data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Distinct ways to distribute $7$ identical balls into $4$ distinct boxes such that no box is empty. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "37600",
"split": "train"
} |
math | [
{
"content": "Last year, 10% of the net income from our school's ball was allocated to clubs for purchases, and the remaining part covered the rental cost of the sports field. This year, we cannot sell more tickets, and the rental cost remains the same, so increasing the share for the clubs can only be achieved... | {
"ground_truth": "12.5",
"style": "rule"
} | {
"index": "37601",
"split": "train"
} |
math | [
{
"content": "Victor has a drawer with 6 socks of 3 different types: 2 complex socks, 2 synthetic socks, and 2 trigonometric socks. He repeatedly draws 2 socks at a time from the drawer at random, and stops if the socks are of the same type. However, Victor is 'synthetic-complex type-blind', so he also stops if... | {
"ground_truth": "\\frac{3}{7}",
"style": "rule"
} | {
"index": "37602",
"split": "train"
} |
math | [
{
"content": "Given that $P$ is any point on the circle $C$: $(x-2)^{2}+(y-2)^{2}=1$, and $Q$ is any point on the line $l$: $x+y=1$, find the minimum value of $| \\overrightarrow{OP}+ \\overrightarrow{OQ}|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5 \\sqrt{2}-2}{2}",
"style": "rule"
} | {
"index": "37603",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=2\\sin \\omega x\\cos \\omega x-2\\sqrt{3}\\cos^{2}\\omega x+\\sqrt{3}$ ($\\omega > 0$), and the distance between two adjacent axes of symmetry of the graph of $y=f(x)$ is $\\frac{\\pi}{2}$.\n\n(Ⅰ) Find the interval of monotonic increase for the function $f(x)$;\n\n(Ⅱ) Giv... | {
"ground_truth": "3\\sqrt{3}",
"style": "rule"
} | {
"index": "37604",
"split": "train"
} |
math | [
{
"content": "Alexio has 120 cards numbered 1-120, inclusive, and places them in a box. Alexio then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 3, 4, or 7? Express your answer as a common fraction. Please output the final answer within \... | {
"ground_truth": "\\dfrac{69}{120}",
"style": "rule"
} | {
"index": "37605",
"split": "train"
} |
math | [
{
"content": "Let $a \\star b=ab-2$. Compute the remainder when $(((579 \\star 569) \\star 559) \\star \\cdots \\star 19) \\star 9$ is divided by 100. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "37606",
"split": "train"
} |
math | [
{
"content": "Let $A$ and $B$ be the endpoints of a semicircular arc of radius $2$. The arc is divided into seven congruent arcs by six equally spaced points $C_1$, $C_2$, $\\dots$, $C_6$. All chords of the form $\\overline {AC_i}$ or $\\overline {BC_i}$ are drawn. Find the product of the lengths of these twelv... | {
"ground_truth": "28672",
"style": "rule"
} | {
"index": "37607",
"split": "train"
} |
math | [
{
"content": "In acute triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given that $a \\neq b$, $c = \\sqrt{3}$, and $\\sqrt{3} \\cos^2 A - \\sqrt{3} \\cos^2 B = \\sin A \\cos A - \\sin B \\cos B$.\n\n(I) Find the measure of angle $C$;\n\n(II) If $\\sin A = \... | {
"ground_truth": "\\frac{24\\sqrt{3} + 18}{25}",
"style": "rule"
} | {
"index": "37608",
"split": "train"
} |
math | [
{
"content": "In Pascal's Triangle, each entry is the sum of the two entries above it. The first few rows of the triangle are shown below.\n\\[\\begin{array}{c@{\\hspace{8em}} c@{\\hspace{6pt}}c@{\\hspace{6pt}}c@{\\hspace{6pt}}c@{\\hspace{4pt}}c@{\\hspace{2pt}} c@{\\hspace{2pt}}c@{\\hspace{2pt}}c@{\\hspace{2pt}... | {
"ground_truth": "62",
"style": "rule"
} | {
"index": "37609",
"split": "train"
} |
math | [
{
"content": "In square $ABCD$ with side length $2$ , let $M$ be the midpoint of $AB$ . Let $N$ be a point on $AD$ such that $AN = 2ND$ . Let point $P$ be the intersection of segment $MN$ and diagonal $AC$ . Find the area of triangle $BPM$ .\n\n*Proposed by Jacob Xu* Please output the final ans... | {
"ground_truth": "2/7",
"style": "rule"
} | {
"index": "37610",
"split": "train"
} |
math | [
