data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Let the positive divisors of \\( 2014^2 \\) be \\( d_{1}, d_{2}, \\cdots, d_{k} \\). Then\n$$\n\\frac{1}{d_{1}+2014}+\\frac{1}{d_{2}+2014}+\\cdots+\\frac{1}{d_{k}+2014} = \n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{27}{4028}",
"style": "rule"
} | {
"index": "38300",
"split": "train"
} |
math | [
{
"content": "If $p$, $q$, $r$, $s$, $t$, and $u$ are integers such that $1728x^3 + 64 = (px^2 + qx + r)(sx^2 + tx + u)$ for all $x$, then what is $p^2+q^2+r^2+s^2+t^2+u^2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23456",
"style": "rule"
} | {
"index": "38301",
"split": "train"
} |
math | [
{
"content": "Let \\( a, b, c, d \\) be positive integers such that \\( \\gcd(a, b) = 24 \\), \\( \\gcd(b, c) = 36 \\), \\( \\gcd(c, d) = 54 \\), and \\( 70 < \\gcd(d, a) < 100 \\). Which of the following numbers is a factor of \\( a \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "38302",
"split": "train"
} |
math | [
{
"content": "Find the sum of the infinite series $1+2\\left(\\dfrac{1}{1998}\\right)+3\\left(\\dfrac{1}{1998}\\right)^2+4\\left(\\dfrac{1}{1998}\\right)^3+\\cdots$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3992004}{3988009}",
"style": "rule"
} | {
"index": "38303",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, where $C$ is an obtuse angle. Given $\\cos(A + B - C) = \\frac{1}{4}$, $a = 2$, and $\\frac{\\sin(B + A)}{\\sin A} = 2$.\n\n(1) Find the value of $\\cos C$;\n\n(2) Find the length of $b$. Pleas... | {
"ground_truth": "\\sqrt{6}",
"style": "rule"
} | {
"index": "38304",
"split": "train"
} |
math | [
{
"content": "Let the circles $S_1$ and $S_2$ meet at the points $A$ and $B$. A line through $B$ meets $S_1$ at a point $D$ other than $B$ and meets $S_2$ at a point $C$ other than $B$. The tangent to $S_1$ through $D$ and the tangent to $S_2$ through $C$ meet at $E$. If $|AD|=15$, $|AC|=16$, $|AB|=10$, what is... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "38305",
"split": "train"
} |
math | [
{
"content": "Given an ellipse $C:\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with foci $F_{1}$ and $F_{2}$, point $A$ lies on $C$, point $B$ lies on the $y$-axis, and satisfies $\\overrightarrow{AF_{1}}⊥\\overrightarrow{BF_{1}}$, $\\overrightarrow{AF_{2}}=\\frac{2}{3}\\overrightarrow{F_{2}B}$. ... | {
"ground_truth": "\\frac{\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "38306",
"split": "train"
} |
math | [
{
"content": "Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "38307",
"split": "train"
} |
math | [
{
"content": "Simplify first, then evaluate: $4(3a^{2}b-ab^{2})-(2ab^{2}+3a^{2}b)$, where $a=\\frac{1}{3}$ and $b=-1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "38308",
"split": "train"
} |
math | [
{
"content": "Solve for $x$: $0.04x + 0.05(25 + x) = 13.5$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "136.\\overline{1}",
"style": "rule"
} | {
"index": "38309",
"split": "train"
} |
math | [
{
"content": "Find $a$ such that $ax^2+15x+4$ is the square of a binomial. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{225}{16}",
"style": "rule"
} | {
"index": "38310",
"split": "train"
} |
math | [
{
"content": "For how many primes \\( p < 50 \\) is \\( p^{4} + 5p^{3} + 4 \\) divisible by 5? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "38311",
"split": "train"
} |
math | [
{
"content": "The side of the base of a regular quadrilateral pyramid \\( \\operatorname{ABCDP} \\) (with \\( P \\) as the apex) is \\( 4 \\sqrt{2} \\), and the angle between adjacent lateral faces is \\( 120^{\\circ} \\). Find the area of the cross-section of the pyramid by a plane passing through the diagonal... | {
"ground_truth": "4\\sqrt{6}",
"style": "rule"
} | {
"index": "38312",
"split": "train"
} |
math | [
{
"content": "The first 14 terms of the sequence $\\{a_n\\}$ are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38. According to this pattern, find $a_{16}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "46",
