data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Let \\( f(n) \\) be the sum of the squares of the digits of positive integer \\( n \\) (in decimal). For example, \\( f(123) = 1^{2} + 2^{2} + 3^{2} = 14 \\). Define \\( f_{1}(n) = f(n) \\), and \\( f_{k+1}(n) = f\\left(f_{k}(n)\\right) \\) for \\( k = 1, 2, 3, \\ldots \\). What is the value of \\... | {
"ground_truth": "145",
"style": "rule"
} | {
"index": "2900",
"split": "train"
} |
math | [
{
"content": "The sequence ${a_n}$ satisfies the equation $a_{n+1} + (-1)^n a_n = 2n - 1$. Determine the sum of the first 80 terms of the sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3240",
"style": "rule"
} | {
"index": "2901",
"split": "train"
} |
math | [
{
"content": "A rectangle with dimensions $8 \\times 2 \\sqrt{2}$ and a circle with a radius of 2 have a common center. What is the area of their overlapping region? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$2 \\pi + 4$",
"style": "rule"
} | {
"index": "2902",
"split": "train"
} |
math | [
{
"content": "I have written a strictly increasing sequence of six positive integers, such that each number (besides the first) is a multiple of the one before it, and the sum of all six numbers is 79 . What is the largest number in my sequence? Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "2903",
"split": "train"
} |
math | [
{
"content": "Simplify \\[\\frac{1}{\\dfrac{1}{\\sqrt{2}+1} + \\dfrac{2}{\\sqrt{3}-1}}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt3-\\sqrt2",
"style": "rule"
} | {
"index": "2904",
"split": "train"
} |
math | [
{
"content": "The graph of \\(y^2 + 2xy + 60|x| = 900\\) partitions the plane into several regions. What is the area of the bounded region? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1800",
"style": "rule"
} | {
"index": "2905",
"split": "train"
} |
math | [
{
"content": "The polynomial $Q(x) = 3x^3 + dx^2 + ex + f$ has the property that the mean of its zeros, the product of its zeros, and the sum of the coefficients are all equal. The $y$-intercept of the graph of $y = Q(x)$ is 9. What is $e$? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "-42",
"style": "rule"
} | {
"index": "2906",
"split": "train"
} |
math | [
{
"content": "Given a constant function on the interval $(0,1)$, $f(x)$, which satisfies: when $x \\notin \\mathbf{Q}$, $f(x)=0$; and when $x=\\frac{p}{q}$ (with $p, q$ being integers, $(p, q)=1, 0<p<q$), $f(x)=\\frac{p+1}{q}$. Determine the maximum value of $f(x)$ on the interval $\\left(\\frac{7}{8}, \\frac{8... | {
"ground_truth": "$\\frac{16}{17}$",
"style": "rule"
} | {
"index": "2907",
"split": "train"
} |
math | [
{
"content": "In a certain math competition, there are 10 multiple-choice questions. Each correct answer earns 4 points, no answer earns 0 points, and each wrong answer deducts 1 point. If the total score becomes negative, the grading system automatically sets the total score to zero. How many different total s... | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "2908",
"split": "train"
} |
math | [
{
"content": "How many roots does the equation\n\n$$\n\\overbrace{f(f(\\ldots f}^{10 \\text{ times } f}(x) \\ldots)) + \\frac{1}{2} = 0\n$$\n\nhave, where \\( f(x) = |x| - 1 \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "2909",
"split": "train"
} |
math | [
{
"content": "Given that the function $f(x)$ is defined on $\\mathbb{R}$ and has a period of $4$ as an even function. When $x \\in [2,4]$, $f(x) =|\\log_4 \\left(x-\\frac{3}{2}\\right)|$. Determine the value of $f\\left(\\frac{1}{2}\\right)$. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "2910",
"split": "train"
} |
math | [
{
"content": "If $\\theta \\in \\left[\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2}\\right]$ and $\\sin 2\\theta = \\dfrac{3\\sqrt{7}}{8}$, find the value of $\\sin \\theta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{3}{4}",
"style": "rule"
} | {
"index": "2911",
