data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "In a diagram, there is an equilateral triangle with a side length of $10$ m. Calculate both the perimeter and the height of the triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5\\sqrt{3}",
"style": "rule"
} | {
"index": "39800",
"split": "train"
} |
math | [
{
"content": "A positive integer $n$ not exceeding $100$ is chosen in such a way that if $n\\le 50$, then the probability of choosing $n$ is $p$, and if $n > 50$, then the probability of choosing $n$ is $3p$. The probability that a perfect square is chosen is Please output the final answer within \\boxed{}.",
... | {
"ground_truth": ".08",
"style": "rule"
} | {
"index": "39801",
"split": "train"
} |
math | [
{
"content": "I have modified my walking game. On move 1, I still do nothing, but for each move $n$ where $2 \\le n \\le 30$, I walk one step backward if $n$ is prime and three steps forward if $n$ is composite. After all 30 moves, I assess how far I am from my starting point. How many total steps would I need ... | {
"ground_truth": "47",
"style": "rule"
} | {
"index": "39802",
"split": "train"
} |
math | [
{
"content": "Each vertex of a convex hexagon $ABCDEF$ is to be assigned a color. There are $7$ colors to choose from, and no two adjacent vertices can have the same color, nor can the vertices at the ends of each diagonal. Calculate the total number of different colorings possible. Please output the final answ... | {
"ground_truth": "5040",
"style": "rule"
} | {
"index": "39803",
"split": "train"
} |
math | [
{
"content": "How many ways can the integers from $-7$ to $7$ inclusive be arranged in a sequence such that the absolute value of the numbers in the sequence does not decrease? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "128",
"style": "rule"
} | {
"index": "39804",
"split": "train"
} |
math | [
{
"content": "Determine the smallest positive integer $n \\geq 3$ for which $$A \\equiv 2^{10 n}\\left(\\bmod 2^{170}\\right)$$ where $A$ denotes the result when the numbers $2^{10}, 2^{20}, \\ldots, 2^{10 n}$ are written in decimal notation and concatenated (for example, if $n=2$ we have $A=10241048576$). Plea... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "39805",
"split": "train"
} |
math | [
{
"content": "We toss a die 30 times. Determine, with an accuracy of 8 decimal places, the probability that there will be at least one six among the numbers rolled, and that no five or four appears before the first six. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.33333333",
"style": "rule"
} | {
"index": "39806",
"split": "train"
} |
math | [
{
"content": "What is the sum of the coordinates of the midpoint of the segment with endpoints $(6, 12)$ and $(0, -6)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "39807",
"split": "train"
} |
math | [
{
"content": "A store decides to discount a certain type of clothing due to the change of season. If each piece of clothing is sold at 50% of the marked price, there will be a loss of 20 yuan per piece. However, if sold at 80% of the marked price, there will be a profit of 40 yuan per piece. \nQuestion: (1) Wh... | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "39808",
"split": "train"
} |
math | [
{
"content": "A positive integer is *bold* iff it has $8$ positive divisors that sum up to $3240$ . For example, $2006$ is bold because its $8$ positive divisors, $1$ , $2$ , $17$ , $34$ , $59$ , $118$ , $1003$ and $2006$ , sum up to $3240$ . Find the smallest positive bold number. Please outpu... | {
"ground_truth": "1614",
"style": "rule"
} | {
"index": "39809",
"split": "train"
} |
math | [
{
"content": "In a table tennis singles match between players A and B, the match is played in a best-of-seven format (i.e., the first player to win four games wins the match, and the match ends). Assume that both players have an equal chance of winning each game. \n(1) Calculate the probability that player B w... | {
"ground_truth": "\\frac{5}{16}",
"style": "rule"
} | {