"style": "rule"
} | {
"index": "38313",
"split": "train"
} |
math | [
{
"content": "In the diagram, $ABCD$ is a trapezoid with an area of $18.$ $CD$ is three times the length of $AB.$ What is the area of $\\triangle ABD?$\n\n[asy]\ndraw((0,0)--(1,4)--(9,4)--(18,0)--cycle);\ndraw((9,4)--(0,0));\nlabel(\"$D$\",(0,0),W);\nlabel(\"$A$\",(1,4),NW);\nlabel(\"$B$\",(9,4),NE);\nlabel(\"$... | {
"ground_truth": "4.5",
"style": "rule"
} | {
"index": "38314",
"split": "train"
} |
math | [
{
"content": "Veronica put on five rings: one on her little finger, one on her middle finger, and three on her ring finger. In how many different orders can she take them all off one by one? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "38315",
"split": "train"
} |
math | [
{
"content": "If you roll four fair 6-sided dice, what is the probability that at least three of them will show the same value? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{72}",
"style": "rule"
} | {
"index": "38316",
"split": "train"
} |
math | [
{
"content": "The following diagram shows equilateral triangle $\\vartriangle ABC$ and three other triangles congruent to it. The other three triangles are obtained by sliding copies of $\\vartriangle ABC$ a distance $\\frac18 AB$ along a side of $\\vartriangle ABC$ in the directions from $A$ to $B$... | {
"ground_truth": "768",
"style": "rule"
} | {
"index": "38317",
"split": "train"
} |
math | [
{
"content": "Given the ellipse $C:\\dfrac{x^2}{m^2}+y^2=1$ (where $m > 1$ is a constant), $P$ is a moving point on curve $C$, and $M$ is the right vertex of curve $C$. The fixed point $A$ has coordinates $(2,0)$.\n\n$(1)$ If $M$ coincides with $A$, find the coordinates of the foci of curve $C$;\n\n$(2)$ If $m=... | {
"ground_truth": "\\dfrac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "38318",
"split": "train"
} |
math | [
{
"content": "Find the minimum value of\n\\[3x^2 + 3xy + y^2 - 3x + 3y + 9\\]\nover all real numbers $x$ and $y.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{45}{8}",
"style": "rule"
} | {
"index": "38319",
"split": "train"
} |
math | [
{
"content": "Find \\(a\\) in the following sequence: \\(1, 8, 27, 64, a, 216, \\ldots \\ldots\\)\n\\[1^{3}, 2^{3}, 3^{3}, 4^{3}, a, 6^{3}, \\ldots \\ldots\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "38320",
"split": "train"
} |
math | [
{
"content": "The picture shows several circles connected by segments. Tanya chooses a natural number \\( n \\) and places different natural numbers in the circles so that the following property holds for all these numbers:\n\nIf numbers \\( a \\) and \\( b \\) are not connected by a segment, then the sum \\( a... | {
"ground_truth": "65",
"style": "rule"
} | {
"index": "38321",
"split": "train"
} |
math | [
{
"content": "Find the minimum value of \n\\[x^3 + 12x + \\frac{81}{x^4}\\]\nfor $x > 0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "38322",
"split": "train"
} |
math | [
{
"content": "Each of $1000$ elves has a hat, red on the inside and blue on the outside or vise versa. An elf with a hat that is red outside can only lie, and an elf with a hat that is blue outside can only tell the truth. One day every elf tells every other elf, “Your hat is red on the outside.” During that ... | {
"ground_truth": "998",
"style": "rule"
} | {
"index": "38323",
"split": "train"
} |
math | [
{
"content": "The expression $y^2+10y+33$ can be written as a combination of a square of a binomial and an integer. Find the integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "38324",
"split": "train"
} |
math | [
{
"content": "Let $x_{1}, \\ldots, x_{100}$ be defined so that for each $i, x_{i}$ is a (uniformly) random integer between 1 and 6 inclusive. Find the expected number of integers in the set $\\{x_{1}, x_{1}+x_{2}, \\ldots, x_{1}+x_{2}+\\ldots+x_{100}\\}$ that are multiples of 6. Please output the final answer w... | {