{
"content": "There are positive integers that have these properties:\n\n$\\bullet$ I. The sum of the squares of their digits is $50,$ and\n\n$\\bullet$ II. Each digit is larger than the one on its left.\n\nWhat is the product of the digits of the largest integer with both properties? Please output the final an... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "37611",
"split": "train"
} |
math | [
{
"content": "Given real numbers $a$ and $b \\gt 0$, if $a+2b=1$, then the minimum value of $\\frac{3}{b}+\\frac{1}{a}$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7 + 2\\sqrt{6}",
"style": "rule"
} | {
"index": "37612",
"split": "train"
} |
math | [
{
"content": "Let $A B C D$ be a rectangle, and let $E$ and $F$ be points on segment $A B$ such that $A E=E F=F B$. If $C E$ intersects the line $A D$ at $P$, and $P F$ intersects $B C$ at $Q$, determine the ratio of $B Q$ to $C Q$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "37613",
"split": "train"
} |
math | [
{
"content": "The real numbers $a,$ $b,$ $c,$ and $d$ satisfy\n\\[a^2 + b^2 + c^2 + 1 = d + \\sqrt{a + b + c - d}.\\]Find $d.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{4}",
"style": "rule"
} | {
"index": "37614",
"split": "train"
} |
math | [
{
"content": "Investigate the function $f(x)=e^{x}-e^{-x}+\\sin x+1$, and let $a=\\frac{2023+2024}{2}=2023.5$. Determine the value of $f(-2023)+f(-2022)+\\ldots +f(2022)+f(2023)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4047",
"style": "rule"
} | {
"index": "37615",
"split": "train"
} |
math | [
{
"content": "$ABCD$ is a square that is made up of two identical rectangles and two squares of area $4 \\mathrm{~cm}^{2}$ and $16 \\mathrm{cm}^{2}$. What is the area, in $\\mathrm{cm}^{2}$, of the square $ABCD$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "37616",
"split": "train"
} |
math | [
{
"content": "How many positive odd integers greater than 1 and less than 200 are square-free? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "37617",
"split": "train"
} |
math | [
{
"content": "When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "37618",
"split": "train"
} |
math | [
{
"content": "A function \\( f \\) satisfies the equation \\((n - 2019) f(n) - f(2019 - n) = 2019\\) for every integer \\( n \\).\nWhat is the value of \\( f(2019) \\)?\nA) 0\nB) 1\nC) \\(2018 \\times 2019\\)\nD) \\(2019^2\\)\nE) \\(2019 \\times 2020\\) Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "2019 \\times 2018",
"style": "rule"
} | {
"index": "37619",
"split": "train"
} |
math | [
{
"content": "Find the value of $x$ such that $\\sqrt{x+ 7} = 9$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "74",
"style": "rule"
} | {
"index": "37620",
"split": "train"
} |
math | [
{
"content": "The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle? Express your answer in terms of pi and in simplest radical form. Please output the final answer within \\boxed{... | {
"ground_truth": "\\frac{3\\sqrt{3}}{\\pi}",
"style": "rule"
} | {
"index": "37621",
"split": "train"
} |
math | [
{
"content": "How many unordered pairs of edges of a given octahedron determine a plane? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "66",
"style": "rule"
} | {
"index": "37622",
"split": "train"
} |
math | [
{
"content": "Given a cube with an unknown volume, two of its dimensions are increased by $1$ and the third is decreased by $2$, and the volume of the resulting rectangular solid is $27$ less than that of the cube. Determine the volume of the original cube. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "37623",
"split": "train"
} |
math | [
{
"content": "Find the largest positive integer $N $ for which one can choose $N $ distinct numbers from the set ${1,2,3,...,100}$ such that neither the sum nor the product of any two different chosen numbers is divisible by $100$ .\n\nProposed by Mikhail Evdokimov Please output the final answer within \... | {
"ground_truth": "44",
"style": "rule"
} | {
"index": "37624",
"split": "train"
} |
math | [
{
