data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "The number \\( a \\) is a root of the equation \\( x^{11} + x^{7} + x^{3} = 1 \\). Specify all natural values of \\( n \\) for which the equality \\( a^{4} + a^{3} = a^{n} + 1 \\) holds. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "5700",
"split": "train"
} |
math | [
{
"content": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl}\nax+3 & \\text{ if }x>0, \\\\\nab & \\text{ if }x=0, \\\\\nbx+c & \\text{ if }x<0.\n\\end{array}\n\\right.\\]If $f(2)=5$, $f(0)=5$, and $f(-2)=-10$, and $a$, $b$, and $c$ are nonnegative integers, then what is $a+b+c$? Please output the final answer with... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "5701",
"split": "train"
} |
math | [
{
"content": "Given that Carl has 24 fence posts and places one on each of the four corners, with 3 yards between neighboring posts, where the number of posts on the longer side is three times the number of posts on the shorter side, determine the area, in square yards, of Carl's lawn. Please output the final a... | {
"ground_truth": "243",
"style": "rule"
} | {
"index": "5702",
"split": "train"
} |
math | [
{
"content": "The image shows a grid consisting of 25 small equilateral triangles. How many rhombuses can be formed from two adjacent small triangles? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "5703",
"split": "train"
} |
math | [
{
"content": "Solve for $x$: $\\frac{6x^2 + 111x +1}{2x+37} = 3x + 1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-18",
"style": "rule"
} | {
"index": "5704",
"split": "train"
} |
math | [
{
"content": "Given a parallelepiped $A B C D A_1 B_1 C_1 D_1$, points $M, N, K$ are midpoints of edges $A B$, $B C$, and $D D_1$ respectively. Construct the cross-sectional plane of the parallelepiped with the plane $MNK$. In what ratio does this plane divide the edge $C C_1$ and the diagonal $D B_1$? Please o... | {
"ground_truth": "3:7",
"style": "rule"
} | {
"index": "5705",
"split": "train"
} |
math | [
{
"content": "Given non-zero vectors \\\\(a\\\\) and \\\\(b\\\\) satisfying \\\\(|b|=2|a|\\\\) and \\\\(a \\perp (\\sqrt{3}a+b)\\\\), find the angle between \\\\(a\\\\) and \\\\(b\\\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{5\\pi}{6}",
"style": "rule"
} | {
"index": "5706",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $a\\cos B=(3c-b)\\cos A$.\n\n$(1)$ Find the value of $\\cos A$;\n\n$(2)$ If $b=3$, and point $M$ is on the line segment $BC$, $\\overrightarrow{AB} + \\overrighta... | {
"ground_truth": "7\\sqrt {2}",
"style": "rule"
} | {
"index": "5707",
"split": "train"
} |
math | [
{
"content": "Let $S$ be a subset of $\\{1,2,3, \\ldots, 12\\}$ such that it is impossible to partition $S$ into $k$ disjoint subsets, each of whose elements sum to the same value, for any integer $k \\geq 2$. Find the maximum possible sum of the elements of $S$. Please output the final answer within \\boxed{}.... | {
"ground_truth": "77",
"style": "rule"
} | {
"index": "5708",
"split": "train"
} |
math | [
{
"content": "A massive vertical plate is fixed to a car moving at a speed of $5 \\, \\text{m/s}$. A ball is flying towards it at a speed of $6 \\, \\text{m/s}$ with respect to the ground. Determine the speed of the ball with respect to the ground after a perfectly elastic normal collision. Please output the fi... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "5709",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = \\sin(\\omega x + \\phi)$ ($\\omega > 0$, $0 \\leq \\phi \\leq \\pi$) is an even function, and the distance between a neighboring highest point and lowest point on its graph is $\\sqrt{4+\\pi^2}$. \n(1) Find the analytical expression of the function $f(x)$. \n(2) If $\... | {
"ground_truth": "-\\frac{5}{9}",
"style": "rule"
} | {
"index": "5710",
"split": "train"
} |
math | [
{
"content": "Find the number of solutions to\n\\[\\sin x = \\left( \\frac{1}{2} \\right)^x\\]on the interval $(0,100 \\pi).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "5711",
"split": "train"
} |
math | [