"ground_truth": "\\frac{50}{3}",
"style": "rule"
} | {
"index": "38325",
"split": "train"
} |
math | [
{
"content": "A polynomial with integer coefficients is of the form\n\\[x^3 + a_2 x^2 + a_1 x - 11 = 0.\\]Enter all the possible integer roots of this polynomial, separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-11, -1, 1, 11",
"style": "rule"
} | {
"index": "38326",
"split": "train"
} |
math | [
{
"content": "Given $\\frac{cos2α}{sin(α-\\frac{π}{4})}=-\\frac{\\sqrt{6}}{2}$, express $cos(α-\\frac{π}{4})$ in terms of radicals. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{6}}{4}",
"style": "rule"
} | {
"index": "38327",
"split": "train"
} |
math | [
{
"content": "In $\\triangle XYZ$, a triangle $\\triangle MNO$ is inscribed such that vertices $M, N, O$ lie on sides $YZ, XZ, XY$, respectively. The circumcircles of $\\triangle XMO$, $\\triangle YNM$, and $\\triangle ZNO$ have centers $P_1, P_2, P_3$, respectively. Given that $XY = 26, YZ = 28, XZ = 27$, and ... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "38328",
"split": "train"
} |
math | [
{
"content": "Each day 289 students are divided into 17 groups of 17. No two students are ever in the same group more than once. What is the largest number of days that this can be done? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "38329",
"split": "train"
} |
math | [
{
"content": "Let $g(n)$ be the product of the proper positive integer divisors of $n$. (Recall that a proper divisor of $n$ is a divisor other than $n$.) For how many values of $n$ does $n$ not divide $g(n)$, given that $2 \\le n \\le 50$? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "38330",
"split": "train"
} |
math | [
{
"content": "Find the number of integer values of $k$ in the closed interval $[-500,500]$ for which the equation $\\log(kx)=2\\log(x+2)$ has exactly one real solution. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "501",
"style": "rule"
} | {
"index": "38331",
"split": "train"
} |
math | [
{
"content": "On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let $R$ be the region form... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "38332",
"split": "train"
} |
math | [
{
"content": "Chords $\\overline{A B}$ and $\\overline{C D}$ of circle $\\omega$ intersect at $E$ such that $A E=8, B E=2, C D=10$, and $\\angle A E C=90^{\\circ}$. Let $R$ be a rectangle inside $\\omega$ with sides parallel to $\\overline{A B}$ and $\\overline{C D}$, such that no point in the interior of $R$ l... | {
"ground_truth": "26+6 \\sqrt{17}",
"style": "rule"
} | {
"index": "38333",
"split": "train"
} |
math | [
{
"content": "A dart board is shaped like a regular dodecagon divided into regions, with a square at its center. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?\n**A)** $\\frac{2+\\sqrt{3}}{6}$ \n**B)... | {
"ground_truth": "\\frac{2 - \\sqrt{3}}{6}",
"style": "rule"
} | {
"index": "38334",
"split": "train"
} |
math | [
{
"content": "A bin has 5 green balls and $k$ purple balls in it, where $k$ is an unknown positive integer. A ball is drawn at random from the bin. If a green ball is drawn, the player wins 2 dollars, but if a purple ball is drawn, the player loses 2 dollars. If the expected amount won for playing the game i... | {
"ground_truth": "k=3",
"style": "rule"
} | {
"index": "38335",
"split": "train"
} |
math | [
{
"content": "If the space vectors $\\overrightarrow{{e_1}}$ and $\\overrightarrow{{e_2}}$ satisfy $|\\overrightarrow{{e_1}}|=|2\\overrightarrow{{e_1}}+\\overrightarrow{{e_2}}|=3$, determine the maximum value of the projection of $\\overrightarrow{{e_1}}$ in the direction of $\\overrightarrow{{e_2}}$. Please ou... | {
"ground_truth": "-\\frac{3\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "38336",
"split": "train"
} |
math | [
{
"content": "Given $0 < a < 1$, and $\\left[a+\\frac{1}{2020}\\right]+\\left[a+\\frac{2}{2020}\\right]+\\cdots+\\left[a+\\frac{2019}{2020}\\right] = 2018$, calculate the value of $[1010 a]$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1009",