"split": "train"
} |
math | [
{
"content": "A factory conducted a survey on the defective parts produced by a team. The number of defective parts produced by the team each day in 7 days is as follows (unit: pieces): 3, 3, 0, 2, 3, 0, 3. What is the value of the variance of the number of defective parts produced by the team in 7 days? Please... | {
"ground_truth": "\\frac{12}{7}",
"style": "rule"
} | {
"index": "2912",
"split": "train"
} |
math | [
{
"content": "How many of the numbers from the set $\\{1,\\ 2,\\ 3,\\ldots,\\ 100\\}$ have a perfect cube factor other than one? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "2913",
"split": "train"
} |
math | [
{
"content": "A 16-step path is to go from $(-4,-4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $-2 < x < 2$, $-2 < y < 2$ at each step? Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "1698",
"style": "rule"
} | {
"index": "2914",
"split": "train"
} |
math | [
{
"content": "A particle starts from the origin on the number line, and at each step, it can move either 1 unit in the positive direction or 1 unit in the negative direction. After 10 steps, if the distance between the particle and the origin is 4, then the total number of distinct ways the particle can move is... | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "2915",
"split": "train"
} |
math | [
{
"content": "The sum of two numbers is $30$. If we double the larger number, and subtract three times the smaller number, the result is 5. What is the positive difference between the two numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "2916",
"split": "train"
} |
math | [
{
"content": "Find the sum of the arithmetic series\n\\[20+20\\frac{1}{5}+20\\frac{2}{5}+\\cdots+40\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3030",
"style": "rule"
} | {
"index": "2917",
"split": "train"
} |
math | [
{
"content": "If \\( p \\) is the smallest positive prime number such that for some integer \\( n \\), \\( p \\) divides \\( n^{2} + 5n + 23 \\), then \\( p = \\) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "2918",
"split": "train"
} |
math | [
{
"content": "A point $(x, y)$ is selected uniformly at random from the unit square $S=\\{(x, y) \\mid 0 \\leq x \\leq 1,0 \\leq y \\leq 1\\}$. If the probability that $(3x+2y, x+4y)$ is in $S$ is $\\frac{a}{b}$, where $a, b$ are relatively prime positive integers, compute $100a+b$. Please output the final answ... | {
"ground_truth": "820",
"style": "rule"
} | {
"index": "2919",
"split": "train"
} |
math | [
{
"content": "A triangle \\( ABC \\) is given. It is known that \\( AB=4 \\), \\( AC=2 \\), and \\( BC=3 \\). The angle bisector of \\( \\angle BAC \\) intersects side \\( BC \\) at point \\( K \\). A line passing through point \\( B \\) parallel to \\( AC \\) intersects the extension of the bisector \\( AK \\)... | {
"ground_truth": "2 \\sqrt{6}",
"style": "rule"
} | {
"index": "2920",
"split": "train"
} |
math | [
{
"content": "A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$... | {
"ground_truth": "683",
"style": "rule"
} | {
"index": "2921",
"split": "train"
} |
math | [
{
"content": "What is the least five-digit positive integer which is congruent to 6 (mod 17)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10,017",
"style": "rule"
} | {
"index": "2922",
"split": "train"
} |
math | [
{
"content": "Participation in the local soccer league is $10\\%$ higher this year than it was last year. Also, from last year to this year, the number of males increased by $5\\%$ and the number of females increased by $20\\%$. There were $20$ males participating in the league last year. What fraction of the l... | {
"ground_truth": "\\frac{4}{11}",
"style": "rule"
} | {
"index": "2923",
"split": "train"
} |
math | [
{
"content": "Find the number of integers $n$ such that \\[1+\\left\\lfloor\\dfrac{100n}{101}\\right\\rfloor=\\left\\lceil\\dfrac{99n}{100}\\right\\rceil.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10100",
