data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given an ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$) with a vertex coordinate at $B(0,1)$, and the eccentricity of the ellipse being $\\frac{\\sqrt{3}}{2}$,\n$(I)$ find the equation of the ellipse;\n$(II)$ let point $Q$ be a point on ellipse $C$ below the x-axis, and $F_1$, ... | {
"ground_truth": "\\frac{\\sqrt{3}}{7}",
"style": "rule"
} | {
"index": "3900",
"split": "train"
} |
math | [
{
"content": "On 40 squares of an $8 \\times 8$ chessboard, a stone was placed on each square. The product of the number of stones on white squares and the number of stones on black squares was calculated. Find the minimum possible value of this product. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "256",
"style": "rule"
} | {
"index": "3901",
"split": "train"
} |
math | [
{
"content": "Given $f(x) = \\sin x + a\\cos x$, \n(1) If $a= \\sqrt{3}$, find the maximum value of $f(x)$ and the corresponding value of $x$. \n(2) If $f\\left(\\frac{\\pi}{4}\\right) = 0$ and $f(x) = \\frac{1}{5}$ $(0 < x < \\pi)$, find the value of $\\tan x$. Please output the final answer within \\boxed{}... | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "3902",
"split": "train"
} |
math | [
{
"content": "Given the digits 0, 1, 2, 3, 4, 5, how many unique six-digit numbers greater than 300,000 can be formed where the digit in the thousand's place is less than 3? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "216",
"style": "rule"
} | {
"index": "3903",
"split": "train"
} |
math | [
{
"content": "Find the sum of all the integer solutions of $x^4 - 25x^2 + 144 = 0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "3904",
"split": "train"
} |
math | [
{
"content": "Given a positive integer $n$, let $M(n)$ be the largest integer $m$ such that \\[ \\binom{m}{n-1} > \\binom{m-1}{n}. \\] Evaluate \\[ \\lim_{n \\to \\infty} \\frac{M(n)}{n}. \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3+\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "3905",
"split": "train"
} |
math | [
{
"content": "Interior numbers begin in the third row of Pascal's Triangle. Calculate the sum of the squares of the interior numbers in the eighth row. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3430",
"style": "rule"
} | {
"index": "3906",
"split": "train"
} |
math | [
{
"content": "Given that $a$, $b$, and $c$ are the lengths of the sides opposite the angles $A$, $B$, and $C$ in $\\triangle ABC$ respectively, with $a=2$, and $$\\frac{\\sin A - \\sin B}{\\sin C} = \\frac{c - b}{2 + b}.$$ Find the maximum area of $\\triangle ABC$. Please output the final answer within \\boxed{... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "3907",
"split": "train"
} |
math | [
{
"content": "Find the smallest three-digit number such that the following holds:\nIf the order of digits of this number is reversed and the number obtained by this is added to the original number, the resulting number consists of only odd digits. Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "209",
"style": "rule"
} | {
"index": "3908",
"split": "train"
} |
math | [
{
"content": "The figures $F_1$, $F_2$, $F_3$, and $F_4$ shown are the first in a sequence of figures. For $n\\ge3$, $F_n$ is constructed from $F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n - 1}$ had on each side of its outside square. For exam... | {
"ground_truth": "761",
"style": "rule"
} | {
"index": "3909",
"split": "train"
} |
math | [
{
"content": "Let $x,$ $y,$ $z$ be real numbers such that $x + 2y + z = 4.$ Find the maximum value of\n\\[xy + xz + yz.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "3910",
"split": "train"
} |
math | [
{
"content": "Let $a, b$, and $c$ be the roots of the cubic polynomial $2x^3 - 3x^2 + 165x - 4$. Compute \\[(a+b-1)^3 + (b+c-1)^3 + (c+a-1)^3.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "117",
"style": "rule"
} | {
"index": "3911",
"split": "train"
} |
math | [
{
"content": "The number of natural numbers from 1 to 1992 that are multiples of 3, but not multiples of 2 or 5, is calculated. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "266",