{
"content": "Given that \\( x \\) and \\( y \\) are real numbers satisfying the following equations:\n\\[\nx + xy + y = 2 + 3 \\sqrt{2} \\quad \\text{and} \\quad x^2 + y^2 = 6,\n\\]\nfind the value of \\( |x + y + 1| \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3 + \\sqrt{2}",
"style": "rule"
} | {
"index": "5712",
"split": "train"
} |
math | [
{
"content": "Triangle $\\triangle ABC$ in the figure has area $10$. Points $D, E$ and $F$, all distinct from $A, B$ and $C$, \nare on sides $AB, BC$ and $CA$ respectively, and $AD = 2, DB = 3$. If triangle $\\triangle ABE$ and quadrilateral $DBEF$ \nhave equal areas, then that area is Please output the final a... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "5713",
"split": "train"
} |
math | [
{
"content": "Let $f$ be a quadratic function which satisfies the following condition. Find the value of $\\frac{f(8)-f(2)}{f(2)-f(1)}$ .\n\nFor two distinct real numbers $a,b$ , if $f(a)=f(b)$ , then $f(a^2-6b-1)=f(b^2+8)$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "5714",
"split": "train"
} |
math | [
{
"content": "Given a moving circle $M$ that passes through the fixed point $F(0,-1)$ and is tangent to the line $y=1$. The trajectory of the circle's center $M$ forms a curve $C$. Let $P$ be a point on the line $l$: $x-y+2=0$. Draw two tangent lines $PA$ and $PB$ from point $P$ to the curve $C$, where $A$ and ... | {
"ground_truth": "\\frac{9}{2}",
"style": "rule"
} | {
"index": "5715",
"split": "train"
} |
math | [
{
"content": "In right triangle $PQR$, we have $\\angle Q = \\angle R$ and $QR = 6\\sqrt{2}$. What is the area of $\\triangle PQR$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "5716",
"split": "train"
} |
math | [
{
"content": "If 7:30 a.m. was 16 minutes ago, how many minutes will it be until 8:00 a.m.? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "5717",
"split": "train"
} |
math | [
{
"content": "Five students, A, B, C, D, and E, participated in a labor skills competition. A and B asked about the results. The respondent told A, \"Unfortunately, neither you nor B got first place.\" To B, the respondent said, \"You certainly are not the worst.\" Determine the number of different possible ran... | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "5718",
"split": "train"
} |
math | [
{
"content": "The average of the numbers $1, 2, 3,\\dots, 49, 50,$ and $x$ is $80x$. What is $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1275}{4079}",
"style": "rule"
} | {
"index": "5719",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC,$ $AB = 9,$ $BC = 10,$ and $AC = 11.$ If $D$ and $E$ are chosen on $\\overline{AB}$ and $\\overline{AC}$ so that $AD = 4$ and $AE = 7,$ then find the area of triangle $ADE.$\n\n[asy]\nunitsize (1 cm);\n\npair A, B, C, D, E;\n\nA = (2,3);\nB = (0,0);\nC = (6,0);\nD = interp(A,B,0.... | {
"ground_truth": "\\frac{280 \\sqrt{2}}{33}",
"style": "rule"
} | {
"index": "5720",
"split": "train"
} |
math | [
{
"content": "A sign has 31 spaces on a single line. The word RHOMBUS is written from left to right in 7 consecutive spaces. There is an equal number of empty spaces on each side of the word. Counting from the left, in what space number should the letter $R$ be put? Please output the final answer within \\boxed... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "5721",
"split": "train"
} |
math | [
{
"content": "Find the number of pairs $(m,n)$ of positive integers with $1\\le m<n\\le 30$ such that there exists a real number $x$ satisfying \\[\\sin(mx)+\\sin(nx)=2.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "63",
"style": "rule"
} | {
"index": "5722",
"split": "train"
} |
math | [
{
"content": "The diagram depicts a bike route through a park, along with the lengths of some of its segments in kilometers. What is the total length of the bike route in kilometers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "52",
"style": "rule"
} | {
"index": "5723",
"split": "train"
} |
math | [
{