"style": "rule"
} | {
"index": "38337",
"split": "train"
} |
math | [
{
"content": "LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid A dollars and Bernardo had paid B dollars, where $A < B.... | {
"ground_truth": "\\frac{B-A}{2}",
"style": "rule"
} | {
"index": "38338",
"split": "train"
} |
math | [
{
"content": "The points $(0,4)$ and $(1,3)$ lie on a circle whose center is on the $x$-axis. What is the radius of the circle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "38339",
"split": "train"
} |
math | [
{
"content": "What is the value of $x + y$ if the sequence $3, ~9, ~x, ~y, ~30$ is an arithmetic sequence? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "38340",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=x^{3}- \\frac {3}{2}x^{2}+ \\frac {3}{4}x+ \\frac {1}{8}$, find the value of $\\sum\\limits_{k=1}^{2016}f( \\frac {k}{2017})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "504",
"style": "rule"
} | {
"index": "38341",
"split": "train"
} |
math | [
{
"content": "Define an ordered triple $(A, B, C)$ of sets to be minimally intersecting if $|A \\cap B| = |B \\cap C| = |C \\cap A| = 1$ and $A \\cap B \\cap C = \\emptyset$. For example, $(\\{1,2\\},\\{2,3\\},\\{1,3,4\\})$ is a minimally intersecting triple. Let $N$ be the number of minimally intersecting orde... | {
"ground_truth": "760",
"style": "rule"
} | {
"index": "38342",
"split": "train"
} |
math | [
{
"content": "How many positive divisors of 30! are prime? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "38343",
"split": "train"
} |
math | [
{
"content": "In the picture, there is a grid consisting of 25 small equilateral triangles. How many rhombuses can be made from two adjacent small triangles? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "38344",
"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": "38345",
"split": "train"
} |
math | [
{
"content": "Suppose that \\[\\operatorname{lcm}(1024,2016)=\\operatorname{lcm}(1024,2016,x_1,x_2,\\ldots,x_n),\\] with $x_1$ , $x_2$ , $\\cdots$ , $x_n$ are distinct postive integers. Find the maximum value of $n$ .\n\n*Proposed by Le Duc Minh* Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "38346",
"split": "train"
} |
math | [
{
"content": "A 4-inch by 6-inch picture is enlarged for framing by tripling its dimensions. A 2-inch-wide border is then placed around each side of the enlarged picture, as shown. Thin metal framing is sold only in increments of one foot. What is the minimum number of linear feet of framing that must b... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "38347",
"split": "train"
} |
math | [
{
"content": "Evaluate $$\\lceil\\sqrt{10}\\rceil + \\lceil\\sqrt{11}\\rceil + \\lceil\\sqrt{12}\\rceil + \\cdots + \\lceil\\sqrt{34}\\rceil$$\nNote: For a real number $x,$ $\\lceil x \\rceil$ denotes the smallest integer that is greater than or equal to $x.$ Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "127",
"style": "rule"
} | {
"index": "38348",
"split": "train"
} |
math | [
{
"content": "Matt now needs to place five identical, dotless dominoes (shaded 1 by 2 rectangles) on a 6 by 5 grid. He wishes to create a path from the upper left-hand corner $C$ to the lower right-hand corner $D$. In the path, consecutive dominoes must touch at their sides and not just at their corners. No dom... | {
"ground_truth": "126",
"style": "rule"
} | {
"index": "38349",
"split": "train"
} |
math | [
{
"content": "These two spinners are divided into thirds and quarters, respectively. If each of these spinners is spun once, what is the probability that the product of the results of the two spins will be an even number? Express your answer as a common fraction.\n\n[asy]\n\nsize(5cm,5cm);\n\ndraw(Circle((0,0),... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "38350",
"split": "train"
} |
math | [
{
"content": "Calculate $45 \\cdot 68 \\cdot 99 \\equiv m \\pmod{25}$, where $0 \\leq m < 25$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "38351",