"style": "rule"
} | {
"index": "3912",
"split": "train"
} |
math | [
{
"content": "A severe earthquake in Madrid caused a total of €50 million in damages. At the time of the earthquake, the exchange rate was such that 2 Euros were worth 3 American dollars. How much damage did the earthquake cause in American dollars? Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "75,000,000",
"style": "rule"
} | {
"index": "3913",
"split": "train"
} |
math | [
{
"content": "A polynomial $p(x)$ leaves a remainder of $-1$ when divided by $x - 1,$ a remainder of 3 when divided by $x - 2,$ and a remainder of 4 when divided by $x + 3.$ Let $r(x)$ be the remainder when $p(x)$ is divided by $(x - 1)(x - 2)(x + 3).$ Find $r(6).$ Please output the final answer within \\boxe... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "3914",
"split": "train"
} |
math | [
{
"content": "Call a positive integer $n$ extra-distinct if the remainders when $n$ is divided by $2, 3, 4, 5,$ and $6$ are distinct. Find the number of extra-distinct positive integers less than $1000$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "049",
"style": "rule"
} | {
"index": "3915",
"split": "train"
} |
math | [
{
"content": "Find the smallest integer \\( k \\) such that when \\( 5 \\) is multiplied by a number consisting of \\( k \\) digits of \\( 7 \\) (i.e., \\( 777\\ldots7 \\) with \\( k \\) sevens), the resulting product has digits summing to \\( 800 \\).\nA) 86\nB) 87\nC) 88\nD) 89\nE) 90 Please output the final ... | {
"ground_truth": "88",
"style": "rule"
} | {
"index": "3916",
"split": "train"
} |
math | [
{
"content": "Determine the number of ways to arrange the letters of the word \"PERCEPTION\". Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "907,200",
"style": "rule"
} | {
"index": "3917",
"split": "train"
} |
math | [
{
"content": "Given points $A(-2, -3)$ and $B(5, 3)$ on the $xy$-plane, find the point $C(x, n)$ such that $AC + CB$ is minimized, where $x = 2$. Find the value of $n$.\nA) $\\frac{6}{7}$\nB) $\\frac{12}{7}$\nC) $6.5$\nD) $\\frac{25}{6}$\nE) $\\frac{13}{2}$ Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{13}{2}",
"style": "rule"
} | {
"index": "3918",
"split": "train"
} |
math | [
{
"content": "Given a cone and a cylinder with equal base radii and heights, if the axis section of the cone is an equilateral triangle, calculate the ratio of the lateral surface areas of this cone and cylinder. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "3919",
"split": "train"
} |
math | [
{
"content": "Given that point $A(-2,α)$ lies on the terminal side of angle $α$ and $\\sin α=- \\dfrac{ \\sqrt{5}}{5}$.\n\n(1) Find the values of $α$, $\\cos α$, and $\\tan α$.\n\n(2) Find the value of $\\dfrac{\\cos ( \\dfrac{π}{2}+α)\\sin (-π-α)}{\\cos ( \\dfrac{11π}{2}-α)\\sin ( \\dfrac{9π}{2}+α)}$. Please o... | {
"ground_truth": "\\dfrac {1}{2}",
"style": "rule"
} | {
"index": "3920",
"split": "train"
} |
math | [
{
"content": "On a Cartesian coordinate plane, points $(2,1)$ and $(3, 4)$ are adjacent points on a square. What is the area of the square? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "3921",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, the equation of curve $C_{1}$ is $x^{2}+y^{2}=1$. Taking the origin $O$ of the Cartesian coordinate system $xOy$ as the pole and the positive half of the $x$-axis as the polar axis, a polar coordinate system is established with the same unit length. It is ... | {
"ground_truth": "2 \\sqrt {5}",
"style": "rule"
} | {
"index": "3922",
"split": "train"
} |
math | [
{
"content": "Does there exist a positive integer \\( m \\) such that the equation \\(\\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} + \\frac{1}{abc} = \\frac{m}{a+b+c}\\) has infinitely many solutions in positive integers \\( (a, b, c) \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "3923",
"split": "train"
} |
math | [
{