"index": "39810",
"split": "train"
} |
math | [
{
"content": "The clock runs 3 minutes fast every 24 hours. The clock was set accurately. After what minimum time will the clock show the correct time again? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "39811",
"split": "train"
} |
math | [
{
"content": "Convert the binary number $110110100_2$ to base 4. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31220_4",
"style": "rule"
} | {
"index": "39812",
"split": "train"
} |
math | [
{
"content": "Let $n \\geq 3$ be an integer. Rowan and Colin play a game on an $n \\times n$ grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid and Colin is allowed to permute the columns. A grid coloring is [i]orderly[/i] if: [list] [*]no matter h... | {
"ground_truth": "2(n! + 1)",
"style": "rule"
} | {
"index": "39813",
"split": "train"
} |
math | [
{
"content": "Container A holds 4 red balls and 6 green balls; container B holds 8 red balls and 6 green balls; container C holds 8 red balls and 6 green balls; container D holds 3 red balls and 7 green balls. A container is selected at random and then a ball is randomly selected from that container. What is th... | {
"ground_truth": "\\frac{13}{28}",
"style": "rule"
} | {
"index": "39814",
"split": "train"
} |
math | [
{
"content": "Find the value of $a$ so that the lines described by\n\\[\\begin{pmatrix} 2 \\\\ -1 \\\\ 0 \\end{pmatrix} + t \\begin{pmatrix} a \\\\ -2 \\\\ 1 \\end{pmatrix}\\]and\n\\[\\begin{pmatrix} 1 \\\\ -3/2 \\\\ -5 \\end{pmatrix} + u \\begin{pmatrix} 1 \\\\ 3/2 \\\\ 2 \\end{pmatrix}\\]are perpendicular. Pl... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "39815",
"split": "train"
} |
math | [
{
"content": "Compute the number of ordered pairs of integers $(x,y)$ with $1\\le x<y\\le 150$ such that $i^x+i^y$ is a real number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3515",
"style": "rule"
} | {
"index": "39816",
"split": "train"
} |
math | [
{
"content": "Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had $56$ dollars. The absolute difference between the amounts Ashley and Betty had to spend was $19$ dollars. The absolute difference between the amounts Betty and Carlos had was $7... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "39817",
"split": "train"
} |
math | [
{
"content": "Let $C$ be the circle with equation $x^2-6y-3=-y^2-4x$. If $(a,b)$ is the center of $C$ and $r$ is its radius, what is the value of $a+b+r$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "39818",
"split": "train"
} |
math | [
{
"content": "(1) Given $ \\frac {\\pi}{2} < \\beta < \\alpha < \\frac {3\\pi}{4}$, $\\cos (\\alpha-\\beta)= \\frac {12}{13}$, $\\sin (\\alpha+\\beta)=- \\frac {3}{5}$, find the value of $\\sin 2\\alpha$. \n(2) Given $ \\frac {\\pi}{2} < \\alpha < \\pi$, $0 < \\beta < \\frac {\\pi}{2}$, $\\tan \\alpha=- \\frac... | {
"ground_truth": "\\frac {63}{65}",
"style": "rule"
} | {
"index": "39819",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_n\\}$, the first $m(m\\geqslant 4)$ terms form an arithmetic sequence with a common difference of $2$. Starting from the $(m-1)$-th term, $a_{m-1}$, $a_{m}$, $a_{m+1}$, ... form a geometric sequence with a common ratio of $2$. If $a_{1}=-2$, then $m=$ ______, and the sum of ... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "39820",
"split": "train"
} |
math | [
{
"content": "In triangle \\(ABC\\), side \\(BC = 28\\). The angle bisector \\(BL\\) is divided by the intersection point of the angle bisectors of the triangle in the ratio \\(4:3\\) from the vertex. Find the radius of the circumscribed circle around triangle \\(ABC\\) if the radius of the inscribed circle is ... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "39821",
"split": "train"
} |
math | [
{
"content": "Let $m=\\underbrace{22222222}_{\\text{8 digits}}$ and $n=\\underbrace{444444444}_{\\text{9 digits}}$.\n\nWhat is $\\gcd(m,n)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "39822",
"split": "train"
} |
math | [
{
