data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given the hyperbola $$E: \\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$$ with left and right vertices A and B, respectively. Let M be a point on the hyperbola such that ∆ABM is an isosceles triangle, and the area of its circumcircle is 4πa², then the eccentricity of the hyperbola ... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "4200",
"split": "train"
} |
math | [
{
"content": "Given that \\( z_{1} \\) and \\( z_{2} \\) are complex numbers with \\( \\left|z_{1}\\right|=3 \\), \\( \\left|z_{2}\\right|=5 \\), and \\( \\left|z_{1}+z_{2}\\right|=7 \\), find the value of \\( \\arg \\left(\\left(\\frac{z_{2}}{z_{1}}\\right)^{3}\\right) \\). Please output the final answer withi... | {
"ground_truth": "\\pi",
"style": "rule"
} | {
"index": "4201",
"split": "train"
} |
math | [
{
"content": "The Nanjing Youth Olympic Games are about to open, and a clothing store owner, Mr. Chen, spent 3600 yuan to purchase two types of sportswear, A and B, and sold them out quickly. When Mr. Chen went to purchase the same types and quantities of clothing again, he found that the purchase prices of typ... | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "4202",
"split": "train"
} |
math | [
{
"content": "Given that $60$ people are wearing sunglasses, $40$ people are wearing caps, and a randomly selected person wearing a cap has a $\\frac{2}{5}$ probability of also wearing sunglasses, find the probability that a person wearing sunglasses is also wearing a cap. Please output the final answer within ... | {
"ground_truth": "\\frac{4}{15}",
"style": "rule"
} | {
"index": "4203",
"split": "train"
} |
math | [
{
"content": "In a class, no two boys were born on the same day of the week and no two girls were born in the same month. If another child were to join the class, this would no longer be true. How many children are there in the class? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "4204",
"split": "train"
} |
math | [
{
"content": "Given the hyperbola $C$: $\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1$ $(a>0, b>0)$, with left and right foci $F_{1}$, $F_{2}$, and the origin $O$, a perpendicular line is drawn from $F_{1}$ to a asymptote of $C$, with the foot of the perpendicular being $D$, and $|DF_{2}|=2\\sqrt{2}|OD|$. Find the eccent... | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "4205",
"split": "train"
} |
math | [
{
"content": "The complete graph of $y=f(x)$, which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is $1$.)\n\nWhat is the sum of the $x$-coordinates of all points where $f(x) = x+1$? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "4206",
"split": "train"
} |
math | [
{
"content": "Let $g$ be a function taking the positive integers to the positive integers, such that:\n\n(i) $g$ is increasing (i.e., $g(n + 1) > g(n)$ for all positive integers $n$)\n(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n$,\n(iii) if $m \\neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) ... | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "4207",
"split": "train"
} |
math | [
{
"content": "For a real number $r$, the quadratics $x^{2}+(r-1)x+6$ and $x^{2}+(2r+1)x+22$ have a common real root. The sum of the possible values of $r$ can be expressed as $\\frac{a}{b}$, where $a, b$ are relatively prime positive integers. Compute $100a+b$. Please output the final answer within \\boxed{}.",... | {
"ground_truth": "405",
"style": "rule"
} | {
"index": "4208",
"split": "train"
} |
math | [
{
"content": "What is the value of the expression $2 + \\frac{1}{2 + \\frac{1}{2 + \\frac{1}{2}}}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{29}{12}",
"style": "rule"
} | {
"index": "4209",
"split": "train"
} |
math | [
{
"content": "Determine the least real number $M$ such that the inequality \\[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \\leq M(a^{2}+b^{2}+c^{2})^{2}\\] holds for all real numbers $a$, $b$ and $c$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "M=\\frac 9{16\\sqrt 2}",
"style": "rule"
} | {
"index": "4210",
"split": "train"
} |
math | [
{
"content": "Given that $\\cos(\\frac{\\pi}{6} - \\alpha) = \\frac{3}{5}$, find the value of $\\cos(\\frac{5\\pi}{6} + \\alpha)$:\n\nA) $\\frac{3}{5}$\nB) $-\\frac{3}{5}$\nC) $\\frac{4}{5}$\nD) $-\\frac{4}{5}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{5}",
"style": "rule"
} | {
"index": "4211",
"split": "train"
} |
math | [
{