"style": "rule"
} | {
"index": "2924",
"split": "train"
} |
math | [
{
"content": "A solid right prism $PQRSTU$ has a height of 20. Its bases are equilateral triangles with side length 10. Points $M$, $N$, and $O$ are the midpoints of edges $PQ$, $QR$, and $RT$, respectively. Calculate the perimeter of triangle $MNO$. Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "5 + 10\\sqrt{5}",
"style": "rule"
} | {
"index": "2925",
"split": "train"
} |
math | [
{
"content": "The matrix\n\\[\\mathbf{M} = \\begin{pmatrix} 0 & 2y & z \\\\ x & y & -z \\\\ x & -y & z \\end{pmatrix}\\]satisfies $\\mathbf{M}^T \\mathbf{M} = \\mathbf{I}.$ Find $x^2 + y^2 + z^2.$\n\nNote: For a matrix $\\mathbf{A},$ $\\mathbf{A}^T$ is the transpose of $\\mathbf{A},$ which is generated by refl... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "2926",
"split": "train"
} |
math | [
{
"content": "A square $EFGH$ is inscribed in the region bound by the parabola $y = x^2 - 8x + 16$ and the $x$-axis. Find the area of square $EFGH$.\n\n[asy]\nunitsize(0.8 cm);\n\nreal parab2 (real x) {\n return(x^2 - 8*x + 16);\n}\n\npair E, F, G, H;\nreal x = -2 + 2*sqrt(3);\n\nE = (4 - x,0);\nF = (4 + x,0);... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "2927",
"split": "train"
} |
math | [
{
"content": "Calculate: \\(\\frac{1}{2 \\cos \\frac{2 \\pi}{7}}+\\frac{1}{2 \\cos \\frac{4 \\pi}{7}}+\\frac{1}{2 \\cos \\frac{6 \\pi}{7}}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "2928",
"split": "train"
} |
math | [
{
"content": "Gloria's grandmother cycled a total of $\\frac{3}{6} + \\frac{4}{4} + \\frac{3}{3} + \\frac{2}{8}$ hours at different speeds, and she cycled a total of $\\frac{3}{5} + \\frac{4}{5} + \\frac{3}{5} + \\frac{2}{5}$ hours at a speed of 5 miles per hour. Calculate the difference in the total time she s... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "2929",
"split": "train"
} |
math | [
{
"content": "Given that $a$, $b$, and $c$ are the roots of the equation $x^3-3x^2+mx+24=0$, and that $-a$ and $-b$ are the roots of the equation $x^2+nx-6=0$, then the value of $n$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "2930",
"split": "train"
} |
math | [
{
"content": "Every day, 12 tons of potatoes are delivered to the city using one type of transport from three collective farms. The price per ton is 4 rubles from the first farm, 3 rubles from the second farm, and 1 ruble from the third farm. To ensure timely delivery, the loading process for the required 12 to... | {
"ground_truth": "24.6667",
"style": "rule"
} | {
"index": "2931",
"split": "train"
} |
math | [
{
"content": "In any permutation of the numbers \\(1, 2, 3, \\ldots, 18\\), we can always find a set of 6 consecutive numbers whose sum is at least \\(m\\). Find the maximum value of the real number \\(m\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "57",
"style": "rule"
} | {
"index": "2932",
"split": "train"
} |
math | [
{
"content": "At 7:10 in the morning, Xiao Ming's mother wakes him up and asks him to get up. However, Xiao Ming sees the time in the mirror and thinks that it is not yet time to get up. He tells his mother, \"It's still early!\" Xiao Ming mistakenly believes that the time is $\\qquad$ hours $\\qquad$ minutes. ... | {
"ground_truth": "4:50",
"style": "rule"
} | {
"index": "2933",
"split": "train"
} |
math | [
{
"content": "Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\\text{9:99}$ just before midnight, $\\text{0:00}$ at midnight, $\\tex... | {
"ground_truth": "275",
"style": "rule"
} | {
"index": "2934",
"split": "train"
} |
math | [
{
"content": "Given that the coefficient of the $x^3$ term in the expansion of $\\left(x+a\\right)\\left(x-2\\right)^5$ is $-60$, find the value of $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "2935",
"split": "train"
} |
math | [
{