"content": "Three prime numbers are randomly selected without replacement from the first ten prime numbers. What is the probability that the sum of the three selected numbers is odd? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{10}",
"style": "rule"
} | {
"index": "39823",
"split": "train"
} |
math | [
{
"content": "Find the area of the region bounded by the graph of $r = \\sec \\theta,$ the graph of $r = \\csc \\theta,$ the $x$-axis, and the $y$-axis. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "39824",
"split": "train"
} |
math | [
{
"content": "John has saved up $5235_9$ dollars for a trip to Japan. A round-trip airline ticket costs $1250_8$ dollars. In base ten, how many dollars will he have left for lodging and food? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3159",
"style": "rule"
} | {
"index": "39825",
"split": "train"
} |
math | [
{
"content": "If the sum of the squares of nonnegative real numbers $a,b,$ and $c$ is $39$, and $ab + bc + ca = 21$, then what is the sum of $a,b,$ and $c$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "39826",
"split": "train"
} |
math | [
{
"content": "Find $X+Y$ (in base 10), given the following addition problem in base 7:\n\\[ \\begin{array}{c@{}c@{\\;}c@{}c@{}c@{}c}\n& & & 5 & X & Y_{7}\\\\\n&+& & & 5 & 2_{7}\\\\\n\\cline{2-6}\n& & & 6 & 4 & X_{7}\\\\\n\\end{array} \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "39827",
"split": "train"
} |
math | [
{
"content": "In a tetrahedron \\( ABCD \\), \\( AB = AC = AD = 5 \\), \\( BC = 3 \\), \\( CD = 4 \\), \\( DB = 5 \\). Find the volume of this tetrahedron. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5\\sqrt{3}",
"style": "rule"
} | {
"index": "39828",
"split": "train"
} |
math | [
{
"content": "Convert \\\\(101 111 011_{(2)}\\\\) to its decimal value. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "379",
"style": "rule"
} | {
"index": "39829",
"split": "train"
} |
math | [
{
"content": "A 1$\\%$ late charge was added to Jenna's bill on the $30^{\\text{th}}$ day past its due date. The resulting total was then increased by 1$\\%$ because she did not pay the bill in the next 30 days either. Her original bill was $\\$400$. Exactly how much is the bill now? Please output the final ans... | {
"ground_truth": "408.04",
"style": "rule"
} | {
"index": "39830",
"split": "train"
} |
math | [
{
"content": "The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle? Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "39831",
"split": "train"
} |
math | [
{
"content": "A classroom has 9 desks arranged in a row. Two students, Alex and Jessie, choose their desks at random. What is the probability that they do not sit next to each other? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{9}",
"style": "rule"
} | {
"index": "39832",
"split": "train"
} |
math | [
{
"content": "The sequence \\(\\left\\{a_{n}\\right\\}\\) satisfies: \\(a_{1}=1\\), and for each \\(n \\in \\mathbf{N}^{*}\\), \\(a_{n}\\) and \\(a_{n+1}\\) are the two roots of the equation \\(x^{2}+3nx+b_{n}=0\\). Find \\(\\sum_{k=1}^{20} b_{k}\\). Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "6385",
"style": "rule"
} | {
"index": "39833",
"split": "train"
} |
math | [
{
"content": "In trapezoid $KLMN$ with bases $KN$ and $LN$, the angle $\\angle LMN$ is $60^{\\circ}$. A circle is circumscribed around triangle $KLN$ and is tangent to the lines $LM$ and $MN$. Find the radius of the circle, given that the perimeter of triangle $KLN$ is 12. Please output the final answer within ... | {
"ground_truth": "\\frac{4 \\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "39834",
"split": "train"
} |
math | [
{
"content": "Calculate the definite integral:\n\n$$\n\\int_{0}^{4} e^{\\sqrt{(4-x) /(4+x)}} \\cdot \\frac{d x}{(4+x) \\sqrt{16-x^{2}}}\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}(e-1)",
"style": "rule"
} | {
"index": "39835",
"split": "train"
} |
math | [
{