"content": "Given the parabola $C: y^{2}=2px\\left(p \\lt 0\\right)$ passing through the point $A\\left(-2,-4\\right)$. \n$(1)$ Find the equation of the parabola $C$ and its directrix equation. \n$(2)$ A line passing through the focus of the parabola, making an angle of $60^{\\circ}$ with the $x$-axis, inte... | {
"ground_truth": "\\frac{32}{3}",
"style": "rule"
} | {
"index": "3924",
"split": "train"
} |
math | [
{
"content": "What is the greatest three-digit multiple of 19? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "988",
"style": "rule"
} | {
"index": "3925",
"split": "train"
} |
math | [
{
"content": "Six distinguishable players are participating in a tennis tournament. Each player plays one match of tennis against every other player. There are no ties in this tournament; each tennis match results in a win for one player and a loss for the other. Suppose that whenever $A$ and $B$ are players in... | {
"ground_truth": "2048",
"style": "rule"
} | {
"index": "3926",
"split": "train"
} |
math | [
{
"content": "Given: $\\sqrt{23.6}=4.858$, $\\sqrt{2.36}=1.536$, then calculate the value of $\\sqrt{0.00236}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.04858",
"style": "rule"
} | {
"index": "3927",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=ax^{2}+bx+c(a\\\\neq 0)$, its graph intersects with line $l$ at two points $A(t,t^{3}-t)$, $B(2t^{2}+3t,t^{3}+t^{2})$, where $t\\\\neq 0$ and $t\\\\neq -1$. Find the value of $f{{'}}(t^{2}+2t)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac {1}{2}",
"style": "rule"
} | {
"index": "3928",
"split": "train"
} |
math | [
{
"content": "For a permutation $\\pi$ of the integers from 1 to 10, define\n\\[ S(\\pi) = \\sum_{i=1}^{9} (\\pi(i) - \\pi(i+1))\\cdot (4 + \\pi(i) + \\pi(i+1)), \\]\nwhere $\\pi (i)$ denotes the $i$ th element of the permutation. Suppose that $M$ is the maximum possible value of $S(\\pi)$ over all per... | {
"ground_truth": "40320",
"style": "rule"
} | {
"index": "3929",
"split": "train"
} |
math | [
{
"content": "What is $\\dbinom{n}{n-1}$ for any positive integer $n$? (Answer in terms of $n$.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "n",
"style": "rule"
} | {
"index": "3930",
"split": "train"
} |
math | [
{
"content": "How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1 \\le x \\le 5$ and $1 \\le y \\le 3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "416",
"style": "rule"
} | {
"index": "3931",
"split": "train"
} |
math | [
{
"content": "The function $f(x)$ satisfies\n\\[ f(xy) = 3xf(y) \\]\nfor all real numbers $x$ and $y$. If $f(1) = 10$, find $f(5)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "3932",
"split": "train"
} |
math | [
{
"content": "Given that the center of the ellipse C is at the origin, its eccentricity is equal to $\\frac{1}{2}$, and one of its minor axis endpoints is the focus of the parabola x²=$4\\sqrt{3}y$.\n1. Find the standard equation of the ellipse C.\n2. The left and right foci of the ellipse are $F_1$ and $F_2$, ... | {
"ground_truth": "\\frac{9}{16}\\pi",
"style": "rule"
} | {
"index": "3933",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer $M$ such that the three numbers $M$, $M+1$, and $M+2$, one of them is divisible by $3^2$, one of them is divisible by $5^2$, and one is divisible by $7^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "98",
"style": "rule"
} | {
"index": "3934",
"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 planning 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": "12\\%",
"style": "rule"
} | {
"index": "3935",
"split": "train"
} |
math | [
{
"content": "The rails on a railroad are $30$ feet long. As the train passes over the point where the rails are joined, there is an audible click. \nThe speed of the train in miles per hour is approximately the number of clicks heard in: Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "20 seconds",
"style": "rule"
} | {
"index": "3936",
"split": "train"
} |
math | [
{
"content": "In a finite sequence of real numbers, the sum of any 7 consecutive terms is negative, while the sum of any 11 consecutive terms is positive. What is the maximum number of terms this sequence can have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "3937",