"content": "Find the number of diagonals in a polygon with 120 sides, and calculate the length of one diagonal of this polygon assuming it is regular and each side length is 5 cm. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "2936",
"split": "train"
} |
math | [
{
"content": "In a club election, 10 officer positions are available. There are 20 candidates, of which 8 have previously served as officers. Determine how many different slates of officers include at least one of the past officers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "184,690",
"style": "rule"
} | {
"index": "2937",
"split": "train"
} |
math | [
{
"content": "A gives B as many cents as B has and C as many cents as C has. Similarly, B then gives A and C as many cents as each then has. C, similarly, then gives A and B as many cents as each then has. After this, each person gives half of what they have to each other person. If each finally has 24 cents, c... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "2938",
"split": "train"
} |
math | [
{
"content": "A point \\((x, y)\\) is randomly selected such that \\(0 \\leq x \\leq 4\\) and \\(0 \\leq y \\leq 5\\). What is the probability that \\(x + y \\leq 5\\)? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "2939",
"split": "train"
} |
math | [
{
"content": "Given that the sequence {a<sub>n</sub>} is an arithmetic sequence, if $\\frac {a_{11}}{a_{10}}$ < -1, and its first n terms sum S<sub>n</sub> has a maximum value, determine the maximum value of n that makes S<sub>n</sub> > 0. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "2940",
"split": "train"
} |
math | [
{
"content": "In two days, 50 financiers raised funds to combat a new virus. Each of them contributed a one-time whole number amount in thousands of rubles, not exceeding 100. Each contribution on the first day did not exceed 50 thousand rubles, while each on the second day was greater than this amount; and no ... | {
"ground_truth": "2525",
"style": "rule"
} | {
"index": "2941",
"split": "train"
} |
math | [
{
"content": "Given the set \\( A = \\{0, 1, 2, 3, 4, 5, 6, 7\\} \\), how many mappings \\( f \\) from \\( A \\) to \\( A \\) satisfy the following conditions?\n1. For all \\( i, j \\in A \\) with \\( i \\neq j \\), \\( f(i) \\neq f(j) \\).\n2. For all \\( i, j \\in A \\) with \\( i + j = 7 \\), \\( f(i) + f(j)... | {
"ground_truth": "384",
"style": "rule"
} | {
"index": "2942",
"split": "train"
} |
math | [
{
"content": "Determine how many roots of the equation\n\n$$\n4 \\sin 2x + 3 \\cos 2x - 2 \\sin x - 4 \\cos x + 1 = 0\n$$\n\nare located in the interval \\(\\left[10^{2014!} \\pi, 10^{2014!+2015} \\pi\\right]\\). In the answer, write down the sum of all the digits of the found number. Please output the final an... | {
"ground_truth": "18135",
"style": "rule"
} | {
"index": "2943",
"split": "train"
} |
math | [
{
"content": "$ABCD$ is a rectangle; $P$ and $Q$ are the mid-points of $AB$ and $BC$ respectively. $AQ$ and $CP$ meet at $R$. If $AC = 6$ and $\\angle ARC = 150^{\\circ}$, find the area of $ABCD$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8\\sqrt{3}",
"style": "rule"
} | {
"index": "2944",
"split": "train"
} |
math | [
{
"content": "How many multiples of 5 are between 80 and 375? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "59",
"style": "rule"
} | {
"index": "2945",
"split": "train"
} |
math | [
{
"content": "For how many values of $c$ in the interval $[0, 1000]$ does the equation \\[5 \\lfloor x \\rfloor + 3 \\lceil x \\rceil = c\\]have a solution for $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "251",
"style": "rule"
} | {
"index": "2946",
"split": "train"
} |
math | [
{
"content": "Alice and Bob each arrive at a party at a random time between 1:00 and 2:00. If Alice arrives after Bob, what is the probability that Bob arrived before 1:30? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "2947",
"split": "train"
} |
math | [
{