"content": "The letters L, K, R, F, O and the digits 1, 7, 8, 9 are \"cycled\" separately and put together in a numbered list. Determine the line number on which LKRFO 1789 will appear for the first time. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "5724",
"split": "train"
} |
math | [
{
"content": "From the 10 numbers $0, 1, 2, \\cdots, 9$, select 3 such that their sum is an even number not less than 10. How many different ways are there to make such a selection? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "5725",
"split": "train"
} |
math | [
{
"content": "What is the value of $(5 \nabla 1)+(4 \nabla 1)$ if the operation $k \nabla m$ is defined as $k(k-m)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "5726",
"split": "train"
} |
math | [
{
"content": "In a factor tree, each value is the product of the two values below it, unless a value is a prime number or a preset integer product of primes. Using this structure, calculate the value of $X$ in the factor tree provided:\n\n[asy]\ndraw((-2,-.3)--(0,0)--(2,-.3),linewidth(1));\ndraw((-3,-1.3)--(-2,... | {
"ground_truth": "6776",
"style": "rule"
} | {
"index": "5727",
"split": "train"
} |
math | [
{
"content": "Determine the number of positive integers $n$ satisfying:\n\n\n\n- $n<10^6$\n- $n$ is divisible by 7\n- $n$ does not contain any of the digits 2,3,4,5,6,7,8. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "104",
"style": "rule"
} | {
"index": "5728",
"split": "train"
} |
math | [
{
"content": "Jenna bakes a $24$-inch by $15$-inch pan of chocolate cake. The cake pieces are made to measure $3$ inches by $2$ inches each. Calculate the number of pieces of cake the pan contains. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "5729",
"split": "train"
} |
math | [
{
"content": "Let $a,$ $b,$ $c,$ $d,$ $e,$ $f$ be positive real numbers such that $a + b + c + d + e + f = 7.$ Find the minimum value of\n\\[\\frac{1}{a} + \\frac{4}{b} + \\frac{9}{c} + \\frac{16}{d} + \\frac{25}{e} + \\frac{36}{f}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "63",
"style": "rule"
} | {
"index": "5730",
"split": "train"
} |
math | [
{
"content": "Given circles $P, Q,$ and $R$ where $P$ has a radius of 1 unit, $Q$ a radius of 2 units, and $R$ a radius of 1 unit. Circles $Q$ and $R$ are tangent to each other externally, and circle $R$ is tangent to circle $P$ externally. Compute the area inside circle $Q$ but outside circle $P$ and circle $R... | {
"ground_truth": "2\\pi",
"style": "rule"
} | {
"index": "5731",
"split": "train"
} |
math | [
{
"content": "What is the sum of the proper divisors of 256? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "255",
"style": "rule"
} | {
"index": "5732",
"split": "train"
} |
math | [
{
"content": "Each vertex of a cube is to be labeled with an integer 1 through 8, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered ... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "5733",
"split": "train"
} |
math | [
{
"content": "Kolya started playing WoW when the hour and minute hands were opposite each other. He finished playing after a whole number of minutes, at which point the minute hand coincided with the hour hand. How long did he play (assuming he played for less than 12 hours)? Please output the final answer with... | {
"ground_truth": "360",
"style": "rule"
} | {
"index": "5734",
"split": "train"
} |
math | [
{
"content": "A production team in a factory is manufacturing a batch of parts. Initially, when each worker is on their own original position, the task can be completed in 9 hours. If the positions of workers $A$ and $B$ are swapped, and other workers' efficiency remains the same, the task can be completed one ... | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "5735",
"split": "train"
} |
math | [
{
"content": "A shopper buys a $100$ dollar coat on sale for $20\\%$ off. An additional $5$ dollars are taken off the sale price by using a discount coupon. A sales tax of $8\\%$ is paid on the final selling price. The total amount the shopper pays for the coat is Please output the final answer within \\boxe... | {
"ground_truth": "81.00",