"split": "train"
} |
math | [
{
"content": "Given the ellipse $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}=1(a＞b＞0)$$ with eccentricity $$e= \\frac { \\sqrt {3}}{2}$$, A and B are the left and right vertices of the ellipse, respectively, and P is a point on the ellipse different from A and B. The angles of inclination of lines PA and PB a... | {
"ground_truth": "\\frac {3}{5}",
"style": "rule"
} | {
"index": "3938",
"split": "train"
} |
math | [
{
"content": "A cuckoo clock rings \"cuckoo\" every hour, with the number of rings corresponding to the hour shown by the hour hand (e.g., at 7:00, it rings 7 times). One morning, Maxim approached the clock at 9:05 and started moving the minute hand until 7 hours had passed. How many times did the clock ring \"... | {
"ground_truth": "43",
"style": "rule"
} | {
"index": "3939",
"split": "train"
} |
math | [
{
"content": "Given the line $l:3x-\\sqrt{3}y-6=0$, and the curve $C:\\rho -4\\sin \\theta =0$ in the polar coordinate system with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis.\n\n(I) Write the line $l$ in parametric form $\\begin{cases} x=2+t\\cos \\alpha \\\\ y=t\\sin \\alpha... | {
"ground_truth": "6+2\\sqrt{3}",
"style": "rule"
} | {
"index": "3940",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\sqrt{3}\\cos (\\frac{\\pi }{2}+x)\\bullet \\cos x+\\sin^{2}x$, where $x\\in R$.\n(I) Find the interval where $f(x)$ is monotonically increasing.\n(II) In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $B=\\frac{\\pi }{4}$,... | {
"ground_truth": "\\frac{3+\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "3941",
"split": "train"
} |
math | [
{
"content": "Solve for $y$: $4+2.3y = 1.7y - 20$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-40",
"style": "rule"
} | {
"index": "3942",
"split": "train"
} |
math | [
{
"content": "Let \\( n \\in \\mathbf{Z}_{+} \\), and\n$$\n\\begin{array}{l}\na, b, c \\in \\{x \\mid x \\in \\mathbf{Z} \\text{ and } x \\in [1,9]\\}, \\\\\nA_{n} = \\underbrace{\\overline{a a \\cdots a}}_{n \\text{ digits}}, B_{n} = \\underbrace{b b \\cdots b}_{2n \\text{ digits}}, C_{n} = \\underbrace{c c \\... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "3943",
"split": "train"
} |
math | [
{
"content": "How many numbers are in the list $$ 1, 4, 7, \\ldots, 2005, 2008 ?$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "670",
"style": "rule"
} | {
"index": "3944",
"split": "train"
} |
math | [
{
"content": "In convex quadrilateral $KLMN$ side $\\overline{MN}$ is perpendicular to diagonal $\\overline{KM}$, side $\\overline{KL}$ is perpendicular to diagonal $\\overline{LN}$, $MN = 65$, and $KL = 28$. The line through $L$ perpendicular to side $\\overline{KN}$ intersects diagonal $\\overline{KM}$ at $O$... | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "3945",
"split": "train"
} |
math | [
{
"content": "Find the minimum value of\n$$\n\\begin{aligned}\nA & =\\sqrt{\\left(1264-z_{1}-\\cdots-z_{n}\\right)^{2}+x_{n}^{2}+y_{n}^{2}}+ \\\\\n& \\sqrt{z_{n}^{2}+x_{n-1}^{2}+y_{n-1}^{2}}+\\cdots+\\sqrt{z_{2}^{2}+x_{1}^{2}+y_{1}^{2}}+ \\\\\n& \\sqrt{z_{1}^{2}+\\left(948-x_{1}-\\cdots-x_{n}\\right)^{2}+\\left... | {
"ground_truth": "1975",
"style": "rule"
} | {
"index": "3946",
"split": "train"
} |
math | [
{
"content": "In a kindergarten class, there are two (small) Christmas trees and five children. The teachers want to divide the children into two groups to form a ring around each tree, with at least one child in each group. The teachers distinguish the children but do not distinguish the trees: two configurati... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "3947",
"split": "train"
} |
math | [
{
"content": "How many ways are there to put 4 distinguishable balls into 2 indistinguishable boxes? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "3948",
"split": "train"
} |
math | [
{
"content": "How many sets of two or more consecutive positive integers have a sum of $15$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "3949",