"content": "Among all integers that alternate between 1 and 0, starting and ending with 1 (e.g., 101, 10101, 10101…), how many are prime numbers? Why? And list all the prime numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "101",
"style": "rule"
} | {
"index": "37625",
"split": "train"
} |
math | [
{
"content": "Given that $y = f(x) + x^2$ is an odd function, and $f(1) = 1$, if $g(x) = f(x) + 2$, then $g(-1) = \\_\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "37626",
"split": "train"
} |
math | [
{
"content": "Using systematic sampling to select 20 students from 1000, the students are randomly numbered from 000 to 999 and grouped: the first group ranges from 000 to 049, the second group from 050 to 099, ..., and the twentieth group from 950 to 999. If the number 122 from the third group is selected, the... | {
"ground_truth": "872",
"style": "rule"
} | {
"index": "37627",
"split": "train"
} |
math | [
{
"content": "A positive integer $n$ is a multiple of 7. The square root of $n$ is between 17 and 18. How many possible values of $n$ are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "37628",
"split": "train"
} |
math | [
{
"content": "Given that $P$ is a point inside rectangle $ABCD$, the distances from $P$ to the vertices of the rectangle are $PA = 5$ inches, $PD = 12$ inches, and $PC = 13$ inches. Find $PB$, which is $x$ inches. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5\\sqrt{2}",
"style": "rule"
} | {
"index": "37629",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer $k$ such that, for any subset $A$ of $S=\\{1,2,\\ldots,2012\\}$ with $|A|=k$ , there exist three elements $x,y,z$ in $A$ such that $x=a+b$ , $y=b+c$ , $z=c+a$ , where $a,b,c$ are in $S$ and are distinct integers.\n\n*Proposed by Huawei Zhu* Plea... | {
"ground_truth": "1008",
"style": "rule"
} | {
"index": "37630",
"split": "train"
} |
math | [
{
"content": "Compute\n\\[\\left( 1 + \\cos \\frac {\\pi}{8} \\right) \\left( 1 + \\cos \\frac {3 \\pi}{8} \\right) \\left( 1 + \\cos \\frac {5 \\pi}{8} \\right) \\left( 1 + \\cos \\frac {7 \\pi}{8} \\right).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "37631",
"split": "train"
} |
math | [
{
"content": "A palindrome between $1000$ and $10000$ is chosen at random. What is the probability that it is divisible by $7$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{5}",
"style": "rule"
} | {
"index": "37632",
"split": "train"
} |
math | [
{
"content": "Find the number of positive integers $n$ that satisfy\n\\[(n - 1)(n - 3)(n - 5) \\dotsm (n - 97) < 0.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "37633",
"split": "train"
} |
math | [
{
"content": "Arrange 8 people, including A and B, to work for 4 days. If 2 people are arranged each day, the probability that A and B are arranged on the same day is ______. (Express the result as a fraction) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{7}",
"style": "rule"
} | {
"index": "37634",
"split": "train"
} |
math | [
{
"content": "$DEB$ is a chord of a circle such that $DE=3$ and $EB=5$ . Let $O$ be the centre of the circle. Join $OE$ and extend $OE$ to cut the circle at $C$ . (See diagram). Given $EC=1$ , find the radius of the circle.\n\n[asy]\nsize(6cm);\n\npair O = (0,0), B = dir(110), D = dir(30), E = 0.4 *... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "37635",
"split": "train"
} |
math | [
{
"content": "Given that a positive integer \\( A \\) can be factorized as \\( A = 2^{\\alpha} \\times 3^{\\beta} \\times 5^{\\gamma} \\), where \\( \\alpha \\), \\( \\beta \\), and \\( \\gamma \\) are natural numbers, and given that half of \\( A \\) is a perfect square, one-third of \\( A \\) is a perfect cub... | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "37636",
"split": "train"
} |
math | [
{
"content": "What is the sum of the two solutions to the equation $54-15x-x^2=0$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-15",
"style": "rule"
} | {
"index": "37637",
"split": "train"
} |
math | [
{