"style": "rule"
} | {
"index": "5736",
"split": "train"
} |
math | [
{
"content": "Given \\( f(x) = \\max \\left| x^3 - a x^2 - b x - c \\right| \\) for \\( 1 \\leq x \\leq 3 \\), find the minimum value of \\( f(x) \\) as \\( a, b, \\) and \\( c \\) range over all real numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/4",
"style": "rule"
} | {
"index": "5737",
"split": "train"
} |
math | [
{
"content": "For how many two-digit natural numbers \\( n \\) are exactly two of the following three statements true: (A) \\( n \\) is odd; (B) \\( n \\) is not divisible by 3; (C) \\( n \\) is divisible by 5? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "5738",
"split": "train"
} |
math | [
{
"content": "Let $ A$ , $ B$ be the number of digits of $ 2^{1998}$ and $ 5^{1998}$ in decimal system. $ A \\plus B \\equal{} ?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1999",
"style": "rule"
} | {
"index": "5739",
"split": "train"
} |
math | [
{
"content": "What is the units digit of the product of the first three positive composite numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "5740",
"split": "train"
} |
math | [
{
"content": "Let $a$, $b$, and $c$ be three positive real numbers whose sum is 2. If no one of these numbers is more than three times any other, find the minimum value of the product $abc$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{9}",
"style": "rule"
} | {
"index": "5741",
"split": "train"
} |
math | [
{
"content": "Mr. Harman needs to know the combined weight in pounds of three boxes he wants to mail. However, the only available scale is not accurate for weights less than $100$ pounds or more than $150$ pounds. So the boxes are weighed in pairs in every possible way. The results are $122$, $125$ and $127$ po... | {
"ground_truth": "187",
"style": "rule"
} | {
"index": "5742",
"split": "train"
} |
math | [
{
"content": "Doraemon and Nobita are playing the game \"rock, paper, scissors.\" The rules state that the winner of each round receives two dorayakis, while the loser gets none. If there is a tie, each player receives one dorayaki. Nobita knows that Doraemon can only play \"rock,\" but he still wants to share ... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "5743",
"split": "train"
} |
math | [
{
"content": "Find $\\tan Y$ in the right triangle shown below.\n\n[asy]\npair X,Y,Z;\nX = (0,0);\nY = (24,0);\nZ = (0,7);\ndraw(X--Y--Z--X);\ndraw(rightanglemark(Y,X,Z,23));\nlabel(\"$X$\",X,SW);\nlabel(\"$Y$\",Y,SE);\nlabel(\"$Z$\",Z,N);\nlabel(\"$25$\",(Y+Z)/2,NE);\nlabel(\"$24$\",Y/2,S);\n[/asy] Please outp... | {
"ground_truth": "\\frac{7}{24}",
"style": "rule"
} | {
"index": "5744",
"split": "train"
} |
math | [
{
"content": "Given the parabola \\(\\Gamma: y^{2}=8 x\\) with focus \\(F\\), a line \\(l\\) passing through \\(F\\) intersects parabola \\(\\Gamma\\) at points \\(A\\) and \\(B\\). Tangents to parabola \\(\\Gamma\\) at \\(A\\) and \\(B\\) intersect the \\(y\\)-axis at points \\(P\\) and \\(Q\\) respectively. F... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "5745",
"split": "train"
} |
math | [
{
"content": "Find $10110_2\\times10100_2\\div10_2$. Express your answer in base 2. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11011100_2",
"style": "rule"
} | {
"index": "5746",
"split": "train"
} |
math | [
{
"content": "Martians love dancing dances that require holding hands. In the dance \"Pyramid,\" no more than 7 Martians can participate, each with no more than three hands. What is the maximum number of hands that can be involved in the dance if each hand of one Martian holds exactly one hand of another Martia... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "5747",
"split": "train"
} |
math | [
{
"content": "Solve $x=\\sqrt{x-\\frac{1}{x}}+\\sqrt{1-\\frac{1}{x}}$ for $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1+\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "5748",
"split": "train"
} |
math | [
{