"content": "6 books of different types are to be divided into 3 groups, one group containing 4 books and the other two groups containing 1 book each. Calculate the number of different ways this can be done. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "4212",
"split": "train"
} |
math | [
{
"content": "Let's call an integer \"extraordinary\" if it has exactly one even divisor other than 2. How many extraordinary numbers exist in the interval $[1 ; 75]$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "4213",
"split": "train"
} |
math | [
{
"content": "Find the number of ordered pairs of integers $(a, b)$ such that $a, b$ are divisors of 720 but $a b$ is not. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2520",
"style": "rule"
} | {
"index": "4214",
"split": "train"
} |
math | [
{
"content": "Given two moving points \\( A(x_1, y_1) \\) and \\( B(x_2, y_2) \\) on the parabola \\( y^2 = 6x \\), where \\( x_1 \\neq x_2 \\) and \\( x_1 + x_2 = 4 \\). The perpendicular bisector of segment \\( AB \\) intersects the \\( x \\)-axis at point \\( C \\). Find the maximum area of triangle \\( \\tr... | {
"ground_truth": "\\frac{14}{3} \\sqrt{7}",
"style": "rule"
} | {
"index": "4215",
"split": "train"
} |
math | [
{
"content": "Let $c_i$ denote the $i$ th composite integer so that $\\{c_i\\}=4,6,8,9,...$ Compute\n\\[\\prod_{i=1}^{\\infty} \\dfrac{c^{2}_{i}}{c_{i}^{2}-1}\\]\n(Hint: $\\textstyle\\sum^\\infty_{n=1} \\tfrac{1}{n^2}=\\tfrac{\\pi^2}{6}$ ) Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "\\frac{12}{\\pi^2}",
"style": "rule"
} | {
"index": "4216",
"split": "train"
} |
math | [
{
"content": "Find the product of all values of $t$ such that $t^2 = 36$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-36",
"style": "rule"
} | {
"index": "4217",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be a triangle with $AC = 28$ , $BC = 33$ , and $\\angle ABC = 2\\angle ACB$ . Compute the length of side $AB$ .\n\n*2016 CCA Math Bonanza #10* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "4218",
"split": "train"
} |
math | [
{
"content": "Let $A$ be the area of the triangle with sides of length $25, 25$, and $30$. Let $B$ be the area of the triangle with sides of length $25, 25,$ and $40$. What is the relationship between $A$ and $B$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "A=B",
"style": "rule"
} | {
"index": "4219",
"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": "4220",
"split": "train"
} |
math | [
{
"content": "The product of the positive integer divisors of a positive integer \\( n \\) is 1024, and \\( n \\) is a perfect power of a prime. Find \\( n \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1024",
"style": "rule"
} | {
"index": "4221",
"split": "train"
} |
math | [
{
"content": "In rectangle $ABCD,$ $P$ is a point on side $\\overline{BC}$ such that $BP = 9$ and $CP = 27.$ If $\\tan \\angle APD = 2,$ then find $AB.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "4222",
"split": "train"
} |
math | [
{
"content": "If the line that passes through the points $(2,7)$ and $(a, 3a)$ has a slope of 2, what is the value of $a$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "4223",
"split": "train"
} |
math | [
{
"content": "Let $C = (3, 5)$ be the midpoint of $AB,$ where $A = (1, 8)$ and $B = (x, y).$ What is the value of $xy$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "4224",
"split": "train"
} |
math | [
{
"content": "Points $P$ and $Q$ are midpoints of adjacent sides of a rectangle with length $l$ and width $w$. What fraction of the interior of the rectangle is shaded if a white triangle is formed by vertex $A$ (one of the corners of the rectangle) and points $P$ and $Q$?\n\n[asy]\nfilldraw((0,0)--(4,0)--(4,2)... | {
"ground_truth": "\\frac{7}{8}",
"style": "rule"
} | {
"index": "4225",
"split": "train"
} |
math | [
{
"content": "Find the number of 11-digit positive integers such that the digits from left to right are non-decreasing. (For example, 12345678999, 55555555555, 23345557889.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "75582",
"style": "rule"
} | {
"index": "4226",
"split": "train"
} |
math | [
{
"content": "Let $m$ be the smallest positive, three-digit integer congruent to 5 (mod 11). Let $n$ be the smallest positive, four-digit integer congruent to 5 (mod 11). What is $n-m$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "902",
"style": "rule"
} | {
"index": "4227",
"split": "train"
} |
math | [
{
