data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Regular octagon $ABCDEFGH$ has area $n$. Let $m$ be the area of quadrilateral $ACEG$. What is $\\frac{m}{n}?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "34700",
"split": "train"
} |
math | [
{
"content": "How many 3-letter words can we make from the letters A, B, C, and D, if we are allowed to repeat letters, and we must use the letter A at least once? (Here, a word is an arbitrary sequence of letters.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "34701",
"split": "train"
} |
math | [
{
"content": "Find the value of the expression $$ f\\left( \\frac{1}{2000} \\right)+f\\left( \\frac{2}{2000} \\right)+...+ f\\left( \\frac{1999}{2000} \\right)+f\\left( \\frac{2000}{2000} \\right)+f\\left( \\frac{2000}{1999} \\right)+...+f\\left( \\frac{2000}{1} \\right) $$ assuming $f(x) =\\frac{x^2}{1 + x^... | {
"ground_truth": "1999.5",
"style": "rule"
} | {
"index": "34702",
"split": "train"
} |
math | [
{
"content": "What is the remainder when the sum $1^2 + 2^2 + 3^2 + \\dots + 10^2$ is divided by 11? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "34703",
"split": "train"
} |
math | [
{
"content": "Let \\( n, m \\) be positive integers. Define the sets \\( A = \\{1, 2, \\ldots, n\\} \\) and \\( B_{n}^{m} = \\{ (a_1, a_2, \\ldots, a_m) \\mid a_i \\in A, i = 1, 2, \\ldots, m \\} \\) that satisfy:\n1. \\( |a_i - a_{i+1}| \\neq n-1 \\) for \\( i = 1, 2, \\ldots, m-1 \\);\n2. Among \\( a_1, a_2, ... | {
"ground_truth": "104",
"style": "rule"
} | {
"index": "34704",
"split": "train"
} |
math | [
{
"content": "Given that the sequence of positive integers $a_{1}, a_{2}, a_{3}, a_{4}$ forms a geometric progression with a common ratio $r$ which is not an integer and $r > 1$, find the minimum possible value of $a_{4}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "34705",
"split": "train"
} |
math | [
{
"content": "How many foonies are in a stack that has a volume of $50 \\mathrm{~cm}^{3}$, given that each foonie has a volume of $2.5 \\mathrm{~cm}^{3}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "34706",
"split": "train"
} |
math | [
{
"content": "A cell phone plan costs $20$ dollars each month, plus $5$ cents per text message sent, plus $10$ cents for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "28.00",
"style": "rule"
} | {
"index": "34707",
"split": "train"
} |
math | [
{
"content": "The distance from platform $A$ to platform $B$ was covered by an electric train in $X$ minutes ($0<X<60$). Find $X$ if it is known that at both the moment of departure from $A$ and the moment of arrival at $B$, the angle between the hour and minute hands of the clock was $X$ degrees. Please output... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "34708",
"split": "train"
} |
math | [
{
"content": "A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\\%$. Find the maximum discount rate that can be offered. Please output the final answer within \\boxed... | {
"ground_truth": "8.8\\%",
"style": "rule"
} | {
"index": "34709",
"split": "train"
} |
math | [
{
"content": "For an arbitrary positive integer $m$, not divisible by $3$, consider the permutation $x \\mapsto 3x \\pmod{m}$ on the set $\\{ 1,2,\\dotsc ,m-1\\}$. This permutation can be decomposed into disjointed cycles; for instance, for $m=10$ the cycles are $(1\\mapsto 3\\to 9,\\mapsto 7,\\mapsto 1)$, $(2\... | {
"ground_truth": "m \\equiv 2, 5, 7, 10 \\pmod{12}",
"style": "rule"
} | {
"index": "34710",
"split": "train"
} |
math | [
{
"content": "\"Sun\", \"Xing\", and \"Zhe\" each represent a digit. The product of the two three-digit numbers 'SunXingZhe' and 'ZheXingSun' is 78445. Can you find the sum of these two three-digit numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "686",
"style": "rule"
} | {
"index": "34711",
"split": "train"
} |
math | [
{
"content": "Given a set of data: $10.1$, $9.8$, $10$, $x$, $10.2$, the average of these data is $10$. Calculate the variance of this set of data. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.02",
"style": "rule"
} | {
"index": "34712",
"split": "train"
} |
math | [
{
"content": "As shown in the figure below, a regular dodecahedron (the polyhedron consisting of $12$ congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom f... | {
"ground_truth": "810",
"style": "rule"
} | {
"index": "34713",
"split": "train"
} |
math | [
{
