ 0 \\end{pmatrix}$ and $\\mathbf{b} = \\begin{pmatrix} -1 \\\\ 1 \\\\ -1 \\end{pmatrix}.$ Find the unit vector $\\mathbf{v}$ so that $\\mathbf{b}$ bisects the angle between $\\mathbf{a}$ and $\\mathbf{v}.$ Please output the final answer within \\bo... | {
"ground_truth": "\\begin{pmatrix} -11/15 \\\\ -2/3 \\\\ -2/15 \\end{pmatrix}",
"style": "rule"
} | {
"index": "6441",
"split": "train"
} |
math | [
{
"content": "The greatest common divisor of two positive integers is $(x+7)$ and their least common multiple is $x(x+7)$, where $x$ is a positive integer. If one of the integers is 56, what is the smallest possible value of the other one? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "294",
"style": "rule"
} | {
"index": "6442",
"split": "train"
} |
math | [
{
"content": "How many such five-digit Shenma numbers exist, where the middle digit is the smallest, the digits increase as they move away from the middle, and all the digits are different? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1512",
"style": "rule"
} | {
"index": "6443",
"split": "train"
} |
math | [
{
"content": "Two teams, Team A and Team B, are playing in a basketball finals series that uses a \"best of seven\" format (the first team to win four games wins the series and the finals end). Based on previous game results, Team A's home and away schedule is arranged as \"home, home, away, away, home, away, h... | {
"ground_truth": "0.18",
"style": "rule"
} | {
"index": "6444",
"split": "train"
} |
math | [
{
"content": "Three cards, each with a positive integer written on it, are lying face-down on a table. Casey, Stacy, and Tracy are told that\n\n(a) the numbers are all different,\n(b) they sum to $13$, and\n(c) they are in increasing order, left to right.\nFirst, Casey looks at the number on the leftmost card a... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "6445",
"split": "train"
} |
math | [
{
"content": "If four coins are tossed at the same time, what is the probability of getting exactly three heads and one tail? Follow this by rolling a six-sided die. What is the probability that the die shows a number greater than 4? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{12}",
"style": "rule"
} | {
"index": "6446",
"split": "train"
} |
math | [
{
"content": "A 0-1 sequence of length $2^k$ is given. Alice can pick a member from the sequence, and reveal it (its place and its value) to Bob. Find the largest number $s$ for which Bob can always pick $s$ members of the sequence, and guess all their values correctly.\n\nAlice and Bob can discuss a strategy b... | {
"ground_truth": "k+1",
"style": "rule"
} | {
"index": "6447",
"split": "train"
} |
math | [
{
"content": "In the diagram, the circle has radius \\(\\sqrt{5}\\). Rectangle \\(ABCD\\) has \\(C\\) and \\(D\\) on the circle, \\(A\\) and \\(B\\) outside the circle, and \\(AB\\) tangent to the circle. What is the area of \\(ABCD\\) if \\(AB = 4AD\\)? Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "16/5",
"style": "rule"
} | {
"index": "6448",
"split": "train"
} |
math | [
{
"content": "If \\( x \\) is a real number and \\( \\lceil x \\rceil = 14 \\), how many possible values are there for \\( \\lceil x^2 \\rceil \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "6449",
"split": "train"
} |
math | [
{
"content": "In the next 3 days, a meteorological station forecasts the weather with an accuracy rate of 0.8. The probability that the forecast is accurate for at least two consecutive days is ___. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.768",
"style": "rule"
} | {
"index": "6450",
"split": "train"
} |
math | [
{
"content": "A list of seven positive integers has a median of 5 and a mean of 15. What is the maximum possible value of the list's largest element? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "87",
"style": "rule"
} | {
"index": "6451",
"split": "train"
} |
math | [
{