"content": "A four-digit integer \\( p \\) and the four-digit integer obtained by reversing the order of the digits of \\( p \\) are both divisible by \\( 63 \\) (product of primes \\( 7 \\) and \\( 9 \\)). Additionally, \\( p \\) must be divisible by \\( 19 \\). Determine the greatest possible value of \\( p... | {
"ground_truth": "5985",
"style": "rule"
} | {
"index": "2948",
"split": "train"
} |
math | [
{
"content": "Simplify $\\frac{1}{1+\\sqrt{2}}\\cdot\\frac{1}{1-\\sqrt{2}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "2949",
"split": "train"
} |
math | [
{
"content": "What is the cube of the square of the second smallest prime number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "729",
"style": "rule"
} | {
"index": "2950",
"split": "train"
} |
math | [
{
"content": "What is the greatest integer value of $x$ for which $5-4x>17$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "2951",
"split": "train"
} |
math | [
{
"content": "$A, B, C$ , and $D$ are all on a circle, and $ABCD$ is a convex quadrilateral. If $AB = 13$ , $BC = 13$ , $CD = 37$ , and $AD = 47$ , what is the area of $ABCD$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "504",
"style": "rule"
} | {
"index": "2952",
"split": "train"
} |
math | [
{
"content": "If the markings on the number line are equally spaced, what is the number $\\text{y}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "2953",
"split": "train"
} |
math | [
{
"content": "Find the maximum and minimum values of the function $f(x)=\\frac{1}{3}x^3-4x$ on the interval $\\left[-3,3\\right]$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{16}{3}",
"style": "rule"
} | {
"index": "2954",
"split": "train"
} |
math | [
{
"content": "Compute $\\cos 135^\\circ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "2955",
"split": "train"
} |
math | [
{
"content": "Find a nonzero $p$ such that $px^2-12x+4=0$ has only one solution. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "2956",
"split": "train"
} |
math | [
{
"content": "I have chosen five of the numbers $\\{1,2,3,4,5,6,7\\}$. If I told you what their product was, that would not be enough information for you to figure out whether their sum was even or odd. What is their product? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "420",
"style": "rule"
} | {
"index": "2957",
"split": "train"
} |
math | [
{
"content": "A merchant buys $n$ radios for $d$ dollars, where $d$ is a positive integer. The merchant sells two radios at half the cost price to a charity sale, and the remaining radios at a profit of 8 dollars each. If the total profit is 72 dollars, what is the smallest possible value of $n$? Please output ... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "2958",
"split": "train"
} |
math | [
{
"content": "Let $L$ be the number formed by $2022$ digits equal to $1$, that is, $L=1111\\dots 111$.\nCompute the sum of the digits of the number $9L^2+2L$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4044",
"style": "rule"
} | {
"index": "2959",
"split": "train"
} |
math | [
{
"content": "Given the parabola $y^{2}=4x$ and the circle $E:(x-4)^{2}+y^{2}=12$. Let $O$ be the origin. The line passing through the center of circle $E$ intersects the circle at points $A$ and $B$. The lines $OA$ and $OB$ intersect the parabola at points $P$ and $Q$ (points $P$ and $Q$ are not coincident wit... | {
"ground_truth": "\\frac{9}{16}",
"style": "rule"
} | {
"index": "2960",
"split": "train"
} |
math | [
{
"content": "The set of $x$-values satisfying the inequality $2 \\leq |x-1| \\leq 5$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4\\leq x\\leq-1\\text{ or }3\\leq x\\leq 6",
"style": "rule"
} | {
"index": "2961",
"split": "train"
} |
math | [
{
"content": "When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that the product of these three numbers, $abc$, equals 8. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{216}",
"style": "rule"
} | {
"index": "2962",
"split": "train"
} |
math | [
{