"content": "In a variant of SHORT BINGO, a $5\\times5$ card is modified such that the middle square is marked as FREE, and the remaining 24 squares are populated with distinct numbers. The modification includes:\n- First column contains 5 distinct numbers from the set $1-15$.\n- Second column contains 5 disti... | {
"ground_truth": "360360",
"style": "rule"
} | {
"index": "34714",
"split": "train"
} |
math | [
{
"content": "Using the method of base prime representation, where the place of each digit represents an exponent in the prime factorization (starting with the smallest prime on the right), convert the number $196$ into base prime. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2002",
"style": "rule"
} | {
"index": "34715",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be a cyclic quadrilateral with circumradius $100\\sqrt{3}$ and $AC=300$ . If $\\angle DBC = 15^{\\circ}$ , then find $AD^2$ .\n\n*Proposed by Anand Iyer* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60000",
"style": "rule"
} | {
"index": "34716",
"split": "train"
} |
math | [
{
"content": "Jordan ran 2 miles in half the time it took Steve to run 3 miles. If it took Steve 24 minutes to run 3 miles, using the same rates, how many minutes would it take Jordan to run 5 miles? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "34717",
"split": "train"
} |
math | [
{
"content": "In an infinite increasing sequence of natural numbers, each number is divisible by at least one of the numbers 1005 and 1006, but none is divisible by 97. Additionally, any two consecutive numbers differ by no more than $k$. What is the smallest possible $k$ for this scenario? Please output the fi... | {
"ground_truth": "2011",
"style": "rule"
} | {
"index": "34718",
"split": "train"
} |
math | [
{
"content": "How many four-digit whole numbers are there such that the leftmost digit is odd, the second digit is even, and all four digits are different? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1400",
"style": "rule"
} | {
"index": "34719",
"split": "train"
} |
math | [
{
"content": "Fisherman Vasya caught several fish. He placed the three largest fish, which constitute 35% of the total weight of the catch, in the refrigerator. He gave the three smallest fish, which constitute 5/13 of the weight of the remaining fish, to the cat. Vasya ate all the rest of the caught fish himse... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "34720",
"split": "train"
} |
math | [
{
"content": "Given triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and the area is $S$. It is given that $6S=a^{2}\\sin A+b^{2}\\sin B$. Express $\\cos C$ in terms of $a$, $b$, and $c$ when the maximum value of $\\frac{{a+b}}{c}$ is achieved. Please ou... | {
"ground_truth": "\\frac{7}{9}",
"style": "rule"
} | {
"index": "34721",
"split": "train"
} |
math | [
{
"content": "How many solutions in natural numbers \\( x, y \\) does the system of equations have\n\n\\[\n\\left\\{\\begin{array}{l}\n\\gcd(x, y)=20! \\\\\n\\text{lcm}(x, y)=30!\n\\end{array} \\quad (\\text{where } n! = 1 \\cdot 2 \\cdot 3 \\cdot \\ldots \\cdot n) ?\\right.\n\\] Please output the final answer ... | {
"ground_truth": "256",
"style": "rule"
} | {
"index": "34722",
"split": "train"
} |
math | [
{
"content": "Suppose $f(x), g(x), h(x), p(x)$ are linear equations where $f(x) = x + 1$, $g(x) = -x + 5$, $h(x) = 4$, and $p(x) = 1$. Define new functions $j(x)$ and $k(x)$ as follows:\n$$j(x) = \\max\\{f(x), g(x), h(x), p(x)\\},$$\n$$k(x)= \\min\\{f(x), g(x), h(x), p(x)\\}.$$\nFind the length squared, $\\ell^... | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "34723",
"split": "train"
} |
math | [
{
"content": "For a positive integer \\( k \\), find the greatest common divisor (GCD) \\( d \\) of all positive even numbers \\( x \\) that satisfy the following conditions:\n1. Both \\( \\frac{x+2}{k} \\) and \\( \\frac{x}{k} \\) are integers, and the difference in the number of digits of these two numbers is... | {
"ground_truth": "1998",
"style": "rule"
} | {
"index": "34724",
"split": "train"
} |
math | [
{
"content": "6 boys and 4 girls are each assigned as attendants to 5 different buses, with 2 attendants per bus. Assuming that boys and girls are separated, and the buses are distinguishable, how many ways can the assignments be made? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5400",
"style": "rule"
} | {
"index": "34725",
"split": "train"
} |
math | [
{
"content": "Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\\sqrt{21}$ and $\\sqrt{31}$. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. ... | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "34726",