"split": "train"
} |
math | [
{
"content": "Given the ellipse \\(\\frac{x^{2}}{16}+\\frac{y^{2}}{9}=1\\) with three points \\(P\\), \\(Q\\), and \\(R\\) on it, where \\(P\\) and \\(Q\\) are symmetric with respect to the origin. Find the maximum value of \\(|RP| + |RQ|\\). Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "38352",
"split": "train"
} |
math | [
{
"content": "According to statistical data, the daily output of a factory does not exceed 200,000 pieces, and the daily defect rate $p$ is approximately related to the daily output $x$ (in 10,000 pieces) by the following relationship:\n$$\np= \\begin{cases} \n\\frac{x^{2}+60}{540} & (0<x\\leq 12) \\\\\n\\frac{... | {
"ground_truth": "\\frac{100}{9}",
"style": "rule"
} | {
"index": "38353",
"split": "train"
} |
math | [
{
"content": "Consider the set $S$ of permutations of $1, 2, \\dots, 2022$ such that for all numbers $k$ in the\npermutation, the number of numbers less than $k$ that follow $k$ is even.\n\nFor example, for $n=4; S = \\{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\\}$ If $|S| = (a!)^b$ where $a, b \... | {
"ground_truth": "2022",
"style": "rule"
} | {
"index": "38354",
"split": "train"
} |
math | [
{
"content": "Let $ a_1 \\equal{} 11^{11}, \\, a_2 \\equal{} 12^{12}, \\, a_3 \\equal{} 13^{13}$, and $ a_n \\equal{} |a_{n \\minus{} 1} \\minus{} a_{n \\minus{} 2}| \\plus{} |a_{n \\minus{} 2} \\minus{} a_{n \\minus{} 3}|, n \\geq 4.$ Determine $ a_{14^{14}}$. Please output the final answer within \\boxed{}.",... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "38355",
"split": "train"
} |
math | [
{
"content": "Given $f(x)=6-12x+x\\,^{3},x\\in\\left[-\\frac{1}{3},1\\right]$, find the maximum and minimum values of the function. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-5",
"style": "rule"
} | {
"index": "38356",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be a rectangle and let $\\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\\overline{DM}$ has integer length, and the lengths of $\\overline{MA}, \\overline{MC},$ and $\\overline{MB}$ are consecutive odd positive integers (in this order). What is the volum... | {
"ground_truth": "24\\sqrt{5}",
"style": "rule"
} | {
"index": "38357",
"split": "train"
} |
math | [
{
"content": "Masha and the bear ate a basket of raspberries and 40 pies, starting and finishing at the same time. Initially, Masha ate raspberries while the bear ate pies, and then (at some moment) they switched. The bear ate both raspberries and pies 3 times faster than Masha. How many pies did Masha eat if t... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "38358",
"split": "train"
} |
math | [
{
"content": "Let $d_1 = a^2 + 2^a + a \\cdot 2^{(a+1)/2} + a^3$ and $d_2 = a^2 + 2^a - a \\cdot 2^{(a+1)/2} + a^3$. If $1 \\le a \\le 300$, how many integral values of $a$ are there such that $d_1 \\cdot d_2$ is a multiple of $3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "38359",
"split": "train"
} |
math | [
{
"content": "In a certain grade with 1000 students, 100 students are selected as a sample using systematic sampling. All students are numbered from 1 to 1000, and are evenly divided into 100 groups (1-10, 11-20, ..., 991-1000 in order). If the number drawn from the first group is 6, then the number that should... | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "38360",
"split": "train"
} |
math | [
{
"content": "In the center of a circular field, there is a geologists' hut. From it, 6 straight roads radiate out, dividing the field into 6 equal sectors. Two geologists leave their hut at a speed of 4 km/h, each choosing a road at random. Determine the probability that the distance between them after one hou... | {
"ground_truth": "0.5",
"style": "rule"
} | {
"index": "38361",
"split": "train"
} |
math | [
{