"content": "Let $ m\\equal{}\\left(abab\\right)$ and $ n\\equal{}\\left(cdcd\\right)$ be four-digit numbers in decimal system. If $ m\\plus{}n$ is a perfect square, find the largest value of $ a\\cdot b\\cdot c\\cdot d$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "600",
"style": "rule"
} | {
"index": "39836",
"split": "train"
} |
math | [
{
"content": "Let $r,$ $s,$ $t$ be the roots of $2x^3 - 7x^2 - 6 = 0.$ Find $rst.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "39837",
"split": "train"
} |
math | [
{
"content": "For what real number \\\\(m\\\\) is the complex number \\\\(z=m^{2}+m-2+(m^{2}-1)i\\\\) \n\\\\((1)\\\\) a real number; \\\\((2)\\\\) an imaginary number; \\\\((3)\\\\) a pure imaginary number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "39838",
"split": "train"
} |
math | [
{
"content": "What is the product of the least common multiple and the greatest common factor of $20$ and $90$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1800",
"style": "rule"
} | {
"index": "39839",
"split": "train"
} |
math | [
{
"content": "A circle centered at $O$ has radius 1 and contains the point $A$. Segment $AB$ is tangent to the circle at $A$ and $\\angle\nAOB=\\theta$. If point $C$ lies on $\\overline{OA}$ and $\\overline{BC}$ bisects $\\angle ABO$, then express $OC$ in terms of $s$ and $c,$ where $s = \\sin \\theta$ and $c =... | {
"ground_truth": "\\frac{1}{1 + s}",
"style": "rule"
} | {
"index": "39840",
"split": "train"
} |
math | [
{
"content": "Find the minimum value for \\(a, b > 0\\) of the expression\n\n$$\n\\frac{|2a - b + 2a(b - a)| + |b + 2a - a(b + 4a)|}{\\sqrt{4a^2 + b^2}}\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "39841",
"split": "train"
} |
math | [
{
"content": "Kate bakes a $20$-inch by $18$-inch pan of cornbread. The cornbread is cut into pieces that measure $2$ inches by $2$ inches. How many pieces of cornbread does the pan contain? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "39842",
"split": "train"
} |
math | [
{
"content": "Find the area of the region in the coordinate plane where the discriminant of the quadratic $ax^2 + bxy + cy^2 = 0$ is not positive. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49 \\pi",
"style": "rule"
} | {
"index": "39843",
"split": "train"
} |
math | [
{
"content": "Fisica and Ritmo discovered a piece of Notalium shaped like a rectangular box, and wanted to find its volume. To do so, Fisica measured its three dimensions using a ruler with infinite precision, multiplied the results and rounded the product to the nearest cubic centimeter, getting a result of 20... | {
"ground_truth": "7174",
"style": "rule"
} | {
"index": "39844",
"split": "train"
} |
math | [
{
"content": "We place planes through each edge and the midpoint of the edge opposite to it in a tetrahedron. Into how many parts do these planes divide the tetrahedron, and what are the volumes of these parts? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "39845",
"split": "train"
} |
math | [
{
"content": "Ali wants to move from point $A$ to point $B$. He cannot walk inside the black areas but he is free to move in any direction inside the white areas (not only the grid lines but the whole plane). Help Ali to find the shortest path between $A$ and $B$. Only draw the path and write its length.\n[img]... | {
"ground_truth": "7 + 5\\sqrt{2}",
"style": "rule"
} | {
"index": "39846",
"split": "train"
} |
math | [
{
"content": "Compute the length of the segment tangent from the point $(1,1)$ to the circle that passes through the points $(4,5),$ $(7,9),$ and $(6,14).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5\\sqrt{2}",
"style": "rule"
} | {
"index": "39847",
"split": "train"
} |
math | [
{
"content": "The solid \\( T \\) consists of all points \\((x,y,z)\\) such that \\(|x| + |y| \\leq 2\\), \\(|x| + |z| \\leq 2\\), and \\(|y| + |z| \\leq 2\\). Find the volume of \\( T \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1664}{81}",
"style": "rule"
} | {
"index": "39848",
"split": "train"
} |
math | [
{
"content": "Given the sets $A=\\{1, 3, 2m-1\\}$ and $B=\\{3, m^2\\}$; if $B \\subseteq A$, find the value of the real number $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "39849",