"split": "train"
} |
math | [
{
"content": "Given an ellipse $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1$ (where $a>0$, $b>0$) with eccentricity $\\frac{\\sqrt{6}}{3}$, the distance from the origin O to the line passing through points A $(0, -b)$ and B $(a, 0)$ is $\\frac{\\sqrt{3}}{2}$. Further, the line $y=kx+m$ ($k \\neq 0$, $m \\ne... | {
"ground_truth": "\\frac{5}{4}",
"style": "rule"
} | {
"index": "34727",
"split": "train"
} |
math | [
{
"content": "In a theater performance of King Lear, the locations of Acts II-V are drawn by lot before each act. The auditorium is divided into four sections, and the audience moves to another section with their chairs if their current section is chosen as the next location. Assume that all four sections are l... | {
"ground_truth": "1/2",
"style": "rule"
} | {
"index": "34728",
"split": "train"
} |
math | [
{
"content": "Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a fifty-cent piece. What is the probability that at least 40 cents worth of coins land on heads? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "34729",
"split": "train"
} |
math | [
{
"content": "In the complex plane, the complex numbers $\\frac {1}{1+i}$ and $\\frac {1}{1-i}$ (where $i$ is the imaginary unit) correspond to points A and B, respectively. If point C is the midpoint of line segment AB, determine the complex number corresponding to point C. Please output the final answer withi... | {
"ground_truth": "\\frac {1}{2}",
"style": "rule"
} | {
"index": "34730",
"split": "train"
} |
math | [
{
"content": "Determine the range of $w(w + x)(w + y)(w + z)$, where $x, y, z$, and $w$ are real numbers such that\n\\[x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "34731",
"split": "train"
} |
math | [
{
"content": "A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to $5$. How many units does the bug crawl altogether? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "34732",
"split": "train"
} |
math | [
{
"content": "Given the function $f\\left(x\\right)=x^{3}+ax^{2}+x+1$ achieves an extremum at $x=-1$. Find:<br/>$(1)$ The equation of the tangent line to $f\\left(x\\right)$ at $\\left(0,f\\left(0\\right)\\right)$;<br/>$(2)$ The maximum and minimum values of $f\\left(x\\right)$ on the interval $\\left[-2,0\\rig... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "34733",
"split": "train"
} |
math | [
{
"content": "What is the $y$-coordinate of the point on the $y$-axis that is equidistant from points $A( -2, 0)$ and $B(-1,4)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{13}{8}",
"style": "rule"
} | {
"index": "34734",
"split": "train"
} |
math | [
{
"content": "The mean, median, and mode of the five numbers 12, 9, 11, 16, x are all equal. Find the value of x. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "34735",
"split": "train"
} |
math | [
{
"content": "What is the greatest four-digit number which is a multiple of 17? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9996",
"style": "rule"
} | {
"index": "34736",
"split": "train"
} |
math | [
{
"content": "Calculate the area bounded by the graph of $y = \\arcsin(\\cos x)$ and the $x$-axis over the interval $0 \\leq x \\leq 2\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\pi^2",
"style": "rule"
} | {
"index": "34737",
"split": "train"
} |
math | [
{
"content": "Determine all real values of the parameter $a$ for which the equation\n\\[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\\]\nhas exactly four distinct real roots that form a geometric progression. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\boxed{a=170}$",
"style": "rule"
} | {
"index": "34738",
"split": "train"
} |
math | [
{
"content": "In the diagram below, \\( \\triangle ABC \\) is a triangle with \\( AB = 39 \\text{ cm}, BC = 45 \\text{ cm}, \\) and \\( CA = 42 \\text{ cm} \\). The tangents at \\( A \\) and \\( B \\) to the circumcircle of \\( \\triangle ABC \\) meet at the point \\( P \\). The point \\( D \\) lies on \\( BC \... | {
"ground_truth": "168",
"style": "rule"
} | {
"index": "34739",
"split": "train"
} |
math | [
{
"content": "Suppose that $ABCD$ is a rectangle with sides of length $12$ and $18$ . Let $S$ be the region of points contained in $ABCD$ which are closer to the center of the rectangle than to any of its vertices. Find the area of $S$ . Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "34740",
"split": "train"
} |
math | [
{