"content": "Given circle $C$: $(x-2)^2 + y^2 = 4$, and line $l$: $x - \\sqrt{3}y = 0$, the probability that the distance from point $A$ on circle $C$ to line $l$ is not greater than $1$ is $\\_\\_\\_\\_\\_\\_\\_.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "5749",
"split": "train"
} |
math | [
{
"content": "Find the value of $x$ that satisfies $\\frac{\\sqrt{3x+5}}{\\sqrt{6x+5}}=\\frac{\\sqrt{5}}{3}$. Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{20}{3}",
"style": "rule"
} | {
"index": "5750",
"split": "train"
} |
math | [
{
"content": "For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$-nice nor $8$-nice. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "749",
"style": "rule"
} | {
"index": "5751",
"split": "train"
} |
math | [
{
"content": "In the right square prism $M-ABCD$, the base $ABCD$ is a rectangle, $MD \\perp$ face $ABCD$ with $MD$ being an integer, and the lengths of $MA$, $MC$, and $MB$ are three consecutive even numbers. What is the volume of the right square prism $M-ABCD$? Please output the final answer within \\boxed{}... | {
"ground_truth": "$24 \\sqrt{5}$",
"style": "rule"
} | {
"index": "5752",
"split": "train"
} |
math | [
{
"content": "Let the function \\( f(x) = x^3 + a x^2 + b x + c \\) (where \\( a, b, c \\) are all non-zero integers). If \\( f(a) = a^3 \\) and \\( f(b) = b^3 \\), then the value of \\( c \\) is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "5753",
"split": "train"
} |
math | [
{
"content": "Let $N$ be the number of ways of distributing $8$ chocolates of different brands among $3$ children such that each child gets at least one chocolate, and no two children get the same number of chocolates. Find the sum of the digits of $N$ . Please output the final answer within \\boxed{}.",... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "5754",
"split": "train"
} |
math | [
{
"content": "Consider the ellipse $C\\_1$: $\\frac{x^{2}}{2}+y^{2}=1$ and the ellipse $C\\_2$: $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1$ ($a > b > 0$). One focus of ellipse $C\\_2$ has the coordinates $(\\sqrt{5},0)$. The line $l$ with slope $1$ intersects ellipse $C\\_2$ at points $A$ and $B$, and the mid... | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "5755",
"split": "train"
} |
math | [
{
"content": "What is the ratio of the numerical value of the area, in square units, of an equilateral triangle of side length 4 units to the numerical value of its perimeter, in units? Express your answer as a common fraction in simplest radical form. Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "5756",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive multiple of $32$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "5757",
"split": "train"
} |
math | [
{
"content": "You are given a number composed of three different non-zero digits, 7, 8, and a third digit which is not 7 or 8. Find the minimum value of the quotient of this number divided by the sum of its digits. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11.125",
"style": "rule"
} | {
"index": "5758",
"split": "train"
} |
math | [
{
"content": "$A_{2n}^{n+3} + A_{4}^{n+1} = \\boxed{\\text{\\_\\_\\_\\_\\_\\_\\_\\_}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "744",
"style": "rule"
} | {
"index": "5759",
"split": "train"
} |
math | [
{
"content": "How many integers $n$ satisfy the inequality $-8\\pi\\le n\\le10\\pi$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "57",
"style": "rule"
} | {
"index": "5760",
"split": "train"
} |
math | [
{
"content": "**Circle $T$ has a circumference of $12\\pi$ inches, and segment $XY$ is a diameter. If the measure of angle $TXZ$ is $45^{\\circ}$, what is the length, in inches, of segment $XZ$?** Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6\\sqrt{2}",
"style": "rule"
} | {
"index": "5761",
"split": "train"
} |
math | [
{
"content": "What is the probability that a triangle formed by randomly selecting 3 vertices from a regular octagon is a right triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{7}",
"style": "rule"
} | {
"index": "5762",
"split": "train"
} |
math | [
{