"split": "train"
} |
math | [
{
"content": "How many license plates can be made if they consist of 3 letters followed by 2 digits, where the first digit must be odd, the second digit must be even, and the first letter must be a vowel? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "84,500",
"style": "rule"
} | {
"index": "3950",
"split": "train"
} |
math | [
{
"content": "The product of the digits of 3214 is 24. How many distinct four-digit positive integers are such that the product of their digits equals 12? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "3951",
"split": "train"
} |
math | [
{
"content": "Given a function $f(x) = 2\\sin x\\cos x + 2\\cos\\left(x + \\frac{\\pi}{4}\\right)\\cos\\left(x - \\frac{\\pi}{4}\\right)$,\n1. Find the interval(s) where $f(x)$ is monotonically decreasing.\n2. If $\\alpha \\in (0, \\pi)$ and $f\\left(\\frac{\\alpha}{2}\\right) = \\frac{\\sqrt{2}}{2}$, find the ... | {
"ground_truth": "\\frac{\\sqrt{6} + \\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "3952",
"split": "train"
} |
math | [
{
"content": "Call a number feared if it contains the digits 13 as a contiguous substring and fearless otherwise. (For example, 132 is feared, while 123 is fearless.) Compute the smallest positive integer $n$ such that there exists a positive integer $a<100$ such that $n$ and $n+10 a$ are fearless while $n+a, n... | {
"ground_truth": "1287",
"style": "rule"
} | {
"index": "3953",
"split": "train"
} |
math | [
{
"content": "For the set $M$, define the function $f_M(x) = \\begin{cases} -1, & x \\in M \\\\ 1, & x \\notin M \\end{cases}$. For two sets $M$ and $N$, define the set $M \\triangle N = \\{x | f_M(x) \\cdot f_N(x) = -1\\}$. Given $A = \\{2, 4, 6, 8, 10\\}$ and $B = \\{1, 2, 4, 8, 16\\}$.\n\n(1) List the elemen... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "3954",
"split": "train"
} |
math | [
{
"content": "If $q_1(x)$ and $r_1$ are the quotient and remainder, respectively, when the polynomial $x^8$ is divided by $x + \\frac{1}{2}$, and if $q_2(x)$ and $r_2$ are the quotient and remainder, respectively, when $q_1(x)$ is divided by $x + \\frac{1}{2}$, then $r_2$ equals Please output the final answer w... | {
"ground_truth": "-\\frac{1}{16}",
"style": "rule"
} | {
"index": "3955",
"split": "train"
} |
math | [
{
"content": "The surface area of a sphere with radius $r$ is $4\\pi r^2$. Including the area of its circular base, what is the total surface area of a hemisphere with radius 6 cm? Express your answer in terms of $\\pi$.\n\n[asy]\n\nimport markers;\nsize(150);\nimport geometry;\ndraw((0,-7)--(0,-1),Arrow);\ndra... | {
"ground_truth": "108\\pi",
"style": "rule"
} | {
"index": "3956",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C_{1}$ is $\\rho \\cos \\theta = 4$. \n$(1)$ Let $M$ be a moving point on the curve $C_{1}$, point $P$ lies on the line s... | {
"ground_truth": "2 + \\sqrt{3}",
"style": "rule"
} | {
"index": "3957",
"split": "train"
} |
math | [
{
"content": "Let $x_1<x_2< \\ldots <x_{2024}$ be positive integers and let $p_i=\\prod_{k=1}^{i}(x_k-\\frac{1}{x_k})$ for $i=1,2, \\ldots, 2024$ . What is the maximal number of positive integers among the $p_i$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1012",
"style": "rule"
} | {
"index": "3958",
"split": "train"
} |
math | [
{
"content": "If $\\left( r + \\frac{1}{r} \\right)^2 = 5,$ then find $r^3 + \\frac{1}{r^3}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{5}",
"style": "rule"
} | {
"index": "3959",
"split": "train"
} |
math | [
{
"content": "Let $f(n)$ be the sum of all the divisors of a positive integer $n$. If $f(f(n)) = n+2$, then call $n$ superdeficient. How many superdeficient positive integers are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "3960",
"split": "train"
} |
math | [
{
"content": "Let $x$ be a real number selected uniformly at random between 100 and 300. If $\\lfloor \\sqrt{x} \\rfloor = 14$, find the probability that $\\lfloor \\sqrt{100x} \\rfloor = 140$.\nA) $\\frac{281}{2900}$\nB) $\\frac{4}{29}$\nC) $\\frac{1}{10}$\nD) $\\frac{96}{625}$\nE) $\\frac{1}{100}$ Please outp... | {