"content": "There are \\( n \\) distinct lines in the plane. One of these lines intersects exactly 5 of the \\( n \\) lines, another one intersects exactly 9 of the \\( n \\) lines, and yet another one intersects exactly 11 of them. Which of the following is the smallest possible value of \\( n \\)? Please ou... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "34741",
"split": "train"
} |
math | [
{
"content": "Let $n$ be the answer to this problem. An urn contains white and black balls. There are $n$ white balls and at least two balls of each color in the urn. Two balls are randomly drawn from the urn without replacement. Find the probability, in percent, that the first ball drawn is white and the secon... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "34742",
"split": "train"
} |
math | [
{
"content": "Given $f(x)=\\sin 2x+\\cos 2x$. \n$(1)$ Find the period and the interval of monotonic increase of $f(x)$. \n$(2)$ Find the maximum and minimum values of the function $f(x)$ on $[0,\\frac{π}{2}]$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "34743",
"split": "train"
} |
math | [
{
"content": "A cylinder has a radius of 5 cm and a height of 10 cm. What is the longest segment, in centimeters, that would fit inside the cylinder? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10\\sqrt{2}",
"style": "rule"
} | {
"index": "34744",
"split": "train"
} |
math | [
{
"content": "The planning committee at school has 10 members. Exactly four of these members are teachers. A four-person subcommittee with at least one member who is a teacher must be formed from the members of the planning committee. How many distinct subcommittees are possible? Please output the final answer ... | {
"ground_truth": "195",
"style": "rule"
} | {
"index": "34745",
"split": "train"
} |
math | [
{
"content": "Find the $x$-intercept point of the line $3x+5y=20$. Provide your answer as an ordered pair. Express the $x$ and $y$ coordinates as common fractions as needed. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left(\\frac{20}{3},0\\right)",
"style": "rule"
} | {
"index": "34746",
"split": "train"
} |
math | [
{
"content": "Given that Ron mistakenly reversed the digits of the two-digit number $a$, and the product of $a$ and $b$ was mistakenly calculated as $221$, determine the correct product of $a$ and $b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "527",
"style": "rule"
} | {
"index": "34747",
"split": "train"
} |
math | [
{
"content": "Given $X \\sim N(5, 4)$, find $P(1 < X \\leq 7)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.9759",
"style": "rule"
} | {
"index": "34748",
"split": "train"
} |
math | [
{
"content": "If we want to write down all the integers from 1 to 10,000, how many times do we have to write a digit, for example, the digit 5? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4000",
"style": "rule"
} | {
"index": "34749",
"split": "train"
} |
math | [
{
"content": "Compute the smallest positive integer such that, no matter how you rearrange its digits (in base ten), the resulting number is a multiple of 63. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "111888",
"style": "rule"
} | {
"index": "34750",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=-x^{3}+ax^{2}+bx$ in the interval $(-2,1)$. The function reaches its minimum value when $x=-1$ and its maximum value when $x=\\frac{2}{3}$.\n(1) Find the equation of the tangent line to the function $y=f(x)$ at $x=-2$.\n(2) Find the maximum and minimum values of the functi... | {
"ground_truth": "-\\frac{3}{2}",
"style": "rule"
} | {
"index": "34751",
"split": "train"
} |
math | [
{
"content": "On a redesigned dartboard, the outer circle radius is increased to $8$ units and the inner circle has a radius of $4$ units. Additionally, two radii divide the board into four congruent sections, each labeled inconsistently with point values as follows: inner sections have values of $3$ and $4$, a... | {
"ground_truth": "\\frac{9}{1024}",
"style": "rule"
} | {
"index": "34752",
"split": "train"
} |
math | [
{
"content": "Let $z$ be a non-real complex number with $z^{23}=1$. Compute $$ \\sum_{k=0}^{22} \\frac{1}{1+z^{k}+z^{2 k}} $$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "46 / 3",
"style": "rule"
} | {
"index": "34753",
"split": "train"
} |
math | [
{
"content": "If 3913 were to be 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": "47",
"style": "rule"
} | {
"index": "34754",
"split": "train"
} |
math | [
{
"content": "Given a geometric sequence $\\{a_n\\}$ with a common ratio $q$, and it satisfies $|q| > 1$, $a_2 + a_7 = 2$, and $a_4 \\cdot a_5 = -15$, find the value of $a_{12}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{25}{3}",