"split": "train"
} |
math | [
{
"content": "Compute the sum:\n\\[\n\\sin^2 3^\\circ + \\sin^2 9^\\circ + \\sin^2 15^\\circ + \\dots + \\sin^2 177^\\circ.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "39850",
"split": "train"
} |
math | [
{
"content": "A sphere intersects the $xy$-plane in a circle centered at $(3, 5, 0)$ with radius 2. The sphere also intersects the $yz$-plane in a circle centered at $(0, 5, -8),$ with radius $r.$ Find $r.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{59}",
"style": "rule"
} | {
"index": "39851",
"split": "train"
} |
math | [
{
"content": "The product of three consecutive even numbers has the form $87XXXXX8$. Provide the 5 missing digits. ($X$ does not necessarily represent the same digits.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "52660",
"style": "rule"
} | {
"index": "39852",
"split": "train"
} |
math | [
{
"content": "In each square of an $11\\times 11$ board, we are to write one of the numbers $-1$ , $0$ , or $1$ in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the boa... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "39853",
"split": "train"
} |
math | [
{
"content": "Alex has 10 different kinds of lunch meat and 9 different kinds of cheese. If he wants to make a sandwich with one kind of meat and two kinds of cheese, how many different sandwiches could he make? (It does not matter in which order he chooses the two types of cheese.) Please output the final ans... | {
"ground_truth": "360",
"style": "rule"
} | {
"index": "39854",
"split": "train"
} |
math | [
{
"content": "Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is $\\frac{1}{29}$ of the original integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "725",
"style": "rule"
} | {
"index": "39855",
"split": "train"
} |
math | [
{
"content": "Given the curve $C$ represented by the equation $\\sqrt {x^{2}+2 \\sqrt {7}x+y^{2}+7}+ \\sqrt {x^{2}-2 \\sqrt {7}x+y^{2}+7}=8$, find the distance from the origin to the line determined by two distinct points on the curve $C$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\dfrac {12}{5}",
"style": "rule"
} | {
"index": "39856",
"split": "train"
} |
math | [
{
"content": "1. Calculate the value of the following expression:\n$(1)(2\\frac{7}{9})^{\\frac{1}{2}} - (2\\sqrt{3} - \\pi)^{0} - (2\\frac{10}{27})^{-\\frac{2}{3}} + 0.25^{-\\frac{3}{2}}$\n2. Given that $x + x^{-1} = 4 (0 < x < 1)$, find the value of ${x^{\\frac{1}{2}}} + {x^{-\\frac{1}{2}}}$. Please output the... | {
"ground_truth": "\\frac{389}{48}",
"style": "rule"
} | {
"index": "39857",
"split": "train"
} |
math | [
{
"content": "Let $S$ be the set of all non-zero real numbers. Define a function $f : S \\to S$ such that for all $x, y \\in S$ with $x + y \\neq 0$, the following holds:\n\\[f(x) + f(y) = f\\left(\\frac{xy}{x+y}f(x+y)\\right).\\]\nDetermine the number of possible values of $f(3)$, denoted as $n$, and the sum o... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "39858",
"split": "train"
} |
math | [
{
"content": "If 3400 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "38",
"style": "rule"
} | {
"index": "39859",
"split": "train"
} |
math | [
{
"content": "Medians $AD$ and $CE$ of $\\triangle ABC$ intersect in $M$. The midpoint of $AE$ is $N$. \nLet the area of $\\triangle MNE$ be $k$ times the area of $\\triangle ABC$. Then $k$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"index": "39860",
"split": "train"
} |
math | [
{
"content": "In a chess-playing club, some of the players take lessons from other players. It is possible (but not necessary) for two players both to take lessons from each other. It so happens that for any three distinct members of the club, $A, B$, and $C$, exactly one of the following three statements is tr... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "39861",
"split": "train"
} |
math | [
{