"content": "In the polar coordinate system, the equation of curve C is $\\rho^2= \\frac{3}{1+2\\sin^2\\theta}$. Point R is at $(2\\sqrt{2}, \\frac{\\pi}{4})$. \nP is a moving point on curve C, and side PQ of rectangle PQRS, with PR as its diagonal, is perpendicular to the polar axis. Find the maximum and min... | {
"ground_truth": "\\frac{\\pi}{6}",
"style": "rule"
} | {
"index": "6452",
"split": "train"
} |
math | [
{
"content": "Find the approximate change in the function \\( y = 3x^{2} + 2 \\) at \\( x = 2 \\) with \\( \\Delta x = 0.001 \\). Determine the absolute and relative errors of the calculation. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.012",
"style": "rule"
} | {
"index": "6453",
"split": "train"
} |
math | [
{
"content": "How many of the first $2018$ numbers in the sequence $101, 1001, 10001, 100001, \\dots$ are divisible by $101$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "505",
"style": "rule"
} | {
"index": "6454",
"split": "train"
} |
math | [
{
"content": "A large sphere has a volume of $288\\pi$ cubic units. A smaller sphere has a volume which is $12.5\\%$ of the volume of the larger sphere. What is the ratio of the radius of the smaller sphere to the radius of the larger sphere? Express your answer as a common fraction. Please output the final a... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "6455",
"split": "train"
} |
math | [
{
"content": "Let \\(p\\) and \\(q\\) be relatively prime positive integers such that \\(\\dfrac pq = \\dfrac1{2^1} + \\dfrac2{4^2} + \\dfrac3{2^3} + \\dfrac4{4^4} + \\dfrac5{2^5} + \\dfrac6{4^6} + \\cdots\\), where the numerators always increase by 1, and the denominators alternate between powers of 2 and 4, w... | {
"ground_truth": "169",
"style": "rule"
} | {
"index": "6456",
"split": "train"
} |
math | [
{
"content": "The square root of $x$ is greater than 3 and less than 4. How many integer values of $x$ satisfy this condition? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "6457",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles A, B, and C are a, b, and c, respectively. If $C= \\frac {\\pi}{3}, b= \\sqrt {2}, c= \\sqrt {3}$, find the measure of angle A. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5\\pi}{12}",
"style": "rule"
} | {
"index": "6458",
"split": "train"
} |
math | [
{
"content": "The opposite of $-23$ is ______; the reciprocal is ______; the absolute value is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "6459",
"split": "train"
} |
math | [
{
"content": "Let $[x]$ represent the greatest integer less than or equal to the real number $x$. Define the sets\n$$\n\\begin{array}{l}\nA=\\{y \\mid y=[x]+[2x]+[4x], x \\in \\mathbf{R}\\}, \\\\\nB=\\{1,2, \\cdots, 2019\\}.\n\\end{array}\n$$\n\nFind the number of elements in the intersection $A \\cap B$. Pleas... | {
"ground_truth": "1154",
"style": "rule"
} | {
"index": "6460",
"split": "train"
} |
math | [
{
"content": "A positive integer \\( n \\) is said to be 'good' if \\( n^2 - 1 \\) can be written as the product of three distinct prime numbers. Find the sum of the five smallest 'good' integers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "104",
"style": "rule"
} | {
"index": "6461",
"split": "train"
} |
math | [
{
"content": "While waiting for their food at a restaurant in Harvard Square, Ana and Banana draw 3 squares $\\square_{1}, \\square_{2}, \\square_{3}$ on one of their napkins. Starting with Ana, they take turns filling in the squares with integers from the set $\\{1,2,3,4,5\\}$ such that no integer is used more... | {
"ground_truth": "\\[ 541 \\]",
"style": "rule"
} | {
"index": "6462",
"split": "train"
} |
math | [
{
"content": "Given $|a|=3$, $|b-2|=9$, and $a+b > 0$, find the value of $ab$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-33",
"style": "rule"
} | {
"index": "6463",
"split": "train"
} |
math | [
{