"content": "One angle of a trapezoid is $60^{\\circ}$. Find the ratio of its bases if it is known that a circle can be inscribed in this trapezoid and a circle can be circumscribed around this trapezoid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1:3",
"style": "rule"
} | {
"index": "4228",
"split": "train"
} |
math | [
{
"content": "Let $A=\\{V, W, X, Y, Z, v, w, x, y, z\\}$. Find the number of subsets of the 2-configuration \\( \\{\\{V, W\\}, \\{W, X\\}, \\{X, Y\\}, \\{Y, Z\\}, \\{Z, V\\}, \\{v, x\\}, \\{v, y\\}, \\{w, y\\}, \\{w, z\\}, \\{x, z\\}, \\{V, v\\}, \\{W, w\\}, \\{X, x\\}, \\{Y, y\\}, \\{Z, z\\}\\} \\) that are co... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "4229",
"split": "train"
} |
math | [
{
"content": "Two regular tetrahedra are inscribed in a cube in such a way that four vertices of the cube serve as the vertices of one tetrahedron, and the remaining four vertices of the cube serve as the vertices of the other. What fraction of the volume of the cube is occupied by the volume of the intersectio... | {
"ground_truth": "1/6",
"style": "rule"
} | {
"index": "4230",
"split": "train"
} |
math | [
{
"content": "Let $n$ be a positive integer, and let $b_0, b_1, \\dots, b_n$ be a sequence of real numbers such that $b_0 = 54$, $b_1 = 81$, $b_n = 0$, and $$ b_{k+1} = b_{k-1} - \\frac{4.5}{b_k} $$ for $k = 1, 2, \\dots, n-1$. Find $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "972",
"style": "rule"
} | {
"index": "4231",
"split": "train"
} |
math | [
{
"content": "A piece of graph paper is folded once so that (0,2) is matched with (4,0), and (7,3) is matched with $(m,n)$. Find $m+n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6.8",
"style": "rule"
} | {
"index": "4232",
"split": "train"
} |
math | [
{
"content": "An icosidodecahedron is a convex polyhedron with 20 triangular faces and 12 pentagonal faces. How many vertices does it have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "4233",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overset{ .}{a}=(\\sin x, \\frac{1}{2})$, $\\overset{ .}{b}=( \\sqrt {3}\\cos x+\\sin x,-1)$, and the function $f(x)= \\overset{ .}{a}\\cdot \\overset{ .}{b}$:\n\n(1) Find the smallest positive period of the function $f(x)$;\n(2) Find the maximum and minimum values of $f(x)$ on the... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "4234",
"split": "train"
} |
math | [
{
"content": "There were no more than 70 mushrooms in the basket, among which 52% were white. If you throw out the three smallest mushrooms, the white mushrooms will become half of the total.\n\nHow many mushrooms are in the basket? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "4235",
"split": "train"
} |
math | [
{
"content": "A softball team played ten games, scoring $1,2,3,4,5,6,7,8,9$, and $10$ runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "4236",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $b=3$, $c=2\\sqrt{3}$, and $A=30^{\\circ}$, find the values of angles $B$, $C$, and side $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "4237",
"split": "train"
} |
math | [
{
"content": "Point $P$ is located inside a square $ABCD$ of side length $10$ . Let $O_1$ , $O_2$ , $O_3$ , $O_4$ be the circumcenters of $P AB$ , $P BC$ , $P CD$ , and $P DA$ , respectively. Given that $P A+P B +P C +P D = 23\\sqrt2$ and the area of $O_1O_2O_3O_4$ is $50$ , the second larges... | {
"ground_truth": "5001",
"style": "rule"
} | {
"index": "4238",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system, points whose x and y coordinates are both integers are called lattice points. How many lattice points \\((x, y)\\) satisfy the inequality \\((|x|-1)^{2}+(|y|-1)^{2}<2\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "4239",
"split": "train"
} |
math | [
{
"content": "Consider the set of all equations $x^3 + a_2x^2 + a_1x + a_0 = 0$, where $a_2$, $a_1$, $a_0$ are real constants and $|a_i| < 2$ for $i = 0,1,2$. Let $r$ be the largest positive real number which satisfies at least one of these equations. Then Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{5}{2} < r < 3",
"style": "rule"
} | {
"index": "4240",
"split": "train"
} |
math | [
{
"content": "Calculate the volume of the tetrahedron with vertices at points \\( A_{1}, A_{2}, A_{3}, A_{4} \\) and its height, dropped from vertex \\( A_{4} \\) onto the face \\( A_{1} A_{2} A_{3} \\).\n\n\\( A_{1}(7, 2, 4) \\) \n\\( A_{2}(7, -1, -2) \\) \n\\( A_{3}(3, 3, 1) \\) \n\\( A_{4}(-4, 2, 1) \\) P... | {
"ground_truth": "21.5",
"style": "rule"
} | {
"index": "4241",
"split": "train"