"content": "Let $b_1$, $b_2$, $b_3$, $c_1$, $c_2$, and $c_3$ be real numbers such that for every real number $x$, we have\n\\[\n x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 = (x^2 + b_1 x + c_1)(x^2 + b_2 x + c_2)(x^2 + b_3 x + c_3). \n\\]Compute $b_1 c_1 + b_2 c_2 + b_3 c_3$. Please output the final answer within \... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "2963",
"split": "train"
} |
math | [
{
"content": "Let $(a_1,a_2,a_3,\\ldots,a_{12})$ be a permutation of $(1,2,3,\\ldots,12)$ for which\n$a_1>a_2>a_3>a_4>a_5>a_6 \\mathrm{\\ and \\ } a_6<a_7<a_8<a_9<a_{10}<a_{11}<a_{12}.$\nAn example of such a permutation is $(6,5,4,3,2,1,7,8,9,10,11,12).$ Find the number of such permutations. Please output the ... | {
"ground_truth": "462",
"style": "rule"
} | {
"index": "2964",
"split": "train"
} |
math | [
{
"content": "In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side lengths $AF=7$ and $FC=9,$ as shown. The area of the shaded region common to the interiors of both rectangles is $\\frac mn$, where $m$ and $n$ are relatively prime positive intege... | {
"ground_truth": "65",
"style": "rule"
} | {
"index": "2965",
"split": "train"
} |
math | [
{
"content": "There are 7 cylindrical logs with a diameter of 5 decimeters each. They are tied together at two places using a rope. How many decimeters of rope are needed at least (excluding the rope length at the knots, and using $\\pi$ as 3.14)? Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "91.4",
"style": "rule"
} | {
"index": "2966",
"split": "train"
} |
math | [
{
"content": "Given a regular 2017-sided polygon \\(A_{1} A_{2} \\cdots A_{2017}\\) inscribed in a unit circle \\(\\odot O\\), choose any two different vertices \\(A_{i}, A_{j}\\). Find the probability that \\(\\overrightarrow{O A_{i}} \\cdot \\overrightarrow{O A_{j}} > \\frac{1}{2}\\). Please output the final ... | {
"ground_truth": "1/3",
"style": "rule"
} | {
"index": "2967",
"split": "train"
} |
math | [
{
"content": "In the expansion of $\\left(a - \\dfrac{1}{\\sqrt{a}}\\right)^7$ the coefficient of $a^{-\\frac{1}{2}}$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-21",
"style": "rule"
} | {
"index": "2968",
"split": "train"
} |
math | [
{
"content": "What is the smallest possible perimeter, in units, of a triangle whose side-length measures are consecutive integer values? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "2969",
"split": "train"
} |
math | [
{
"content": "How many numbers are in the list $165, 159, 153, \\ldots, 30, 24?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "2970",
"split": "train"
} |
math | [
{
"content": "The graph of the function $f(x)=\\sin(2x+\\varphi)$ is translated to the right by $\\frac{\\pi}{6}$ units, and the resulting graph is symmetric about the origin. Determine the value of $\\varphi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": "2971",
"split": "train"
} |
math | [
{
"content": "Given $y=y_{1}+y_{2}$, where $y_{1}$ is directly proportional to $(x+1)$, and $y_{2}$ is inversely proportional to $(x+1)$. When $x=0$, $y=-5$; when $x=2$, $y=-7$. Find the value of $x$ when $y=5$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "- \\frac{5}{2}",
"style": "rule"
} | {
"index": "2972",
"split": "train"
} |
math | [
{
"content": "A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day's job. At what time will the doughnut machine complete the job? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4:30 PM",
"style": "rule"
} | {
"index": "2973",
"split": "train"
} |
math | [
{
"content": "If $\\sin \\left(\\frac{\\pi }{3}+\\alpha \\right)=\\frac{1}{3}$, then find the value of $\\cos \\left(\\frac{\\pi }{3}-2\\alpha \\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{7}{9}",
"style": "rule"
} | {
"index": "2974",
"split": "train"
} |
math | [
{
"content": "A charity sells $140$ benefit tickets for a total of $2001$ dollars. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$782",