"content": "The lengths of the sides of a triangle with positive area are $\\log_{10}12$, $\\log_{10}75$, and $\\log_{10}n$, where $n$ is a positive integer. Find the number of possible values for $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "893",
"style": "rule"
} | {
"index": "6464",
"split": "train"
} |
math | [
{
"content": "Let $d$ be a randomly chosen divisor of 2016. Find the expected value of $\\frac{d^{2}}{d^{2}+2016}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "6465",
"split": "train"
} |
math | [
{
"content": "In triangle \\(ABC\\), the side lengths are 4, 5, and \\(\\sqrt{17}\\). Find the area of the region consisting of those and only those points \\(X\\) inside triangle \\(ABC\\) for which the condition \\(XA^{2} + XB^{2} + XC^{2} \\leq 21\\) is satisfied. Please output the final answer within \\boxe... | {
"ground_truth": "\\frac{5 \\pi}{9}",
"style": "rule"
} | {
"index": "6466",
"split": "train"
} |
math | [
{
"content": "Compute $\\arcsin 0.$ Express your answer in radians. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "6467",
"split": "train"
} |
math | [
{
"content": "What is the median of the numbers in the list $19^{20}, \\frac{20}{19}, 20^{19}, 2019, 20 \\times 19$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2019",
"style": "rule"
} | {
"index": "6468",
"split": "train"
} |
math | [
{
"content": "Find an ordered pair $(a, b)$ of real numbers for which $x^{2}+a x+b$ has a non-real root whose cube is 343. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(7, 49)",
"style": "rule"
} | {
"index": "6469",
"split": "train"
} |
math | [
{
"content": "The gravitational force that Earth exerts on an object is inversely proportional to the square of the distance between the center of the Earth and the object. When Alice is on the surface of Earth, 6,000 miles from the center, the gravitational force is 400 Newtons. What is the gravitational force... | {
"ground_truth": "\\frac{1}{9}",
"style": "rule"
} | {
"index": "6470",
"split": "train"
} |
math | [
{
"content": "1. Given $\\tan \\frac{\\alpha}{2} = \\frac{1}{2}$, find the value of $\\sin\\left(\\alpha + \\frac{\\pi}{6}\\right)$.\n2. Given $\\alpha \\in \\left(\\pi, \\frac{3\\pi}{2}\\right)$ and $\\cos\\alpha = -\\frac{5}{13}$, $\\tan \\frac{\\beta}{2} = \\frac{1}{3}$, find the value of $\\cos\\left(\\frac... | {
"ground_truth": "-\\frac{17\\sqrt{13}}{65}",
"style": "rule"
} | {
"index": "6471",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$, where $|\\overrightarrow{a}|= \\sqrt {2}$, $|\\overrightarrow{b}|=2$, and $(\\overrightarrow{a}-\\overrightarrow{b})\\perp \\overrightarrow{a}$, calculate the angle between vector $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Please... | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "6472",
"split": "train"
} |
math | [
{
"content": "8 people are seated in two rows with 4 people in each row. Among them, 2 specific people must sit in the front row, and 1 specific person must sit in the back row. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5760",
"style": "rule"
} | {
"index": "6473",
"split": "train"
} |
math | [
{
"content": "It is known that none of the digits of a three-digit number is zero, and the sum of all possible two-digit numbers composed of the digits of this number is equal to the number itself. Find the largest such three-digit number. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "396",
"style": "rule"
} | {
"index": "6474",
"split": "train"
} |
math | [
{
"content": "Let $A B C$ be a triangle with $A B=A C=\\frac{25}{14} B C$. Let $M$ denote the midpoint of $\\overline{B C}$ and let $X$ and $Y$ denote the projections of $M$ onto $\\overline{A B}$ and $\\overline{A C}$, respectively. If the areas of triangle $A B C$ and quadrilateral $A X M Y$ are both positive... | {
"ground_truth": "1201",
"style": "rule"
} | {
"index": "6475",
"split": "train"
} |
math | [
{