} |
math | [
{
"content": "$24$ lemons are required to make $32$ gallons of lemonade. How many lemons are needed to make $4$ gallons of lemonade? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "4242",
"split": "train"
} |
math | [
{
"content": "Simplify first and then evaluate:<br />(1) $4(m+1)^2 - (2m+5)(2m-5)$, where $m=-3$.<br />(2) Simplify: $\\frac{x^2-1}{x^2+2x} \\div \\frac{x-1}{x}$, where $x=2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "4243",
"split": "train"
} |
math | [
{
"content": "A person forgot the last digit of a phone number and dialed randomly. Calculate the probability of connecting to the call in no more than 3 attempts. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{3}{10}",
"style": "rule"
} | {
"index": "4244",
"split": "train"
} |
math | [
{
"content": "Triangles $ABC$ and $AEF$ are such that $B$ is the midpoint of $\\overline{EF}.$ Also, $AB = EF = 1,$ $BC = 6,$ $CA = \\sqrt{33},$ and\n\\[\\overrightarrow{AB} \\cdot \\overrightarrow{AE} + \\overrightarrow{AC} \\cdot \\overrightarrow{AF} = 2.\\]Find the cosine of the angle between vectors $\\ove... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "4245",
"split": "train"
} |
math | [
{
"content": "A regular triangular prism \\( A B C A_{1} B_{1} C_{1} \\) is inscribed in a sphere, where \\( A B C \\) is the base and \\( A A_{1}, B B_{1}, C C_{1} \\) are the lateral edges. The segment \\( C D \\) is the diameter of this sphere, and point \\( K \\) is the midpoint of the edge \\( A A_{1} \\).... | {
"ground_truth": "9\\sqrt{2}",
"style": "rule"
} | {
"index": "4246",
"split": "train"
} |
math | [
{
"content": "A rectangular swimming pool is 20 meters long, 15 meters wide, and 3 meters deep. Calculate its surface area. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "4247",
"split": "train"
} |
math | [
{
"content": "Line $l$ in the coordinate plane has equation $3x-5y+40=0$. This line is rotated $45^{\\circ}$ counterclockwise about the point $(20,20)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "4248",
"split": "train"
} |
math | [
{
"content": "What is the smallest five-digit positive integer congruent to $2 \\pmod{17}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10013",
"style": "rule"
} | {
"index": "4249",
"split": "train"
} |
math | [
{
"content": "Ewan writes out a sequence where he counts by 11s starting at 3. Which number will appear in Ewan's sequence? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "113",
"style": "rule"
} | {
"index": "4250",
"split": "train"
} |
math | [
{
"content": "In three-dimensional space, find the number of lattice points that have a distance of 4 from the origin. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "4251",
"split": "train"
} |
math | [
{
"content": "(1) Use the Euclidean algorithm to find the greatest common divisor (GCD) of 117 and 182, and verify it using the method of successive subtraction. \n(2) Use the Horner's method to calculate the value of the polynomial \\\\(f(x)=1-9x+8x^{2}-4x^{4}+5x^{5}+3x^{6}\\\\) at \\\\(x=-1\\\\). Please outp... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "4252",
"split": "train"
} |
math | [
{
"content": "The vertical coordinate of the intersection point of the new graph obtained by shifting the graph of the quadratic function $y=x^{2}+2x+1$ $2$ units to the left and then $3$ units up is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "4253",
"split": "train"
} |
math | [
{
"content": "Two congruent circles centered at points $A$ and $B$ each pass through the other circle's center. The line containing both $A$ and $B$ is extended to intersect the circles at points $C$ and $D$. The circles intersect at two points, one of which is $E$. What is the degree measure of $\\angle CED$? ... | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "4254",
"split": "train"
} |
math | [
{
"content": "A company plans to promote the same car in two locations, A and B. It is known that the relationship between the sales profit (unit: ten thousand yuan) and the sales volume (unit: cars) in the two locations is $y_1=5.06t-0.15t^2$ and $y_2=2t$, respectively, where $t$ is the sales volume ($t\\in\\m... | {
"ground_truth": "45.6",
"style": "rule"
} | {
"index": "4255",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that the area of $\\triangle ABC$ is $3\\sin A$, the perimeter is $4(\\sqrt{2}+1)$, and $\\sin B + \\sin C = \\sqrt{2}\\sin A$.\n\n1. Find the values of $a$ and $\\cos A$.\n2. Find the va... | {