"content": "Out of the 200 natural numbers between 1 and 200, how many numbers must be selected to ensure that there are at least 2 numbers among them whose product equals 238? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "198",
"style": "rule"
} | {
"index": "5763",
"split": "train"
} |
math | [
{
"content": "$AM$ is the angle bisector of triangle $ABC$. $BM = 2$, $CM = 3$, and $D$ is the point where the extension of $AM$ intersects the circumcircle of the triangle. If $MD = 2$, find $AB$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{10}",
"style": "rule"
} | {
"index": "5764",
"split": "train"
} |
math | [
{
"content": "Let $x, y, z$ be real numbers so that \\begin{align*}\ny+z & = 13, \\\\\nz+x & = 14, \\\\\nx+y & = 15.\n\\end{align*} Find $\\sqrt{xyz(x+y+z)}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "84",
"style": "rule"
} | {
"index": "5765",
"split": "train"
} |
math | [
{
"content": "What is the value of $x$ for which $(8-x)^2=x^2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "5766",
"split": "train"
} |
math | [
{
"content": "In the arithmetic sequence $\\{a\\_n\\}$, it is known that $a\\_1 - a\\_4 - a\\_8 - a\\_{12} + a\\_{15} = 2$. Find the value of $S\\_{15}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-30",
"style": "rule"
} | {
"index": "5767",
"split": "train"
} |
math | [
{
"content": "A man went to a bank to cash a check. The cashier made a mistake and gave him the same number of cents instead of dollars and the same number of dollars instead of cents. The man did not count the money, put it in his pocket, and additionally dropped a 5-cent coin. When he got home, he discovered ... | {
"ground_truth": "31.63",
"style": "rule"
} | {
"index": "5768",
"split": "train"
} |
math | [
{
"content": "Given that the domain of the function f(x) is R, f(2x-2) is an even function, f(x-3)+f(-x+1)=0 when x∈[-2,-1], f(x)=\\frac{1}{a^x}-ax-4 where a>0 and a≠1, and f(-2)=4, find Σ from k=1 to 19 of |f(k)|. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "5769",
"split": "train"
} |
math | [
{
"content": "In a geometric progression with a common ratio of 4, denoted as $\\{b_n\\}$, where $T_n$ represents the product of the first $n$ terms of $\\{b_n\\}$, the fractions $\\frac{T_{20}}{T_{10}}$, $\\frac{T_{30}}{T_{20}}$, and $\\frac{T_{40}}{T_{30}}$ form another geometric sequence with a common ratio ... | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "5770",
"split": "train"
} |
math | [
{
"content": "Calculate the sum of the series $(3+13+23+33+43)+(11+21+31+41+51)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "270",
"style": "rule"
} | {
"index": "5771",
"split": "train"
} |
math | [
{
"content": "A piece of cheese is located at $(12,10)$ in a coordinate plane. A mouse is at $(4,-2)$ and is running up the line $y=-5x+18$. At the point $(a,b)$ the mouse starts getting farther from the cheese rather than closer to it. What is $a + b$? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "5772",
"split": "train"
} |
math | [
{
"content": "Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(4,3)$, respectively. What is its area? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "37.5\\sqrt{3}",
"style": "rule"
} | {
"index": "5773",
"split": "train"
} |
math | [
{
"content": "Quadrilateral $EFGH$ has right angles at $F$ and $H$, and $EG = 5$. If $EFGH$ has two sides with distinct integer lengths, and each side length is greater than 1, what is the area of $EFGH$? Express your answer in simplest radical form. Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "5774",
"split": "train"
} |
math | [
{
"content": "Solve the following equations: \n(1) $\\frac {1}{2}x - 2 = 4 + \\frac {1}{3}x$ \n(2) $\\frac {x-1}{4} - 2 = \\frac {2x-3}{6}$ \n(3) $\\frac {1}{3}[x - \\frac {1}{2}(x-1)] = \\frac {2}{3}(x - \\frac {1}{2})$ \n(4) $\\frac {x}{0.7} - \\frac {0.17-0.2x}{0.03} = 1$ Please output the final answer w... | {
"ground_truth": "\\frac {14}{17}",
"style": "rule"
} | {
"index": "5775",
"split": "train"
} |
math | [
{