"content": "Given that the magnitude of the star Altair is $0.75$ and the magnitude of the star Vega is $0$, determine the ratio of the luminosity of Altair to Vega. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10^{-\\frac{3}{10}}",
"style": "rule"
} | {
"index": "6476",
"split": "train"
} |
math | [
{
"content": "There are 4 pieces of part $A$ weighing 5 tons each, 6 pieces of part $B$ weighing 4 tons each, 11 pieces of part $C$ weighing 3 tons each, and 7 pieces of part $D$ weighing 1 ton each. If all the parts are to be transported at once, what is the minimum number of trucks, each with a capacity of 6 ... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "6477",
"split": "train"
} |
math | [
{
"content": "There are 6 locked suitcases and 6 keys for them. However, it is unknown which key opens which suitcase. What is the minimum number of attempts needed to ensure that all suitcases are opened? How many attempts are needed if there are 10 suitcases and 10 keys? Please output the final answer within ... | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "6478",
"split": "train"
} |
math | [
{
"content": "A bowling ball cannot weigh more than 16 pounds and must have a diameter of $8 \\frac{1}{2}$ inches. How many square inches are in the surface area of a bowling ball before the finger holes are drilled? Express your answer as a common fraction in terms of $\\pi$. Please output the final answer wit... | {
"ground_truth": "\\frac{289\\pi}{4}",
"style": "rule"
} | {
"index": "6479",
"split": "train"
} |
math | [
{
"content": "George is about to get a certain amount of change less than one dollar from the cash register. If he gets the most quarters possible and the rest in pennies, he would need to receive 3 pennies to meet the amount. If he gets the most dimes possible and the rest in pennies, he would need to receive ... | {
"ground_truth": "106",
"style": "rule"
} | {
"index": "6480",
"split": "train"
} |
math | [
{
"content": "At the start of this month, Mathilde and Salah each had 100 coins. For Mathilde, this was $25 \\%$ more coins than she had at the start of last month. For Salah, this was $20 \\%$ fewer coins than he had at the start of last month. What was the total number of coins that they had at the start of l... | {
"ground_truth": "205",
"style": "rule"
} | {
"index": "6481",
"split": "train"
} |
math | [
{
"content": "Convert 89 into a binary number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1011001_{(2)}",
"style": "rule"
} | {
"index": "6482",
"split": "train"
} |
math | [
{
"content": "What perfect-square integer is closest to 273? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "289",
"style": "rule"
} | {
"index": "6483",
"split": "train"
} |
math | [
{
"content": "A point $Q$ is chosen in the interior of $\\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\\triangle DEF$, the resulting smaller triangles $u_{1}$, $u_{2}$, and $u_{3}$ have areas $9$, $16$, and $36$, respectively. Find the area of $\\triangle DEF$. Please outp... | {
"ground_truth": "169",
"style": "rule"
} | {
"index": "6484",
"split": "train"
} |
math | [
{
"content": "What is the units digit of $23^{23}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "6485",
"split": "train"
} |
math | [
{
"content": "Cube $ABCDEFGH,$ labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\\overline{AB}$ and $\\overline{CG}$ respectively. The plane divides the cube into two solids. Find the volume of the larger of the two solids.\n\n[asy]\... | {
"ground_truth": "\\frac{41}{48}",
"style": "rule"
} | {
"index": "6486",
"split": "train"
} |
math | [
{
"content": "Suppose that the angles of triangle $ABC$ satisfy\n\\[\\cos 3A + \\cos 3B + \\cos 3C = 1.\\]\nTwo sides of the triangle have lengths 8 and 15. Find the maximum length of the third side assuming one of the angles is $150^\\circ$. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "\\sqrt{289 + 120\\sqrt{3}}",
"style": "rule"
} | {
"index": "6487",