"style": "rule"
} | {
"index": "34755",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer $b$ for which $x^2+bx+2008$ factors into a product of two binomials, each having integer coefficients. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "259",
"style": "rule"
} | {
"index": "34756",
"split": "train"
} |
math | [
{
"content": "The sequence $(a_n)$ satisfies $a_0=0$ and $a_{n + 1} = \\frac85a_n + \\frac65\\sqrt {4^n - a_n^2}$ for $n\\geq 0$. Find the greatest integer less than or equal to $a_{10}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "983",
"style": "rule"
} | {
"index": "34757",
"split": "train"
} |
math | [
{
"content": "A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimetres and its base has radius $30$ centimetres, find the volume of the larg... | {
"ground_truth": "108000",
"style": "rule"
} | {
"index": "34758",
"split": "train"
} |
math | [
{
"content": "There is a beach soccer tournament with 17 teams, where each team plays against every other team exactly once. A team earns 3 points for a win in regular time, 2 points for a win in extra time, and 1 point for a win in a penalty shootout. The losing team earns no points. What is the maximum number... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "34759",
"split": "train"
} |
math | [
{
"content": "The product $8 \\times .25 \\times 2 \\times .125 =$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\frac{1}{2}$",
"style": "rule"
} | {
"index": "34760",
"split": "train"
} |
math | [
{
"content": "Let $T$ be the set of points $(x, y)$ in the Cartesian plane that satisfy\n\\[\\big|\\big| |x|-3\\big|-1\\big|+\\big|\\big| |y|-3\\big|-1\\big|=2.\\]\nWhat is the total length of all the lines that make up $T$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32\\sqrt{2}",
"style": "rule"
} | {
"index": "34761",
"split": "train"
} |
math | [
{
"content": "Three dice are thrown, and the sums of the points that appear on them are counted. In how many ways can you get a total of 5 points and 6 points? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "34762",
"split": "train"
} |
math | [
{
"content": "Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. H... | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "34763",
"split": "train"
} |
math | [
{
"content": "A collector has \\( N \\) precious stones. If he takes away the three heaviest stones, then the total weight of the stones decreases by \\( 35\\% \\). From the remaining stones, if he takes away the three lightest stones, the total weight further decreases by \\( \\frac{5}{13} \\). Find \\( N \\).... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "34764",
"split": "train"
} |
math | [
{
"content": "Given rational numbers $x$ and $y$ satisfy $|x|=9$, $|y|=5$. \n$(1)$ If $x \\lt 0$, $y \\gt 0$, find the value of $x+y$; \n$(2)$ If $|x+y|=x+y$, find the value of $x-y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "34765",
"split": "train"
} |
math | [
{
"content": "What is $8^{15} \\div 64^3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8^9",
"style": "rule"
} | {
"index": "34766",
"split": "train"
} |
math | [
{
"content": "A rectangular pool table has vertices at $(0,0)(12,0)(0,10)$, and $(12,10)$. There are pockets only in the four corners. A ball is hit from $(0,0)$ along the line $y=x$ and bounces off several walls before eventually entering a pocket. Find the number of walls that the ball bounces off of before e... | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "34767",
"split": "train"
} |
math | [
{
"content": "Let the function $f(x)= \\frac{ \\sqrt{3}}{2}- \\sqrt{3}\\sin^2 \\omega x-\\sin \\omega x\\cos \\omega x$ ($\\omega > 0$) and the graph of $y=f(x)$ has a symmetry center whose distance to the nearest axis of symmetry is $\\frac{\\pi}{4}$.\n\n$(1)$ Find the value of $\\omega$; $(2)$ Find the maximu... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "34768",
"split": "train"
} |
math | [
{
"content": "A regular football match is being played. A draw is possible. The waiting time for the next goal is independent of previous events in the match. It is known that the expected total number of goals in football matches of these teams is 2.8. Find the probability that an even number of goals will be ... | {
"ground_truth": "0.502",
"style": "rule"
} | {
"index": "34769",
"split": "train"
} |
math | [
{