"content": "As shown in the diagram, squares \\( a \\), \\( b \\), \\( c \\), \\( d \\), and \\( e \\) are used to form a rectangle that is 30 cm long and 22 cm wide. Find the area of square \\( e \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "39862",
"split": "train"
} |
math | [
{
"content": "Find the distance from the point $(1,2,3)$ to the line described by\n\\[\\begin{pmatrix} 6 \\\\ 7 \\\\ 7 \\end{pmatrix} + t \\begin{pmatrix} 3 \\\\ 2 \\\\ -2 \\end{pmatrix}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "39863",
"split": "train"
} |
math | [
{
"content": "Equilateral $\\triangle ABC$ is inscribed in a circle of radius $2$. Extend $\\overline{AB}$ through $B$ to point $D$ so that $AD=13,$ and extend $\\overline{AC}$ through $C$ to point $E$ so that $AE = 11.$ Through $D,$ draw a line $l_1$ parallel to $\\overline{AE},$ and through $E,$ draw a line $... | {
"ground_truth": "865",
"style": "rule"
} | {
"index": "39864",
"split": "train"
} |
math | [
{
"content": "The diagram shows a square \\(PQRS\\) with edges of length 1, and four arcs, each of which is a quarter of a circle. Arc \\(TRU\\) has center \\(P\\); arc \\(VPW\\) has center \\(R\\); arc \\(UV\\) has center \\(S\\); and arc \\(WT\\) has center \\(Q\\). What is the length of the perimeter of the ... | {
"ground_truth": "(2\\sqrt{2} - 1)\\pi",
"style": "rule"
} | {
"index": "39865",
"split": "train"
} |
math | [
{
"content": "If a positive integer multiple of 864 is picked randomly, with each multiple having the same probability of being picked, what is the probability that it is divisible by 1944? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{9}",
"style": "rule"
} | {
"index": "39866",
"split": "train"
} |
math | [
{
"content": "Let $[ x ]$ denote the greatest integer less than or equal to $x$. For example, $[10.2] = 10$. Calculate the value of $\\left[\\frac{2017 \\times 3}{11}\\right] + \\left[\\frac{2017 \\times 4}{11}\\right] + \\left[\\frac{2017 \\times 5}{11}\\right] + \\left[\\frac{2017 \\times 6}{11}\\right] + \\l... | {
"ground_truth": "6048",
"style": "rule"
} | {
"index": "39867",
"split": "train"
} |
math | [
{
"content": "Triangle \\(ABC\\) is isosceles with \\(AB = AC\\) and \\(BC = 65 \\, \\text{cm}\\). \\(P\\) is a point on \\(BC\\) such that the perpendicular distances from \\(P\\) to \\(AB\\) and \\(AC\\) are \\(24 \\, \\text{cm}\\) and \\(36 \\, \\text{cm}\\), respectively. The area of \\(\\triangle ABC\\), i... | {
"ground_truth": "2535",
"style": "rule"
} | {
"index": "39868",
"split": "train"
} |
math | [
{
"content": "A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "39869",
"split": "train"
} |
math | [
{
"content": "Given the function \\( f(x)=\\sqrt{3} \\sin 2x + 2 \\cos^2 x + a \\), if the minimum value of \\( f(x) \\) on the interval \\(\\left[ 0, \\frac{\\pi}{2} \\right] \\) is \\(-1\\), find the value of \\( a \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "39870",
"split": "train"
} |
math | [
{
"content": "Let $p_{}$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of $5$ heads before one encounters a run of $2$ tails. Given that $p_{}$ can be written in the form $m/n$ where $m_{}$ and $n_{}$ are relatively prime positive integers, find $m+n$. Plea... | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "39871",
"split": "train"
} |
math | [
{
"content": "All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3\\times3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$. What is the number in the center? Ple... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "39872",
"split": "train"
} |
math | [
{
"content": "The product of two positive consecutive integers is 506. What is their sum? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "39873",
"split": "train"
} |
math | [
{
"content": "Given the inequality $\\frac{x-2}{ax-1} > 0$ with the solution set $(-1,2)$, determine the constant term in the expansion of the binomial $(ax-\\frac{1}{x^{2}})^{6}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "39874",
"split": "train"
} |
math | [
{