"style": "rule"
} | {
"index": "2975",
"split": "train"
} |
math | [
{
"content": "Let $P$ units be the increase in circumference of a circle resulting from an increase in $\\pi$ units in the diameter. Then $P$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\pi^2",
"style": "rule"
} | {
"index": "2976",
"split": "train"
} |
math | [
{
"content": "The quadrilateral \\(P Q R S\\) is inscribed in a circle. Diagonals \\(P R\\) and \\(Q S\\) are perpendicular and intersect at point \\(M\\). It is known that \\(P S = 13\\), \\(Q M = 10\\), and \\(Q R = 26\\). Find the area of the quadrilateral \\(P Q R S\\). Please output the final answer within... | {
"ground_truth": "319",
"style": "rule"
} | {
"index": "2977",
"split": "train"
} |
math | [
{
"content": "Suppose the quadratic function $f(x)=ax^2+bx+c$ has a maximum value $M$ and a minimum value $m$ in the interval $[-2,2]$, and the set $A={x|f(x)=x}$.\n\n(1) If $A={1,2}$ and $f(0)=2$, find the values of $M$ and $m$.\n(2) If $A={2}$ and $a\\geqslant 1$, let $g(a)=M+m$, find the minimum value of $g(... | {
"ground_truth": "\\frac{63}{4}",
"style": "rule"
} | {
"index": "2978",
"split": "train"
} |
math | [
{
"content": "Tony will paint 16 right, cylindrical columns. The top face and bottom face of each column will be covered, so those parts will not be painted. Each column is 18 feet tall and has a diameter of 10 feet. One gallon of paint will cover 350 square feet. If paint is sold only in full gallons, how m... | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "2979",
"split": "train"
} |
math | [
{
"content": "Let \\( M = 42 \\cdot 43 \\cdot 75 \\cdot 196 \\). Find the ratio of the sum of the odd divisors of \\( M \\) to the sum of the even divisors of \\( M \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{14}",
"style": "rule"
} | {
"index": "2980",
"split": "train"
} |
math | [
{
"content": "A group of $10$ students from Class 9(1) of a certain school are doing \"pull-up\" training. The number of times they have done it is recorded in the table below. Find the median of the number of times these $10$ students have done.\n\n| Times | $4$ | $5$ | $6$ | $7$ | $8$ |\n|-------|-----|-----|... | {
"ground_truth": "5.5",
"style": "rule"
} | {
"index": "2981",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = x^3 + 3x^2 + 6x + 4$, and given that $f(a) = 14$ and $f(b) = -14$, calculate the value of $a + b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "2982",
"split": "train"
} |
math | [
{
"content": "Through points \\( R \\) and \\( E \\), located on sides \\( AB \\) and \\( AD \\) of parallelogram \\( ABCD \\) respectively, where \\( AR = \\frac{2}{3} AB \\) and \\( AE = \\frac{1}{3} AD \\), a line is drawn.\n\nFind the ratio of the area of the parallelogram to the area of the resulting trian... | {
"ground_truth": "9:1",
"style": "rule"
} | {
"index": "2983",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{v}$ be a vector such that\n\\[\\left\\| \\mathbf{v} + \\begin{pmatrix} 4 \\\\ -2 \\end{pmatrix} \\right\\| = 10.\\]\nFind the smallest possible value of $\\|\\mathbf{v}\\|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10 - 2\\sqrt{5}",
"style": "rule"
} | {
"index": "2984",
"split": "train"
} |
math | [
{
"content": "What is the least positive integer $n$ such that $7350$ is a factor of $n!$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "2985",
"split": "train"
} |
math | [
{
"content": "Jeff decides to play with a Magic 8 Ball. Each time he asks it a question, it has a 2/5 chance of giving him a positive answer. If he asks it 5 questions, what is the probability that it gives him exactly 2 positive answers? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "\\frac{216}{625}",
"style": "rule"
} | {
"index": "2986",
"split": "train"
} |
math | [
{