"content": "What is the smallest integer that can be placed in the box so that $\\frac{1}{2} < \\frac{\\square}{9}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "37638",
"split": "train"
} |
math | [
{
"content": "Given that \\( r, s, t \\) are integers, and the set \\( \\{a \\mid a = 2^r + 2^s + 2^t, 0 \\leq t < s < r\\} \\) forms a sequence \\(\\{a_n\\} \\) from smallest to largest as \\(7, 11, 13, 14, \\cdots\\), find \\( a_{36} \\). Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "131",
"style": "rule"
} | {
"index": "37639",
"split": "train"
} |
math | [
{
"content": "Two of the altitudes of an acute triangle divide the sides into segments of lengths $7, 4, 3$, and $y$ units, as shown. Calculate the value of $y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{12}{7}",
"style": "rule"
} | {
"index": "37640",
"split": "train"
} |
math | [
{
"content": "Find \\(g(2022)\\) if for any real numbers \\(x\\) and \\(y\\) the following equation holds:\n$$\ng(x-y)=2022(g(x)+g(y))-2021 x y .\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2043231",
"style": "rule"
} | {
"index": "37641",
"split": "train"
} |
math | [
{
"content": "Place each of the digits 3, 4, 5, and 6 in exactly one square to make the smallest possible product. Arrange the digits such that two numbers are formed and multiplied. How should the digits be placed, and what is the minimum product? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "1610",
"style": "rule"
} | {
"index": "37642",
"split": "train"
} |
math | [
{
"content": "In triangle ABC, the altitude, angle bisector and median from C divide the angle C into four equal angles. Find angle B. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "37643",
"split": "train"
} |
math | [
{
"content": "In a cube $ABCDEFGH$ where each side has length $2$ units. Find $\\sin \\angle GAC$. (Consider this by extending the calculations needed for finding $\\sin \\angle HAC$) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "37644",
"split": "train"
} |
math | [
{
"content": "The sequence of real numbers \\( a_1, a_2, \\cdots, a_n, \\cdots \\) is defined by the following equation: \\( a_{n+1} = 2^n - 3a_n \\) for \\( n = 0, 1, 2, \\cdots \\).\n1. Find an expression for \\( a_n \\) in terms of \\( a_0 \\) and \\( n \\).\n2. Find \\( a_0 \\) such that \\( a_{n+1} > a_n \... | {
"ground_truth": "\\frac{1}{5}",
"style": "rule"
} | {
"index": "37645",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = x^3 - 3x^2 - 9x + 1$,\n(1) Determine the monotonicity of the function on the interval $[-4, 4]$.\n(2) Calculate the function's local maximum and minimum values as well as the absolute maximum and minimum values on the interval $[-4, 4]$. Please output the final answer wi... | {
"ground_truth": "-75",
"style": "rule"
} | {
"index": "37646",
"split": "train"
} |
math | [
{
"content": "(1) Given that $\\tan \\alpha = -2$, calculate the value of $\\dfrac {3\\sin \\alpha + 2\\cos \\alpha}{5\\cos \\alpha - \\sin \\alpha}$.\n(2) Given that $\\sin \\alpha = \\dfrac {2\\sqrt{5}}{5}$, calculate the value of $\\tan (\\alpha + \\pi) + \\dfrac {\\sin \\left( \\dfrac {5\\pi}{2} + \\alpha \... | {
"ground_truth": "-\\dfrac{5}{2}",
"style": "rule"
} | {
"index": "37647",
"split": "train"
} |
math | [
{
"content": "The six faces of a three-inch wooden cube are each painted red. The cube is then cut into one-inch cubes along the lines shown in the diagram. How many of the one-inch cubes have red paint on at least two faces? [asy]\n\npair A,B,C,D,E,F,G;\n\npair a,c,d,f,g,i,j,l,m,o,p,r,s,u,v,x,b,h;\n\nA=(0.8,1)... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "37648",
"split": "train"
} |
math | [
{
"content": "Square $IJKL$ has one vertex on each side of square $WXYZ$. Point $I$ is on $WZ$ such that $WI = 9 \\cdot IZ$. Determine the ratio of the area of square $IJKL$ to the area of square $WXYZ$.\nA) $\\frac{1}{20}$ \nB) $\\frac{1}{50}$ \nC) $\\frac{1}{40}$ \nD) $\\frac{1}{64}$ \nE) $\\frac{1}{80}$ ... | {
"ground_truth": "\\frac{1}{50}",
"style": "rule"
} | {
"index": "37649",
"split": "train"
} |
math | [
{