"ground_truth": "\\frac{281}{2900}",
"style": "rule"
} | {
"index": "3961",
"split": "train"
} |
math | [
{
"content": "Let $PQRS$ be a convex pentagon, and let $H_P,$ $H_Q,$ $H_R,$ $H_S,$ and $H_T$ denote the centroids of the triangles $QRS,$ $RSP,$ $SPQ,$ $PQR,$ and $QRP$, respectively. Find $\\frac{[H_PH_QH_RH_S]}{[PQRS]}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{9}",
"style": "rule"
} | {
"index": "3962",
"split": "train"
} |
math | [
{
"content": "What is the remainder when 5462 is divided by 9? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "3963",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be a convex quadrilateral with $BC=2$ and $CD=6.$ Suppose that the centroids of $\\triangle ABC, \\triangle BCD,$ and $\\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$? Please output the final answer within \\boxed{}.... | {
"ground_truth": "12+10\\sqrt{3}",
"style": "rule"
} | {
"index": "3964",
"split": "train"
} |
math | [
{
"content": "A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rect... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "3965",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate plane, there are four fixed points \\(A(-3,0), B(1,-1), C(0,3), D(-1,3)\\) and a moving point \\(P\\). What is the minimum value of \\(|PA| + |PB| + |PC| + |PD|\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3\\sqrt{2} + 2\\sqrt{5}",
"style": "rule"
} | {
"index": "3966",
"split": "train"
} |
math | [
{
"content": "Abby, Bernardo, Carl, and Debra play a revised game where each starts with five coins and there are five rounds. In each round, five balls are placed in an urn—two green, two red, and one blue. Each player draws a ball at random without replacement. If a player draws a green ball, they give one co... | {
"ground_truth": "\\frac{64}{15625}",
"style": "rule"
} | {
"index": "3967",
"split": "train"
} |
math | [
{
"content": "Simplify the expression $\\frac{5^5 + 5^3 + 5}{5^4 - 2\\cdot5^2 + 5}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{651}{116}",
"style": "rule"
} | {
"index": "3968",
"split": "train"
} |
math | [
{
"content": "There are 256 players in a tennis tournament who are ranked from 1 to 256, with 1 corresponding to the highest rank and 256 corresponding to the lowest rank. When two players play a match in the tournament, the player whose rank is higher wins the match with probability \\(\\frac{3}{5}\\).\n\nIn e... | {
"ground_truth": "103",
"style": "rule"
} | {
"index": "3969",
"split": "train"
} |
math | [
{
"content": "What is the remainder when 100 ! is divided by 101 ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "3970",
"split": "train"
} |
math | [
{
"content": "Last year, Michael took 8 math tests and received 8 different scores, each an integer between 85 and 95, inclusive. After each test, he noticed that the average of his test scores was an integer. His score on the eighth test was 90. What was his score on the seventh test? Please output the final a... | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "3971",
"split": "train"
} |
math | [
{
"content": "Two friends, Alice and Bob, start cycling towards a park 80 miles away. Alice cycles 3 miles per hour slower than Bob. Upon reaching the park, Bob immediately turns around and starts cycling back, meeting Alice 15 miles away from the park. Find the speed of Alice. Please output the final answer wi... | {
"ground_truth": "6.5",
"style": "rule"
} | {
"index": "3972",
"split": "train"
} |
math | [
{
"content": "Assume $x$, $y$, $z$, and $w$ are positive integers such that $x^3 = y^2$, $z^5 = w^4$, and $z - x = 31$. Determine $w - y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2351",
"style": "rule"
} | {
"index": "3973",
"split": "train"
} |
math | [
{
"content": "If, in the expression $x^2 - 3$, $x$ increases or decreases by a positive amount of $a$, the expression changes by an amount: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\pm 2ax + a^2$",
"style": "rule"
} | {
"index": "3974",
"split": "train"