"content": "What is the remainder when $9^{1995}$ is divided by 7? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "34770",
"split": "train"
} |
math | [
{
"content": "In Pascal's triangle, compute the seventh element in Row 20. Afterward, determine how many times greater this element is compared to the third element in the same row. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "204",
"style": "rule"
} | {
"index": "34771",
"split": "train"
} |
math | [
{
"content": "In triangle $PQR$, let the side lengths be $PQ = 7,$ $PR = 8,$ and $QR = 5$. Calculate:\n\\[\\frac{\\cos \\frac{P - Q}{2}}{\\sin \\frac{R}{2}} - \\frac{\\sin \\frac{P - Q}{2}}{\\cos \\frac{R}{2}}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{16}{7}",
"style": "rule"
} | {
"index": "34772",
"split": "train"
} |
math | [
{
"content": "A cafe has 3 tables and 5 individual counter seats. People enter in groups of size between 1 and 4, inclusive, and groups never share a table. A group of more than 1 will always try to sit at a table, but will sit in counter seats if no tables are available. Conversely, a group of 1 will always tr... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "34773",
"split": "train"
} |
math | [
{
"content": "In a cultural performance, there are already 10 programs arranged in the program list. Now, 3 more programs are to be added, with the requirement that the relative order of the originally scheduled 10 programs remains unchanged. How many different arrangements are there for the program list? (Answ... | {
"ground_truth": "1716",
"style": "rule"
} | {
"index": "34774",
"split": "train"
} |
math | [
{
"content": "Find the number of positive integers $n$ that satisfy\n\\[(n - 2)(n - 4)(n - 6) \\dotsm (n - 98) < 0.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "34775",
"split": "train"
} |
math | [
{
"content": "A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to $5$. How many units does the bug crawl altogether? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "34776",
"split": "train"
} |
math | [
{
"content": "A pizza parlor has six different toppings. How many different one- and two-topping pizzas can you order? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "34777",
"split": "train"
} |
math | [
{
"content": "What is the value of $2468 + 8642 + 6824 + 4286$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22220",
"style": "rule"
} | {
"index": "34778",
"split": "train"
} |
math | [
{
"content": "Given the function f(x) = 2x^3 - ax^2 + 1, where a ∈ R.\n(I) When a = 6, the line y = -6x + m is tangent to f(x). Find the value of m.\n(II) If the function f(x) has exactly one zero in the interval (0, +∞), find the monotonic intervals of the function.\n(III) When a > 0, if the sum of the maximum... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "34779",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be a square of side length $1$ , and let $P$ be a variable point on $\\overline{CD}$ . Denote by $Q$ the intersection point of the angle bisector of $\\angle APB$ with $\\overline{AB}$ . The set of possible locations for $Q$ as $P$ varies along $\\overline{CD}$ is a l... | {
"ground_truth": "3 - 2\\sqrt{2}",
"style": "rule"
} | {
"index": "34780",
"split": "train"
} |
math | [
{
"content": "Square $ABCD$ is inscribed in the region bound by the parabola $y = x^2 - 8x + 12$ and the $x$-axis, as shown below. Find the area of square $ABCD.$\n\n[asy]\nunitsize(0.8 cm);\n\nreal parab (real x) {\n return(x^2 - 8*x + 12);\n}\n\npair A, B, C, D;\nreal x = -1 + sqrt(5);\n\nA = (4 - x,0);\nB ... | {
"ground_truth": "24 - 8 \\sqrt{5}",
"style": "rule"
} | {
"index": "34781",
"split": "train"
} |
math | [
{
"content": "Consider all polynomials of the form\n\\[x^9 + a_8 x^8 + a_7 x^7 + \\dots + a_2 x^2 + a_1 x + a_0,\\]where $a_i \\in \\{0,1\\}$ for all $0 \\le i \\le 8.$ Find the number of such polynomials that have exactly two different integer roots. Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "56",
"style": "rule"
} | {
"index": "34782",
"split": "train"
} |
math | [
{
"content": "Grandma Wang has 6 stools that need to be painted by a painter. Each stool needs to be painted twice. The first coat takes 2 minutes, but there must be a 10-minute wait before applying the second coat. How many minutes will it take to paint all 6 stools? Please output the final answer within \\box... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "34783",
"split": "train"
} |
math | [
{
"content": "What is the total number of digits used when the first 2500 positive even integers are written? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9449",