"content": "The hyperbola given by the equation \\[\\frac{x^2}{100} - \\frac{y^2}{64} = 1\\]has asymptotes $y = \\pm mx,$ where $m$ is positive. Find $m.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac45",
"style": "rule"
} | {
"index": "38362",
"split": "train"
} |
math | [
{
"content": "Given non-negative real numbers $x_{1}, x_{2}, \\cdots, x_{n}$ satisfying the inequality $x_{1} + x_{2} + \\cdots + x_{n} \\leqslant \\frac{1}{2}$, find the minimum value of $f\\left(x_{1}, x_{2}, \\cdots, x_{n}\\right) = \\prod_{i=1}^{n}\\left(1 - x_{i}\\right)$. Please output the final answer wi... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "38363",
"split": "train"
} |
math | [
{
"content": "Given six people are arranged in a row from left to right. Only A or B can be placed at the far left, and A cannot be placed at the far right, calculate the total number of different arrangements. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "216",
"style": "rule"
} | {
"index": "38364",
"split": "train"
} |
math | [
{
"content": "What are the first three digits to the right of the decimal point in the decimal representation of $(10^{2003}+1)^{11/7}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "571",
"style": "rule"
} | {
"index": "38365",
"split": "train"
} |
math | [
{
"content": "For a positive integer $n$, let $1 \\times 2 \\times \\cdots \\times n = n!$. If $\\frac{2017!}{2^{n}}$ is an integer, find the maximum value of $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2010",
"style": "rule"
} | {
"index": "38366",
"split": "train"
} |
math | [
{
"content": "Lyndy conducted a survey of the 300 students in her school. She found that 60 students own dogs, while only 30 students own cats. What percent of the students own cats? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10\\%",
"style": "rule"
} | {
"index": "38367",
"split": "train"
} |
math | [
{
"content": "**p1.** The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were o... | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "38368",
"split": "train"
} |
math | [
{
"content": "Compute the following expression:\n\\[\n\\frac{(1 + 15) \\left( 1 + \\dfrac{15}{2} \\right) \\left( 1 + \\dfrac{15}{3} \\right) \\dotsm \\left( 1 + \\dfrac{15}{21} \\right)}{(1 + 17) \\left( 1 + \\dfrac{17}{2} \\right) \\left( 1 + \\dfrac{17}{3} \\right) \\dotsm \\left( 1 + \\dfrac{17}{19} \\right... | {
"ground_truth": "272",
"style": "rule"
} | {
"index": "38369",
"split": "train"
} |
math | [
{
"content": "Suppose four solid iron balls are placed in a cylinder with the radius of 1 cm, such that every two of the four balls are tangent to each other, and the two balls in the lower layer are tangent to the cylinder base. Now put water into the cylinder. Find, in $\\text{cm}^2$ , the volume of water ne... | {
"ground_truth": "\\frac{4 \\pi}{3}",
"style": "rule"
} | {
"index": "38370",
"split": "train"
} |
math | [
{
"content": "If $$1+6+11+16+21+26+\\cdots+91+96+101 \\equiv n\\pmod{15},$$where $0\\le n<15$, what is the value of $n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "38371",
"split": "train"
} |
math | [
{
"content": "Calculate the following expression:\n$$2(1+2(1+2(1+2(1+2(1+2(1+2(1+2(1+2(1+2(1+2))))))))))$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4094",
"style": "rule"
} | {
"index": "38372",
"split": "train"
} |
math | [
{
"content": "In how many ways can 5 people be seated around a round table? (Two seatings are considered the same if one is a rotation of the other.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "38373",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $\\cos A \\cos B = \\sin A \\sin B - \\frac{\\sqrt{2}}{2}$. \n$(1)$ Find the measure of angle $C$. \n$(2)$ Given $b=4$ and the area of $\\triangle ABC$ is $6$, find: \n... | {
"ground_truth": "\\frac{\\sqrt{2}}{10}",
"style": "rule"
} | {
"index": "38374",
"split": "train"
} |
math | [
{
"content": "What is $(15x^2) \\cdot (6x) \\cdot \\left(\\frac{1}{(3x)^2}\\right)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10x",