} |
math | [
{
"content": "Given that $f(x)$ and $g(x)$ are functions defined on $\\mathbb{R}$ with $g(x) \\neq 0$, $f'(x)g(x) > f(x)g'(x)$. Let $a$ and $b$ be the points obtained from two consecutive throws of the same die. If $f(x) - a^x g(x) = 0$ and $\\frac{f(1)}{g(1)} + \\frac{f(-1)}{g(-1)} \\geq \\frac{10}{3}$, calcul... | {
"ground_truth": "\\frac{13}{36}",
"style": "rule"
} | {
"index": "3975",
"split": "train"
} |
math | [
{
"content": "Select 4 out of 7 different books to distribute to 4 students, one book per student, with the restriction that books A and B cannot be given to student C, and calculate the number of different distribution methods. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "600",
"style": "rule"
} | {
"index": "3976",
"split": "train"
} |
math | [
{
"content": "The points on this graph show the end-of-year sales amounts for each year. During what year after 1994 did sales increase the most number of dollars? Express your answer as a 4-digit year.\n\n[asy]unitsize(101/10);\nreal textsize = 10, largetext = 12;\ndraw((10,0)--(0,0)--(0,10),Arrows(4,20));\np... | {
"ground_truth": "1998",
"style": "rule"
} | {
"index": "3977",
"split": "train"
} |
math | [
{
"content": "If the tangent line to the curve $y=e^{x}$ at $x=1$ is perpendicular to the line $2x+my+1=0$, then calculate the value of $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2e",
"style": "rule"
} | {
"index": "3978",
"split": "train"
} |
math | [
{
"content": "Given a triangle $\\triangle ABC$ with an area of $S$, and $\\overrightarrow{AB} \\cdot \\overrightarrow{AC} = S$.\n\n(I) Find the value of $\\tan 2A$;\n\n(II) If $\\cos C = \\frac{3}{5}$, and $|\\overrightarrow{AC} - \\overrightarrow{AB}| = 2$, find the area $S$ of $\\triangle ABC$. Please output... | {
"ground_truth": "\\frac{8}{5}",
"style": "rule"
} | {
"index": "3979",
"split": "train"
} |
math | [
{
"content": "If in an arithmetic sequence, the sum of the first three terms is 34, the sum of the last three terms is 146, and the sum of all terms is 390, then the number of terms in the sequence is __________. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "3980",
"split": "train"
} |
math | [
{
"content": "Given the sets\n$$\n\\begin{array}{l}\nA=\\left\\{(x, y) \\mid x=m, y=-3m+2, m \\in \\mathbf{Z}_{+}\\right\\}, \\\\\nB=\\left\\{(x, y) \\mid x=n, y=a\\left(a^{2}-n+1\\right), n \\in \\mathbf{Z}_{+}\\right\\},\n\\end{array}\n$$\n\nfind the total number of integers $a$ such that $A \\cap B \\neq \\v... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "3981",
"split": "train"
} |
math | [
{
"content": "In a convex $n$-gon, a diagonal is selected. The selected diagonal is intersected by exactly 14 other diagonals of the $n$-gon. Find the sum of all possible values of $n$. A vertex of the $n$-gon is not considered an intersection. Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "3982",
"split": "train"
} |
math | [
{
"content": "A point has rectangular coordinates $(x,y,z)$ and spherical coordinates $\\left(2, \\frac{8 \\pi}{7}, \\frac{2 \\pi}{9} \\right).$ Find the spherical coordinates of the point with rectangular coordinates $(x,y,-z).$ Enter your answer in the form $(\\rho,\\theta,\\phi),$ where $\\rho > 0,$ $0 \\l... | {
"ground_truth": "\\left( 2, \\frac{8 \\pi}{7}, \\frac{7 \\pi}{9} \\right)",
"style": "rule"
} | {
"index": "3983",
"split": "train"
} |
math | [
{
"content": "Given the point \\( P \\) lies in the plane of the right triangle \\( \\triangle ABC \\) with \\( \\angle BAC = 90^\\circ \\), and \\( \\angle CAP \\) is an acute angle. Also given are the conditions:\n\\[ |\\overrightarrow{AP}| = 2, \\quad \\overrightarrow{AP} \\cdot \\overrightarrow{AC} = 2, \\q... | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "3984",
"split": "train"
} |
math | [
{
"content": "In the decimal representation of $\frac{1}{7}$, the 100th digit to the right of the decimal is? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "3985",
"split": "train"
} |
math | [
{