"content": "If $\\log_8{3}=p$ and $\\log_3{5}=q$, then, in terms of $p$ and $q$, $\\log_{10}{5}$ equals Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3pq}{1+3pq}",
"style": "rule"
} | {
"index": "39875",
"split": "train"
} |
math | [
{
"content": "In the diagram, $ABCD$ is a trapezoid with an area of $18.$ $CD$ is three times the length of $AB.$ Determine the area of $\\triangle ABC.$\n\n[asy]\ndraw((0,0)--(2,5)--(11,5)--(18,0)--cycle);\ndraw((11,5)--(0,0));\nlabel(\"$D$\",(0,0),W);\nlabel(\"$A$\",(2,5),NW);\nlabel(\"$B$\",(11,5),NE);\nlabe... | {
"ground_truth": "4.5",
"style": "rule"
} | {
"index": "39876",
"split": "train"
} |
math | [
{
"content": "Find the number of positive integers $n \\le 1000$ such that $15n$ is a perfect square. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "39877",
"split": "train"
} |
math | [
{
"content": "The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base $10$, where digit a is not zero. Find the three-digit number $abc$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "937",
"style": "rule"
} | {
"index": "39878",
"split": "train"
} |
math | [
{
"content": "Point $F$ is taken in side $AD$ of square $ABCD$. At $C$ a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. The area of $ABCD$ is $256$ square inches and the area of $\\triangle CEF$ is $200$ square inches. Then the number of inches in $BE$ is: Please output the final answer within \\... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "39879",
"split": "train"
} |
math | [
{
"content": "Xiao Ming participated in the \"Inheriting Classics, Building the Future\" themed speech competition. His scores for speech image, speech content, and speech effect were 9, 8, and 9 respectively. If the scores for speech image, speech content, and speech effect are determined in a ratio of 2:5:3 t... | {
"ground_truth": "8.5",
"style": "rule"
} | {
"index": "39880",
"split": "train"
} |
math | [
{
"content": "An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\\angle ABC = \\angle ACB = 3 \\angle D$ and $\\angle BAC = k \\pi$ in radians, then find $k$. Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "\\frac{5}{11}",
"style": "rule"
} | {
"index": "39881",
"split": "train"
} |
math | [
{
"content": "Three faces of a right rectangular prism have areas of 48, 49 and 50 square units. What is the volume of the prism, in cubic units? Express your answer to the nearest whole number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "343",
"style": "rule"
} | {
"index": "39882",
"split": "train"
} |
math | [
{
"content": "Using a table of a certain height, two identical blocks of wood are placed as shown in Figure 1. Length $r$ is found to be $32$ inches. After rearranging the blocks as in Figure 2, length $s$ is found to be $28$ inches. How high is the table? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "39883",
"split": "train"
} |
math | [
{
"content": "For how many values of \\( k \\) is \\( 18^{18} \\) the least common multiple of the positive integers \\( 9^9 \\), \\( 12^{12} \\), and \\( k \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "39884",
"split": "train"
} |
math | [
{
"content": "Given that \\( P \\) is any point on the arc \\( \\overparen{AD} \\) of the circumscribed circle of square \\( ABCD \\), find the value of \\( \\frac{PA + PC}{PB} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "39885",
"split": "train"
} |
math | [
{
"content": "Given the number 826,000,000, express it in scientific notation. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8.26\\times 10^{8}",
"style": "rule"
} | {
"index": "39886",
"split": "train"
} |
math | [
{
"content": "What is the maximum number of rooks that can be placed on a $300 \\times 300$ chessboard such that each rook attacks at most one other rook? (A rook attacks all the squares it can reach according to chess rules without passing through other pieces.) Please output the final answer within \\boxed{}.... | {
"ground_truth": "400",
"style": "rule"
} | {
"index": "39887",
"split": "train"
} |
math | [