"content": "At night, there was a heavy snowfall. In the morning, Xiao Long and his father measured the length of a circular path in the garden by walking. They started from the same point and walked in the same direction. Xiao Long's step length is 54 cm, and his father's step length is 72 cm. Each of them w... | {
"ground_truth": "21.6",
"style": "rule"
} | {
"index": "37650",
"split": "train"
} |
math | [
{
"content": "Three distinct numbers are selected simultaneously and at random from the set $\\{1, 2, 3, 4, 6\\}$. What is the probability that at least one number divides another among the selected numbers? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "\\frac{9}{10}",
"style": "rule"
} | {
"index": "37651",
"split": "train"
} |
math | [
{
"content": "Let $S$ be a subset of $\\{1,2,3,\\ldots,1989\\}$ such that no two members of $S$ differ by $4$ or $7$. What is the largest number of elements $S$ can have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "905",
"style": "rule"
} | {
"index": "37652",
"split": "train"
} |
math | [
{
"content": "Given that there are \\( c \\) prime numbers less than 100 such that their unit digits are not square numbers, find the values of \\( c \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "37653",
"split": "train"
} |
math | [
{
"content": "Six congruent circles form a ring with each circle externally tangent to the two circles adjacent to it. All six circles are internally tangent to a circle $\\cal C$ with radius 30. Let $K$ be the area of the region inside $\\cal C$ and outside all of the six circles in the ring. Find $\\lfloor... | {
"ground_truth": "942",
"style": "rule"
} | {
"index": "37654",
"split": "train"
} |
math | [
{
"content": "Consider a cube $A B C D E F G H$, where $A B C D$ and $E F G H$ are faces, and segments $A E, B F, C G, D H$ are edges of the cube. Let $P$ be the center of face $E F G H$, and let $O$ be the center of the cube. Given that $A G=1$, determine the area of triangle $A O P$. Please output the final a... | {
"ground_truth": "$\\frac{\\sqrt{2}}{24}$",
"style": "rule"
} | {
"index": "37655",
"split": "train"
} |
math | [
{
"content": "Let $f(x)=x^{2}+a x+b$ and $g(x)=x^{2}+c x+d$ be two distinct real polynomials such that the $x$-coordinate of the vertex of $f$ is a root of $g$, the $x$-coordinate of the vertex of $g$ is a root of $f$ and both $f$ and $g$ have the same minimum value. If the graphs of the two polynomials interse... | {
"ground_truth": "-8048",
"style": "rule"
} | {
"index": "37656",
"split": "train"
} |
math | [
{
"content": "Given an equilateral triangle ABC, a student starts from point A and moves the chess piece using a dice-rolling method, where the direction of the movement is determined by the dice roll. Each time the dice is rolled, the chess piece is moved from one vertex of the triangle to another vertex. If t... | {
"ground_truth": "\\frac{21}{64}",
"style": "rule"
} | {
"index": "37657",
"split": "train"
} |
math | [
{
"content": "What is the largest integer \\( n \\) such that \n\n$$\n\\frac{\\sqrt{7}+2 \\sqrt{n}}{2 \\sqrt{7}-\\sqrt{n}}\n$$\n\nis an integer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "343",
"style": "rule"
} | {
"index": "37658",
"split": "train"
} |
math | [
{
"content": "It is known that the variance of each of the given independent random variables does not exceed 4. Determine the number of such variables for which the probability that the deviation of the arithmetic mean of the random variable from the arithmetic mean of their mathematical expectations by no mor... | {
"ground_truth": "6400",
"style": "rule"
} | {
"index": "37659",
"split": "train"
} |
math | [
{
"content": "4.3535… is a decimal, which can be abbreviated as , and the repeating cycle is . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "37660",
"split": "train"
} |
math | [
{
"content": "Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $1500$ are not factorial tails? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "37661",
"split": "train"
} |
math | [
{