"split": "train"
} |
math | [
{
"content": "In a game of Chomp, two players alternately take bites from a 5-by-7 grid of unit squares. To take a bite, a player chooses one of the remaining squares, then removes (\"eats\") all squares in the quadrant defined by the left edge (extended upward) and the lower edge (extended rightward) of the ch... | {
"ground_truth": "792",
"style": "rule"
} | {
"index": "6488",
"split": "train"
} |
math | [
{
"content": "Given the function\\\\(f(x)= \\\\begin{cases} (-1)^{n}\\\\sin \\\\dfrac {πx}{2}+2n,\\\\;x∈\\[2n,2n+1) \\\\\\\\ (-1)^{n+1}\\\\sin \\\\dfrac {πx}{2}+2n+2,\\\\;x∈\\[2n+1,2n+2)\\\\end{cases}(n∈N)\\\\)，if the sequence\\\\(\\\\{a\\_{m}\\\\}\\\\) satisfies\\\\(a\\_{m}=f(m)\\\\;(m∈N^{\\*})\\\\)，and the su... | {
"ground_truth": "909",
"style": "rule"
} | {
"index": "6489",
"split": "train"
} |
math | [
{
"content": "A smaller regular tetrahedron is formed by joining the midpoints of the edges of a larger regular tetrahedron. Determine the ratio of the volume of the smaller tetrahedron to the volume of the larger tetrahedron. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "6490",
"split": "train"
} |
math | [
{
"content": "The vertices of an equilateral triangle lie on the hyperbola $xy=1$, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "6491",
"split": "train"
} |
math | [
{
"content": "Timur and Alexander are counting the trees growing around the house. Both move in the same direction, but they start counting from different trees. What is the number of trees growing around the house if the tree that Timur called the 12th, Alexander counted as $33-m$, and the tree that Timur call... | {
"ground_truth": "76",
"style": "rule"
} | {
"index": "6492",
"split": "train"
} |
math | [
{
"content": "For how many integer values of $n$ between 1 and 474 inclusive does the decimal representation of $\\frac{n}{475}$ terminate? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "6493",
"split": "train"
} |
math | [
{
"content": "Solve the equations. \n$(3+x) \\times 30\\% = 4.8$ \n$5 : x = \\frac{9}{2} : \\frac{8}{5}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{16}{9}",
"style": "rule"
} | {
"index": "6494",
"split": "train"
} |
math | [
{
"content": "In how many ways can a President, a Vice-President, and a Secretary be chosen from a group of 6 people, given the following constraints:\n- The President must be one of the first three members of the group.\n- The Vice-President must be one of the last four members of the group.\n- No person can h... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "6495",
"split": "train"
} |
math | [
{
"content": "Seven test scores have a mean of $85$, a median of $88$, and a mode of $90$. Calculate the sum of the three lowest test scores. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "237",
"style": "rule"
} | {
"index": "6496",
"split": "train"
} |
math | [
{
"content": "What is $\\frac{1}{3}$ of $\\frac{1}{4}$ of $\\frac{1}{5}$ of 60? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "6497",
"split": "train"
} |
math | [
{
"content": "Define a function $f(x)$ on $\\mathbb{R}$ that satisfies $f(x+6)=f(x)$. When $x \\in [-3,3)$, $f(x)=\\begin{cases} -(x+2)^2, & -3 \\leqslant x < -1 \\\\ x, & -1 \\leqslant x < 3 \\end{cases}$. Find $f(4)$ and the value of $f(1)+f(2)+f(3)+\\ldots+f(2016)+f(2017)$. Please output the final answer wit... | {
"ground_truth": "337",
"style": "rule"
} | {
"index": "6498",
"split": "train"
} |
math | [
{
"content": "Given two lines $l_1: x+3y-3m^2=0$ and $l_2: 2x+y-m^2-5m=0$ intersect at point $P$ ($m \\in \\mathbb{R}$). \n(1) Express the coordinates of the intersection point $P$ of lines $l_1$ and $l_2$ in terms of $m$. \n(2) For what value of $m$ is the distance from point $P$ to the line $x+y+3=0$ the sh... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "6499",
"split": "train"
} |
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