"content": "There are a batch of wooden strips with lengths of \\(1, 2, 3, 4, 5, 6, 7, 8, 9, 10,\\) and 11 centimeters, with an adequate quantity of each length. If you select 3 strips appropriately to form a triangle with the requirement that the base is 11 centimeters long, how many different triangles can ... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "5776",
"split": "train"
} |
math | [
{
"content": "An array of integers is arranged in a grid of 7 rows and 1 column with eight additional squares forming a separate column to the right. The sequence of integers in the main column of squares and in each of the two rows form three distinct arithmetic sequences. Find the value of $Q$ if the sequence... | {
"ground_truth": "\\frac{-851}{3}",
"style": "rule"
} | {
"index": "5777",
"split": "train"
} |
math | [
{
"content": "For integers $a$ and $b$ consider the complex number \\[\\frac{\\sqrt{ab+2016}}{ab+100}-\\left({\\frac{\\sqrt{|a+b|}}{ab+100}}\\right)i\\]\n Find the number of ordered pairs of integers $(a,b)$ such that this complex number is a real number. Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "103",
"style": "rule"
} | {
"index": "5778",
"split": "train"
} |
math | [
{
"content": "We call a number greater than $25$, [i] semi-prime[/i] if it is the sum of some two different prime numbers. What is the greatest number of consecutive natural numbers that can be [i]semi-prime[/i]? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "5779",
"split": "train"
} |
math | [
{
"content": "A number $n$ is $b a d$ if there exists some integer $c$ for which $x^{x} \\equiv c(\\bmod n)$ has no integer solutions for $x$. Find the number of bad integers between 2 and 42 inclusive. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "5780",
"split": "train"
} |
math | [
{
"content": "Determine how many five-letter words can be formed such that they begin and end with the same letter and the third letter is always a vowel. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "87880",
"style": "rule"
} | {
"index": "5781",
"split": "train"
} |
math | [
{
"content": "A standard domino game consists of 28 tiles. Each tile is made up of two integers ranging from 0 to 6, inclusive. All possible combinations $(a, b)$, where $a \\leq b$, are listed exactly once. Note that tile $(4,2)$ is listed as tile $(2,4)$, because $2 \\leq 4$. Excluding the piece $(0,0)$, for ... | {
"ground_truth": "\\frac{13}{2}",
"style": "rule"
} | {
"index": "5782",
"split": "train"
} |
math | [
{
"content": "If $\\sin(\\frac{\\pi}{6} - \\alpha) = \\frac{2}{3}$, find the value of $\\cos(\\frac{2\\pi}{3} - \\alpha)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{2}{3}",
"style": "rule"
} | {
"index": "5783",
"split": "train"
} |
math | [
{
"content": "Define $P=\\{\\mathrm{S}, \\mathrm{T}\\}$ and let $\\mathcal{P}$ be the set of all proper subsets of $P$. (A proper subset is a subset that is not the set itself.) How many ordered pairs $(\\mathcal{S}, \\mathcal{T})$ of proper subsets of $\\mathcal{P}$ are there such that (a) $\\mathcal{S}$ is no... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "5784",
"split": "train"
} |
math | [
{
"content": "The endpoints of a line segment $MN$ with a fixed length of $4$ move on the parabola $y^{2}=x$. Let $P$ be the midpoint of the line segment $MN$. The minimum distance from point $P$ to the $y$-axis is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{7}{4}",
"style": "rule"
} | {
"index": "5785",
"split": "train"
} |
math | [
{
"content": "A social club has $2k+1$ members, each of whom is fluent in the same $k$ languages. Any pair of members always talk to each other in only one language. Suppose that there were no three members such that they use only one language among them. Let $A$ be the number of three-member subsets such th... | {
"ground_truth": "\\binom{2k+1}{3} - k(2k+1)",
"style": "rule"
} | {
"index": "5786",
"split": "train"
} |
math | [
{