{
"content": "Let $a \\bowtie b = a+\\sqrt{b+\\sqrt{b+\\sqrt{b+...}}}$. If $7\\bowtie g = 9$, find the value of $g$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "39888",
"split": "train"
} |
math | [
{
"content": "Alice and Bob each arrive at a gathering at a random time between 12:00 noon and 1:00 PM. If Alice arrives after Bob, what is the probability that Bob arrived before 12:45 PM? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.5625",
"style": "rule"
} | {
"index": "39889",
"split": "train"
} |
math | [
{
"content": "Let point $O$ be the origin of a three-dimensional coordinate system, and let points $A,$ $B,$ and $C$ be located on the positive $x,$ $y,$ and $z$ axes, respectively. If $OA = \\sqrt{144}$ and $\\angle BAC = 45^\\circ,$ then compute the area of triangle $ABC.$ Please output the final answer withi... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "39890",
"split": "train"
} |
math | [
{
"content": "Consider the set of all fractions $\\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by $1$, the value of the fraction is increased by $10\\%$? Please output the final answer w... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "39891",
"split": "train"
} |
math | [
{
"content": "A sequence consists of the digits $122333444455555 \\ldots$ such that each positive integer $n$ is repeated $n$ times, in increasing order. Find the sum of the 4501st and 4052nd digits of this sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "39892",
"split": "train"
} |
math | [
{
"content": "Country $X$ has $40\\%$ of the world's population and $60\\%$ of the world's wealth. Country $Y$ has $20\\%$ of the world's population but $30\\%$ of its wealth. Country $X$'s top $50\\%$ of the population owns $80\\%$ of the wealth, and the wealth in Country $Y$ is equally shared among its citize... | {
"ground_truth": "1.6",
"style": "rule"
} | {
"index": "39893",
"split": "train"
} |
math | [
{
"content": "Let $P$ be the least common multiple of all integers from $10$ to $40$, inclusive. Let $Q$ be the least common multiple of $P$ and the integers $41, 42, 43, 44, 45, 46, 47, 48, 49,$ and $50$. What is the value of $\\frac{Q}{P}?$\nA) 1763\nB) 82861\nC) 9261\nD) 14756 Please output the final answer ... | {
"ground_truth": "82861",
"style": "rule"
} | {
"index": "39894",
"split": "train"
} |
math | [
{
"content": "The quadratic $-6x^2+36x+216$ can be written in the form $a(x+b)^2+c$, where $a$, $b$, and $c$ are constants. What is $a+b+c$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "261",
"style": "rule"
} | {
"index": "39895",
"split": "train"
} |
math | [
{
"content": "In 10 boxes, place ping-pong balls such that the number of balls in each box is at least 11, not equal to 17, not a multiple of 6, and all numbers are distinct. What is the minimum number of ping-pong balls needed? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "174",
"style": "rule"
} | {
"index": "39896",
"split": "train"
} |
math | [
{
"content": "Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie? Please output the final ans... | {
"ground_truth": "50^{\\circ}",
"style": "rule"
} | {
"index": "39897",
"split": "train"
} |
math | [
{
"content": "If $\\overrightarrow{a} = (2, 3)$, $\\overrightarrow{b} = (-4, 7)$, and $\\overrightarrow{a} + \\overrightarrow{c} = 0$, then the projection of $\\overrightarrow{c}$ in the direction of $\\overrightarrow{b}$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "-\\frac{\\sqrt{65}}{5}",
"style": "rule"
} | {
"index": "39898",
"split": "train"
} |
math | [
{
"content": "A semipro baseball league has teams with $21$ players each. League rules state that a player must be paid at least $15,000$ dollars, and that the total of all players' salaries for each team cannot exceed $700,000$ dollars. What is the maximum possible salary, in dollars, for a single player? Plea... | {
"ground_truth": "400000",
"style": "rule"
} | {
"index": "39899",
"split": "train"
} |
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