"content": "The volume of a hemispherical soup bowl is 8 liters. How much soup fills the bowl up to half its height? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2.5",
"style": "rule"
} | {
"index": "37662",
"split": "train"
} |
math | [
{
"content": "A positive integer is called *oneic* if it consists of only $1$ 's. For example, the smallest three oneic numbers are $1$ , $11$ , and $111$ . Find the number of $1$ 's in the smallest oneic number that is divisible by $63$ . Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "37663",
"split": "train"
} |
math | [
{
"content": "Given that the sum of the binomial coefficients of all terms in the expansion of ${(a{x}^{2}+\\frac{1}{x})}^{n}$ is $128$, and the sum of all coefficients is $-1$.<br/>$(1)$ Find the values of $n$ and $a$;<br/>$(2)$ Find the constant term in the expansion of $(2x-\\frac{1}{{x}^{2}}){(a{x}^{2}+\\fr... | {
"ground_truth": "448",
"style": "rule"
} | {
"index": "37664",
"split": "train"
} |
math | [
{
"content": "Add $A85_{12}$ and $2B4_{12}$. Express your answer in base $12$, using $A$ for $10$ and $B$ for $11$ if necessary. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1179_{12}",
"style": "rule"
} | {
"index": "37665",
"split": "train"
} |
math | [
{
"content": "If a bag contains only green, yellow, and red marbles in the ratio $3: 4: 2$ and 63 of the marbles are not red, how many red marbles are in the bag? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "37666",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $c=2$ and $b=\\sqrt{2}a$. Find the maximum value of the area of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "37667",
"split": "train"
} |
math | [
{
"content": "Let $S$ be the set of ordered pairs $(a, b)$ of positive integers such that \\operatorname{gcd}(a, b)=1$. Compute $$\\sum_{(a, b) \\in S}\\left\\lfloor\\frac{300}{2 a+3 b}\\right\\rfloor$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7400",
"style": "rule"
} | {
"index": "37668",
"split": "train"
} |
math | [
{
"content": "Given a sequence ${a_n}$ that satisfies $a_1=1$ and $a_n=a_1+ \\frac {1}{2}a_2+ \\frac {1}{3}a_3+…+ \\frac {1}{n-1}a_{n-1}$ for $n\\geqslant 2, n\\in\\mathbb{N}^*$, if $a_k=2017$, then $k=$ \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4034",
"style": "rule"
} | {
"index": "37669",
"split": "train"
} |
math | [
{
"content": "An infinite geometric series has a first term of $12$ and a second term of $4.$ A second infinite geometric series has the same first term of $12,$ a second term of $4+n,$ and a sum of four times that of the first series. Find the value of $n.$ Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "37670",
"split": "train"
} |
math | [
{
"content": "For a positive number such as 3.27, 3 is referred to as the integral part of the number and .27 as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression. Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "\\frac{3 + \\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "37671",
"split": "train"
} |
math | [
{
"content": "We say that a positive real number $d$ is good if there exists an infinite sequence $a_{1}, a_{2}, a_{3}, \\ldots \\in(0, d)$ such that for each $n$, the points $a_{1}, \\ldots, a_{n}$ partition the interval $[0, d]$ into segments of length at most $1 / n$ each. Find $\\sup \\{d \\mid d \\text { i... | {
"ground_truth": "\\ln 2",
"style": "rule"
} | {
"index": "37672",
"split": "train"
} |
math | [
{
"content": "Let $x, y,$ and $z$ be positive real numbers. Find the minimum value of:\n\\[\n\\frac{(x^2 + 4x + 2)(y^2 + 4y + 2)(z^2 + 4z + 2)}{xyz}.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "216",
"style": "rule"
} | {
"index": "37673",
"split": "train"
} |
math | [
{
"content": "Find the total number of positive four-digit integers \\( N \\) satisfying both of the following properties:\n(i) \\( N \\) is divisible by 7, and\n(ii) when the first and last digits of \\( N \\) are interchanged, the resulting positive integer is also divisible by 7. (Note that the resulting int... | {
"ground_truth": "210",