"style": "rule"
} | {
"index": "38375",
"split": "train"
} |
math | [
{
"content": "A line passing through point P(1, 2) is tangent to the circle $x^2+y^2=4$ and perpendicular to the line $ax-y+1=0$. Find the value of the real number $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{4}",
"style": "rule"
} | {
"index": "38376",
"split": "train"
} |
math | [
{
"content": "Two circles have centers that are \\( d \\) units apart, and each has a diameter \\( \\sqrt{d} \\). For any \\( d \\), let \\( A(d) \\) be the area of the smallest circle that contains both of these circles. Find \\( \\lim _{d \\rightarrow \\infty} \\frac{A(d)}{d^{2}} \\). Please output the final ... | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "38377",
"split": "train"
} |
math | [
{
"content": "Compute the unique positive integer $m$ such that\n\\[1 \\cdot 2^1 + 2 \\cdot 2^2 + 3 \\cdot 2^3 + \\dots + m \\cdot 2^m = 2^{m + 8}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "129",
"style": "rule"
} | {
"index": "38378",
"split": "train"
} |
math | [
{
"content": "What is the tenth positive integer that is both odd and a multiple of 3? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "57",
"style": "rule"
} | {
"index": "38379",
"split": "train"
} |
math | [
{
"content": "Given that $\\alpha$ is an obtuse angle and $\\beta$ is also an obtuse angle with $\\cos\\alpha = -\\frac{2\\sqrt{5}}{5}$ and $\\sin\\beta = \\frac{\\sqrt{10}}{10}$, find the value of $\\alpha + \\beta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7\\pi}{4}",
"style": "rule"
} | {
"index": "38380",
"split": "train"
} |
math | [
{
"content": "An ellipse has foci at $(0, 2)$ and $(3, 0)$. It has two $x$-intercepts, one of which is the origin. What is the other one? Enter your answer as an ordered pair. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left(\\tfrac{15}{4},0\\right)",
"style": "rule"
} | {
"index": "38381",
"split": "train"
} |
math | [
{
"content": "Given that spinner A contains the numbers 4, 5, 6, spinner B contains the numbers 1, 2, 3, 4, 5, and spinner C can take numbers from the set 7, 8, 9 if spinner B lands on an odd number and the set {6, 8} if spinner B lands on an even number, find the probability that the sum of the numbers resulti... | {
"ground_truth": "\\frac{4}{15}",
"style": "rule"
} | {
"index": "38382",
"split": "train"
} |
math | [
{
"content": "Find the coefficient of $x^2$ when $3(x^2 - x^3) +2(x - 2x^2 + 3x^5) -(4x^3 - x^2)$ is simplified. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "38383",
"split": "train"
} |
math | [
{
"content": "Given six people, including A, B, and C, are arranged in a row for a photo, and A and B are not on the same side as C, calculate the total number of different arrangements. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "38384",
"split": "train"
} |
math | [
{
"content": "Angle bisectors $AA', BB'$ and $CC'$ are drawn in triangle $ABC$ with angle $\\angle B= 120^o$ . Find $\\angle A'B'C'$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "38385",
"split": "train"
} |
math | [
{
"content": "The circle C is tangent to the y-axis and the line l: y = (√3/3)x, and the circle C passes through the point P(2, √3). Determine the diameter of circle C. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{14}{3}",
"style": "rule"
} | {
"index": "38386",
"split": "train"
} |
math | [
{
"content": "Given the hyperbola $\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1$ and the parabola $y^{2} = 4cx$, where $c = \\sqrt{a^{2} + b^{2}}$, find the eccentricity of the hyperbola given that $|AB| = 4c$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2} + 1",
"style": "rule"
} | {
"index": "38387",
"split": "train"
} |
math | [
{
"content": "Given that \\( a, b, c, d \\) are prime numbers (they can be the same), and \\( abcd \\) is the sum of 35 consecutive positive integers, find the minimum value of \\( a + b + c + d \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "38388",