"content": "In the book \"Nine Chapters on the Mathematical Art,\" a tetrahedron with all four faces being right-angled triangles is called a \"biēnào.\" Given that tetrahedron $ABCD$ is a \"biēnào,\" $AB\\bot $ plane $BCD$, $BC\\bot CD$, and $AB=\\frac{1}{2}BC=\\frac{1}{3}CD$. If the volume of this tetrahedr... | {
"ground_truth": "14\\pi",
"style": "rule"
} | {
"index": "5787",
"split": "train"
} |
math | [
{
"content": "Compute $11^{-1} \\pmod{1105}$, where $1105=5\\cdot 13 \\cdot 17$. Express your answer as a residue from $0$ to $1104$, inclusive. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "201",
"style": "rule"
} | {
"index": "5788",
"split": "train"
} |
math | [
{
"content": "Given the quadratic function $f(x) = mx^2 - 2x - 3$, if the solution set of the inequality $f(x) < 0$ is $(-1, n)$.\n(1) Solve the inequality about $x$: $2x^2 - 4x + n > (m + 1)x - 1$;\n(2) Determine whether there exists a real number $a \\in (0, 1)$, such that the minimum value of the function $y... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "5789",
"split": "train"
} |
math | [
{
"content": "Among the rye seeds, there are $0.4\\%$ weed seeds. What is the probability of detecting 5 weed seeds when randomly selecting 5000 seeds? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.000055",
"style": "rule"
} | {
"index": "5790",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\cos x$, where $x\\in[0,2\\pi]$, there are two distinct zero points $x\\_1$, $x\\_2$, and the equation $f(x)=m$ has two distinct real roots $x\\_3$, $x\\_4$. If these four numbers are arranged in ascending order to form an arithmetic sequence, the value of the real number... | {
"ground_truth": "-\\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "5791",
"split": "train"
} |
math | [
{
"content": "A square array of dots with 10 rows and 10 columns is given. Each dot is coloured either blue or red. Whenever two dots of the same colour are adjacent in the same row or column, they are joined by a line segment of the same colour as the dots. If they are adjacent but of different colours, they a... | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "5792",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer with exactly 16 positive divisors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "210",
"style": "rule"
} | {
"index": "5793",
"split": "train"
} |
math | [
{
"content": "Given the inequality $\\ln (x+1)-(a+2)x\\leqslant b-2$ that always holds, find the minimum value of $\\frac {b-3}{a+2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1-e",
"style": "rule"
} | {
"index": "5794",
"split": "train"
} |
math | [
{
"content": "How many nonnegative integers can be written in the form\\[a_7\\cdot3^7+a_6\\cdot3^6+a_5\\cdot3^5+a_4\\cdot3^4+a_3\\cdot3^3+a_2\\cdot3^2+a_1\\cdot3^1+a_0\\cdot3^0,\\]where $a_i\\in \\{-1,0,1\\}$ for $0\\le i \\le 7$?\n$\\textbf{(A) } 512 \\qquad \\textbf{(B) } 729 \\qquad \\textbf{(C) } 1094 \\q... | {
"ground_truth": "3281",
"style": "rule"
} | {
"index": "5795",
"split": "train"
} |
math | [
{
"content": "The function $y = x^2 + 2x - 1$ attains its minimum value on the interval $[0, 3]$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "5796",
"split": "train"
} |
math | [
{
"content": "Let $a, b, c$ be not necessarily distinct integers between 1 and 2011, inclusive. Find the smallest possible value of $\\frac{a b+c}{a+b+c}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\frac{2}{3}$",
"style": "rule"
} | {
"index": "5797",
"split": "train"
} |
math | [
{
"content": "If the sum of the first $10$ terms and the sum of the first $100$ terms of a given arithmetic progression are $100$ and $10$, respectively, then the sum of first $110$ terms is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-110",
"style": "rule"
} | {
"index": "5798",
"split": "train"
} |
math | [
{
"content": "There are 10 steps, and one can take 1, 2, or 3 steps at a time to complete them in 7 moves. Calculate the total number of different ways to do this. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "77",
"style": "rule"
} | {
"index": "5799",
"split": "train"
} |
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