"style": "rule"
} | {
"index": "37674",
"split": "train"
} |
math | [
{
"content": "$ f\\left( x \\right) \\equal{} \\frac {x^5}{5x^4 \\minus{} 10x^3 \\plus{} 10x^2 \\minus{} 5x \\plus{} 1}$ . $ \\sum_{i \\equal{} 1}^{2009} f\\left( \\frac {i}{2009} \\right) \\equal{} ?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1005",
"style": "rule"
} | {
"index": "37675",
"split": "train"
} |
math | [
{
"content": "A sequence $a_1$, $a_2$, $\\ldots$ of non-negative integers is defined by the rule $a_{n+2}=|a_{n+1}-a_n|$ for $n\\geq1$. If $a_1=999$, $a_2<999$, and $a_{2006}=1$, how many different values of $a_2$ are possible? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "324",
"style": "rule"
} | {
"index": "37676",
"split": "train"
} |
math | [
{
"content": "A book of one hundred pages has its pages numbered from 1 to 100. How many pages in this book have the digit 5 in their numbering? (Note: one sheet has two pages.)\n(a) 13\n(b) 14\n(c) 15\n(d) 16\n(e) 17 Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "37677",
"split": "train"
} |
math | [
{
"content": "In rectangle $PQRS$, $PQ=7$ and $QR =4$. Points $X$ and $Y$ are on $\\overline{RS}$ so that $RX = 2$ and $SY=3$. Lines $PX$ and $QY$ intersect at $Z$. Find the area of $\\triangle PQZ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19.6",
"style": "rule"
} | {
"index": "37678",
"split": "train"
} |
math | [
{
"content": "Let $r(x)$ have a domain $\\{0,1,2,3\\}$ and a range $\\{1,3,5,7\\}$. Let $s(x)$ be defined on the domain $\\{1,2,3,4,5,6\\}$ with the function rule $s(x) = 2x + 1$. Determine the sum of all possible values of $s(r(x))$ where $r(x)$ outputs only odd numbers. Please output the final answer within \... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "37679",
"split": "train"
} |
math | [
{
"content": "A circle with radius $\\frac{\\sqrt{2}}{2}$ and a regular hexagon with side length 1 share the same center. Calculate the area inside the circle, but outside the hexagon. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{2} - \\frac{3\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "37680",
"split": "train"
} |
math | [
{
"content": "Given that \\( a > 0 \\), if \\( f(g(h(a))) = 17 \\), where \\( f(x) = x^2 + 5 \\), \\( g(x) = x^2 - 3 \\), and \\( h(x) = 2x + 1 \\), what is the value of \\( a \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{-1 + \\sqrt{3 + 2\\sqrt{3}}}{2}",
"style": "rule"
} | {
"index": "37681",
"split": "train"
} |
math | [
{
"content": "Darryl has a six-sided die with faces $1, 2, 3, 4, 5, 6$ . He knows the die is weighted so that one face\ncomes up with probability $1/2$ and the other five faces have equal probability of coming up. He\nunfortunately does not know which side is weighted, but he knows each face is equally likel... | {
"ground_truth": "3/26",
"style": "rule"
} | {
"index": "37682",
"split": "train"
} |
math | [
{
"content": "The image depicts a top-down view of a three-layered pyramid made of 14 identical cubes. Each cube is assigned a natural number in such a way that the numbers corresponding to the cubes in the bottom layer are all different, and the number on any other cube is the sum of the numbers on the four ad... | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "37683",
"split": "train"
} |
math | [
{
"content": "let $x,y,z$ be positive reals , such that $x+y+z=1399$ find the\n $$ \\max( [x]y + [y]z + [z]x ) $$ \n( $[a]$ is the biggest integer not exceeding $a$ ) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "652400",
"style": "rule"
} | {
"index": "37684",
"split": "train"
} |
math | [
{
"content": "Evaluate $(x^x)^{(x^x)}$ at $x = 2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "256",
"style": "rule"
} | {
"index": "37685",
"split": "train"
} |
math | [
{
"content": "Divide a circle with a circumference of 24 into 24 equal segments. Select 8 points from the 24 segment points such that the arc length between any two chosen points is not equal to 3 or 8. How many different ways are there to choose such a set of 8 points? Provide reasoning. Please output the fina... | {