"split": "train"
} |
math | [
{
"content": "In a small office, each worker has a probability of being late once every 40 days due to traffic. Calculate the probability that among three randomly chosen workers on a given day, exactly two are late while the third one is on time. Express your answer as a percent to the nearest tenth. Please ou... | {
"ground_truth": "0.2\\%",
"style": "rule"
} | {
"index": "38389",
"split": "train"
} |
math | [
{
"content": "In the following equation, each of the letters represents uniquely a different digit in base ten:\n\\[(YE) \\cdot (ME) = TTT\\]\nThe sum $E+M+T+Y$ equals Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "38390",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $(a+b)(\\sin A - \\sin B) = (c-b)\\sin C.$ \n(1) Determine the measure of angle $A$. \n(2) If $2c=3b$ and the area of $\\triangle ABC$ is $6 \\sqrt{3}$, find the value of $a$. Please output the... | {
"ground_truth": "\\frac{2\\sqrt{21}}{3}",
"style": "rule"
} | {
"index": "38391",
"split": "train"
} |
math | [
{
"content": "Find the $\\emph{positive}$ real number(s) $x$ such that $\\frac{1}{2}\\left( 3x^2-1\\right) = \\left( x^2-50x-10\\right)\\left( x^2+25x+5\\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25 + 2\\sqrt{159}",
"style": "rule"
} | {
"index": "38392",
"split": "train"
} |
math | [
{
"content": "For a given positive integer $n$, we define $\\varphi(n)$ to be the number of positive integers less than or equal to $n$ which share no common prime factors with $n$. Find all positive integers $n$ for which $\\varphi(2019 n)=\\varphi\\left(n^{2}\\right)$. Please output the final answer within \\... | {
"ground_truth": "1346, 2016, 2019",
"style": "rule"
} | {
"index": "38393",
"split": "train"
} |
math | [
{
"content": "Given the functions f(x) = x and g(x) = ax^2 - x, where a > 0. If for all x1 in the interval [1, 2], there exists an x2 in the interval [1, 2] such that f(x1) * f(x2) = g(x1) * g(x2) holds true, then find the value of a. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "38394",
"split": "train"
} |
math | [
{
"content": "Determine the values of $x$ and $y$ given that 15 is the arithmetic mean of the set $\\{8, x, 21, y, 14, 11\\}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "38395",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be a triangle in the plane with $AB=13, BC=14, AC=15$. Let $M_{n}$ denote the smallest possible value of $\\left(AP^{n}+BP^{n}+CP^{n}\\right)^{\\frac{1}{n}}$ over all points $P$ in the plane. Find $\\lim _{n \\rightarrow \\infty} M_{n}$. Please output the final answer within \\boxed{}.",... | {
"ground_truth": "\\frac{65}{8}",
"style": "rule"
} | {
"index": "38396",
"split": "train"
} |
math | [
{
"content": "Let $(a,b,c,d)$ be a solution to the system\\begin{align*}a+b&=15,\\\\ab+c+d&=78,\\\\ad+bc&=160,\\\\cd&=96.\\end{align*}Find the greatest possible value of $a^2+b^2+c^2+d^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "717",
"style": "rule"
} | {
"index": "38397",
"split": "train"
} |
math | [
{
"content": "A line segment is divided into three parts, $x$, $y$, and $z$, such that, $x < y < z$ and $x$ to $y$ is as $y$ to $z$. If all three parts combined form a segment of length $s$, and $x + y = z$, determine the value of the ratio $\\frac{x}{y}$.\nA) $\\frac{-1 - \\sqrt{5}}{2}$\nB) $\\frac{-1 + \\sqrt... | {
"ground_truth": "\\frac{-1 + \\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "38398",
"split": "train"
} |
math | [
{
"content": "Two circles are externally tangent to each other at point \\( A \\), and both are tangent to a third circle at points \\( B \\) and \\( C \\). The extension of chord \\( AB \\) of the first circle intersects the second circle at point \\( D \\), and the extension of chord \\( AC \\) intersects the... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "38399",
"split": "train"
} |
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