"content": "Points \\( K \\) and \\( N \\) are located on the sides \\( AB \\) and \\( AC \\) of triangle \\( ABC \\) respectively, such that \\( AK = BK \\) and \\( AN = 2NC \\).\n\nIn what ratio does the segment \\( KN \\) divide the median \\( AM \\) of triangle \\( ABC \\)? Please output the final answer ... | {
"ground_truth": "4:3",
"style": "rule"
} | {
"index": "3986",
"split": "train"
} |
math | [
{
"content": "Calculate the value of $\\cos 96^\\circ \\cos 24^\\circ - \\sin 96^\\circ \\sin 66^\\circ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "3987",
"split": "train"
} |
math | [
{
"content": "Given two unit vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ with an angle of 120° between them, find the projection of $\\overrightarrow{a} + \\overrightarrow{b}$ onto the direction of $\\overrightarrow{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "3988",
"split": "train"
} |
math | [
{
"content": "Given a moving large circle $\\odot O$ tangent externally to a fixed small circle $\\odot O_{1}$ with radius 3 at point $P$, $AB$ is the common external tangent of the two circles with $A$ and $B$ as the points of tangency. A line $l$ parallel to $AB$ is tangent to $\\odot O_{1}$ at point $C$ and ... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "3989",
"split": "train"
} |
math | [
{
"content": "The Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many conference games are scheduled? Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "3990",
"split": "train"
} |
math | [
{
"content": "Given the sequence: \\(\\frac{2}{3}, \\frac{2}{9}, \\frac{4}{9}, \\frac{6}{9}, \\frac{8}{9}, \\frac{2}{27}, \\frac{4}{27}, \\cdots, \\frac{26}{27}, \\cdots, \\frac{2}{3^{n}}, \\frac{4}{3^{n}}, \\cdots, \\frac{3^{n}-1}{3^{n}}, \\cdots\\). Then, \\(\\frac{2020}{2187}\\) is the \\(n\\)-th term of thi... | {
"ground_truth": "1553",
"style": "rule"
} | {
"index": "3991",
"split": "train"
} |
math | [
{
"content": "If I roll 7 standard 6-sided dice and multiply the number on the face of each die, what is the probability that the result is a composite number and the sum of the numbers rolled is divisible by 3? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "3992",
"split": "train"
} |
math | [
{
"content": "If $\\log_6 (4x)=2$, find $\\log_x 27$. Express your answer in simplest fractional form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac32",
"style": "rule"
} | {
"index": "3993",
"split": "train"
} |
math | [
{
"content": "Rectangle ABCD has AB = 6 and BC = 3. Point M is chosen on side AB so that ∠AMD = ∠CMD. Calculate the degree measure of ∠AMD. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "67.5",
"style": "rule"
} | {
"index": "3994",
"split": "train"
} |
math | [
{
"content": "Let $ a,\\ b$ be the real numbers such that $ 0\\leq a\\leq b\\leq 1$ . Find the minimum value of $ \\int_0^1 |(x\\minus{}a)(x\\minus{}b)|\\ dx$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/12",
"style": "rule"
} | {
"index": "3995",
"split": "train"
} |
math | [
{
"content": "In the product shown, $\\text{B}$ is a digit. The value of $\\text{B}$ is\n$\\begin{array}{rr} &\\text{B}2 \\\\ \\times& 7\\text{B} \\\\ \\hline &6396 \\\\ \\end{array}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "3996",
"split": "train"
} |
math | [
{
"content": "Using a $4 \\times 4$ grid where points are spaced equally at 1 unit apart both horizontally and vertically, how many rectangles are there whose four vertices are points on this grid? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "101",
"style": "rule"
} | {
"index": "3997",
"split": "train"
} |
math | [
{
"content": "Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "95",
"style": "rule"
} | {
"index": "3998",
"split": "train"
} |
math | [
{
"content": "On the side \\( AD \\) of the rhombus \\( ABCD \\), a point \\( M \\) is taken such that \\( MD = 0.3 \\, AD \\) and \\( BM = MC = 11 \\). Find the area of triangle \\( BCM \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20\\sqrt{6}",
"style": "rule"
} | {
"index": "3999",
"split": "train"
} |
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