"content": "In the equation $\\frac{1}{(\\;\\;\\;)} + \\frac{4}{(\\;\\;\\;)} + \\frac{9}{(\\;\\;\\;\\;)} = 1$, fill in the three brackets in the denominators with a positive integer, respectively, such that the equation holds true. The minimum value of the sum of these three positive integers is $\\_\\_\\_\\_... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "2987",
"split": "train"
} |
math | [
{
"content": "Given a convex quadrilateral \\(ABCD\\), \\(X\\) is the midpoint of the diagonal \\(AC\\). It is known that \\(CD \\parallel BX\\). Find \\(AD\\), given that \\(BX = 3\\), \\(BC = 7\\), and \\(CD = 6\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "2988",
"split": "train"
} |
math | [
{
"content": "What is the product of the numerator and the denominator when $0.\\overline{012}$ is expressed as a fraction in lowest terms? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1332",
"style": "rule"
} | {
"index": "2989",
"split": "train"
} |
math | [
{
"content": "Anton thought of a three-digit number, and Alex is trying to guess it. Alex successively guessed the numbers 109, 704, and 124. Anton observed that each of these numbers matches the thought number exactly in one digit place. What number did Anton think of? Please output the final answer within \\b... | {
"ground_truth": "729",
"style": "rule"
} | {
"index": "2990",
"split": "train"
} |
math | [
{
"content": "If \\( AC = 1.5 \\, \\text{cm} \\) and \\( AD = 4 \\, \\text{cm} \\), what is the relationship between the areas of triangles \\( \\triangle ABC \\) and \\( \\triangle DBC \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3/5",
"style": "rule"
} | {
"index": "2991",
"split": "train"
} |
math | [
{
"content": "Anya, Vanya, Danya, and Tanya collected apples. Each of them collected a whole number percentage from the total number of apples, and all these numbers are distinct and more than zero. Then Tanya, who collected the most apples, ate her apples. After that, each of the remaining kids still had a who... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "2992",
"split": "train"
} |
math | [
{
"content": "The number $n$ is a prime number between 20 and 30. If you divide $n$ by 8, the remainder is 5. What is the value of $n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "2993",
"split": "train"
} |
math | [
{
"content": "The distances from a certain point inside a regular hexagon to three of its consecutive vertices are 1, 1, and 2, respectively. What is the side length of this hexagon? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "2994",
"split": "train"
} |
math | [
{
"content": "In the sequence $\\{{a_{n}}\\}$, the adjacent terms ${a_{n}}$ and ${a_{n+1}}$ are the roots of the equation ${x^{2}}+3nx+{{b_{n}}=0}$. Given that ${{a_{10}}=-17}$, find the value of ${{b_{51}}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5840",
"style": "rule"
} | {
"index": "2995",
"split": "train"
} |
math | [
{
"content": "We know that every natural number has factors. For a natural number $a$, we call the positive factors less than $a$ the proper factors of $a$. For example, the positive factors of $10$ are $1$, $2$, $5$, $10$, where $1$, $2$, and $5$ are the proper factors of $10$. The quotient obtained by dividin... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "2996",
"split": "train"
} |
math | [
{
"content": "Find the positive value of $t$ that satisfies $ab = t-2i$ given $|a|=2$ and $|b|=\\sqrt{26}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "2997",
"split": "train"
} |
math | [
{
"content": "Let $Q$ be the product of the first $50$ positive odd integers. Find the largest integer $m$ such that $Q$ is divisible by $5^m.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "2998",
"split": "train"
} |
math | [
{
"content": "Find all positive integers $n$ such that: \n\\[ \\dfrac{n^3+3}{n^2+7} \\]\nis a positive integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 \\text{ and } 5",
"style": "rule"
} | {
"index": "2999",
"split": "train"
} |
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