"style": "rule"
} | {
"index": "34784",
"split": "train"
} |
math | [
{
"content": "Given integers $x$ and $y$ satisfy the equation $2xy + x + y = 83$, find the values of $x + y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-85",
"style": "rule"
} | {
"index": "34785",
"split": "train"
} |
math | [
{
"content": "Let $x$ and $y$ be positive real numbers. Find the minimum value of\n\\[\\left( x + \\frac{1}{y} \\right) \\left( x + \\frac{1}{y} - 1024 \\right) + \\left( y + \\frac{1}{x} \\right) \\left( y + \\frac{1}{x} - 1024 \\right).\\] Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "-524288",
"style": "rule"
} | {
"index": "34786",
"split": "train"
} |
math | [
{
"content": "Two numbers whose sum is $6$ and the absolute value of whose difference is $8$ are roots of the equation: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "x^2-6x-7=0",
"style": "rule"
} | {
"index": "34787",
"split": "train"
} |
math | [
{
"content": "Let $AB$ be a segment of unit length and let $C, D$ be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set $\\{A,B,C,D\\}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{5}}{125}",
"style": "rule"
} | {
"index": "34788",
"split": "train"
} |
math | [
{
"content": "Perform the calculations: \n3.21 - 1.05 - 1.95 \n15 - (2.95 + 8.37) \n14.6 × 2 - 0.6 × 2 \n0.25 × 1.25 × 32 Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "34789",
"split": "train"
} |
math | [
{
"content": "If $x$ is a number between 0 and 1, which of the following represents the smallest value?\n\nA). $x$\nB). $x^2$\nC). $2x$\nD). $\\sqrt{x}$\nE). $\\frac{1}{x}$\n\nExpress your answer as A, B, C, D, or E. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\text{B}",
"style": "rule"
} | {
"index": "34790",
"split": "train"
} |
math | [
{
"content": "Let $A = (1,0),$ $B = (4,3),$ and $C = (p,q)$ be three points on the parabola $y = -x^2 + 6x - 5,$ where $1 \\le p \\le 4.$ Find the largest possible area of triangle $ABC.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{27}{8}",
"style": "rule"
} | {
"index": "34791",
"split": "train"
} |
math | [
{
"content": "How many distinguishable arrangements are there of $1$ brown tile, $1$ purple tile, $2$ green tiles, and $3$ yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "420",
"style": "rule"
} | {
"index": "34792",
"split": "train"
} |
math | [
{
"content": "$2016$ bugs are sitting in different places of $1$ -meter stick. Each bug runs in one or another direction with constant and equal speed. If two bugs face each other, then both of them change direction but not speed. If bug reaches one of the ends of the stick, then it flies away. What is the gr... | {
"ground_truth": "1008^2",
"style": "rule"
} | {
"index": "34793",
"split": "train"
} |
math | [
{
"content": "Determine the radius of the sphere that touches the faces of the unit cube passing through vertex $A$ and the edges passing through vertex $B$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 - \\sqrt{2}",
"style": "rule"
} | {
"index": "34794",
"split": "train"
} |
math | [
{
"content": "Simplify $\\frac{1}{330} + \\frac{19}{30}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{11}",
"style": "rule"
} | {
"index": "34795",
"split": "train"
} |
math | [
{
"content": "Given the set $\\{a, \\frac{b}{a}, 1\\} = \\{a^2, a+b, 0\\}$, find the value of $a^{2015} + b^{2016}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "34796",
"split": "train"
} |
math | [
{
"content": "Given the line $x-y+2=0$ and the circle $C$: $(x-3)^{2}+(y-3)^{2}=4$ intersect at points $A$ and $B$. The diameter through the midpoint of chord $AB$ is $MN$. Calculate the area of quadrilateral $AMBN$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{2}",
"style": "rule"
} | {
"index": "34797",
"split": "train"
} |
math | [
{
"content": "For each prime $p$, a polynomial $P(x)$ with rational coefficients is called $p$-good if and only if there exist three integers $a, b$, and $c$ such that $0 \\leq a<b<c<\\frac{p}{3}$ and $p$ divides all the numerators of $P(a)$, $P(b)$, and $P(c)$, when written in simplest form. Compute the number... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "34798",
"split": "train"
} |
math | [
{
"content": "Evaluate $(7 + 5 + 3) \\div 3 - 2 \\div 3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4 \\frac{1}{3}",
"style": "rule"
} | {
"index": "34799",
"split": "train"
} |
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