"ground_truth": "258",
"style": "rule"
} | {
"index": "37686",
"split": "train"
} |
math | [
{
"content": "In right triangle \\( ABC \\), a point \\( D \\) is on hypotenuse \\( AC \\) such that \\( BD \\perp AC \\). Let \\(\\omega\\) be a circle with center \\( O \\), passing through \\( C \\) and \\( D \\) and tangent to line \\( AB \\) at a point other than \\( B \\). Point \\( X \\) is chosen on \\(... | {
"ground_truth": "8041",
"style": "rule"
} | {
"index": "37687",
"split": "train"
} |
math | [
{
"content": "Find the maximum value of the positive real number \\( A \\) such that for any real numbers \\( x, y, z \\), the inequality\n$$\nx^{4} + y^{4} + z^{4} + x^{2} y z + x y^{2} z + x y z^{2} - A(x y + y z + z x)^{2} \\geq 0\n$$\nholds. Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "37688",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle XYZ$, the medians $\\overline{XM}$ and $\\overline{YN}$ are perpendicular. If $XM=12$ and $YN=18$, then what is the area of $\\triangle XYZ$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "37689",
"split": "train"
} |
math | [
{
"content": "What is the smallest prime number dividing the sum $3^{11}+5^{13}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "37690",
"split": "train"
} |
math | [
{
"content": "A tangent line is drawn to the moving circle $C: x^2 + y^2 - 2ay + a^2 - 2 = 0$ passing through the fixed point $P(2, -1)$. If the point of tangency is $T$, then the minimum length of the line segment $PT$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "\\sqrt {2}",
"style": "rule"
} | {
"index": "37691",
"split": "train"
} |
math | [
{
"content": "Given a hyperbola \\( x^{2} - y^{2} = t \\) (where \\( t > 0 \\)), the right focus is \\( F \\). Any line passing through \\( F \\) intersects the right branch of the hyperbola at points \\( M \\) and \\( N \\). The perpendicular bisector of \\( M N \\) intersects the \\( x \\)-axis at point \\( P... | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "37692",
"split": "train"
} |
math | [
{
"content": "Find the minimum value of $\\sqrt{x^2+y^2}$ if $5x+12y=60$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{60}{13}",
"style": "rule"
} | {
"index": "37693",
"split": "train"
} |
math | [
{
"content": "How many three-digit whole numbers contain at least one digit 6 or at least one digit 8? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "452",
"style": "rule"
} | {
"index": "37694",
"split": "train"
} |
math | [
{
"content": "All positive integers whose digits add up to 12 are listed in increasing order. What is the eleventh number in that list? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "156",
"style": "rule"
} | {
"index": "37695",
"split": "train"
} |
math | [
{
"content": "Compute \\[\\lfloor \\sqrt{1} \\rfloor + \\lfloor \\sqrt{2} \\rfloor + \\lfloor \\sqrt{3} \\rfloor + \\cdots + \\lfloor \\sqrt{25} \\rfloor.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "75",
"style": "rule"
} | {
"index": "37696",
"split": "train"
} |
math | [
{
"content": "The function $f(x)$ satisfies\n\\[b^2 f(a) = a^2 f(b)\\]for all real numbers $a$ and $b.$ If $f(2) \\neq 0,$ find\n\\[\\frac{f(5) - f(1)}{f(2)}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "37697",
"split": "train"
} |
math | [
{
"content": "Find the set of $x$-values satisfying the inequality $|\\frac{5-x}{3}|<2$. [The symbol $|a|$ means $+a$ if $a$ is positive, \n$-a$ if $a$ is negative,$0$ if $a$ is zero. The notation $1<a<2$ means that a can have any value between $1$ and $2$, excluding $1$ and $2$. ] Please output the final answe... | {
"ground_truth": "-1 < x < 11",
"style": "rule"
} | {
"index": "37698",
"split": "train"
} |
math | [
{
"content": "The number $5\\,41G\\,507\\,2H6$ is divisible by $40.$ Determine the sum of all distinct possible values of the product $GH.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "225",
"style": "rule"
} | {
"index": "37699",
"split": "train"
} |
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