"ground_truth": "\\frac{4\\sqrt{6} - 7}{18}",
"style": "rule"
} | {
"index": "4256",
"split": "train"
} |
math | [
{
"content": "Find all positive integers $A$ which can be represented in the form: \\[ A = \\left ( m - \\dfrac 1n \\right) \\left( n - \\dfrac 1p \\right) \\left( p - \\dfrac 1m \\right) \\]\r\nwhere $m\\geq n\\geq p \\geq 1$ are integer numbers. \r\n\r\n*Ioan Bogdan* Please output the final answer within ... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "4257",
"split": "train"
} |
math | [
{
"content": "An American traveling in Italy wishes to exchange American money (dollars) for Italian money (lire). If 3000 lire = 1.60, how much lire will the traveler receive in exchange for 1.00? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1875",
"style": "rule"
} | {
"index": "4258",
"split": "train"
} |
math | [
{
"content": "The line with equation $y = 3x + 5$ is translated 2 units to the right. What is the equation of the resulting line? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "y = 3x - 1",
"style": "rule"
} | {
"index": "4259",
"split": "train"
} |
math | [
{
"content": "Determine $\\sqrt[5]{102030201}$ without a calculator. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "101",
"style": "rule"
} | {
"index": "4260",
"split": "train"
} |
math | [
{
"content": "Marius is entering a wildlife photo contest, and wishes to arrange his nine snow leopards in a row. Each leopard has a unique collar color. The two shortest leopards have inferiority complexes and demand to be placed at the ends of the row. Moreover, the two tallest leopards, which have a superior... | {
"ground_truth": "2880",
"style": "rule"
} | {
"index": "4261",
"split": "train"
} |
math | [
{
"content": "Given $$\\frac {\\sin\\alpha-2\\cos\\alpha}{\\sin\\alpha +2\\cos\\alpha}=3,$$ calculate: \n(1) $$\\frac {\\sin\\alpha+2\\cos\\alpha}{5\\cos\\alpha -\\sin\\alpha};$$ \n(2) $({\\sin\\alpha+\\cos\\alpha})^{2}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{17}",
"style": "rule"
} | {
"index": "4262",
"split": "train"
} |
math | [
{
"content": "A $10$ by $4$ rectangle has the same center as a circle of radius $5$. Calculate the area of the region common to both the rectangle and the circle.\nA) $40 + 2\\pi$\nB) $36 + 4\\pi$\nC) $40 + 4\\pi$\nD) $44 + 4\\pi$\nE) $48 + 2\\pi$ Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "40 + 4\\pi",
"style": "rule"
} | {
"index": "4263",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, the sides opposite angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and it is given that $3a\\cos A = \\sqrt{6}(b\\cos C + c\\cos B)$. \n(1) Calculate the value of $\\tan 2A$. \n(2) If $\\sin\\left(\\frac{\\pi}{2} + B\\right) = \\frac{1}{3}$ and $c = 2\\sqrt{2}$, find the ... | {
"ground_truth": "\\frac{8}{5}\\sqrt{2}",
"style": "rule"
} | {
"index": "4264",
"split": "train"
} |
math | [
{
"content": "Out of 8 shots, 3 hit the target. The total number of ways in which exactly 2 hits are consecutive is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "4265",
"split": "train"
} |
math | [
{
"content": "A square is divided into 2016 triangles, with no vertex of any triangle lying on the sides or inside any other triangle. The sides of the square are sides of some of the triangles in the division. How many total points, which are the vertices of the triangles, are located inside the square? Please... | {
"ground_truth": "1007",
"style": "rule"
} | {
"index": "4266",
"split": "train"
} |
math | [
{
"content": "The side of a triangle are 2, $\\sqrt{6},$ and $1 + \\sqrt{3}.$ Enter the angles of the triangle in degrees, separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "75^\\circ",
"style": "rule"
} | {
"index": "4267",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, where $\\angle C=90^{\\circ}$, $\\angle B=30^{\\circ}$, and $AC=1$, let $M$ be the midpoint of $AB$. $\\triangle ACM$ is folded along $CM$ such that the distance between points $A$ and $B$ is $\\sqrt{2}$. What is the distance from point $A$ to plane $BCM$? Please output the fi... | {
"ground_truth": "\\frac{\\sqrt{6}}{3}",
"style": "rule"
} | {
"index": "4268",
"split": "train"
} |
math | [
{
"content": "Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\\triangle{ABC}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$3\\sqrt{3}$",
"style": "rule"
} | {
"index": "4269",
"split": "train"
} |
math | [
{
"content": "Calculate:\n\n1. $(2\\frac{3}{5})^{0} + 2^{-2}\\cdot(2\\frac{1}{4})^{-\\frac{1}{2}} + (\\frac{25}{36})^{0.5} + \\sqrt{(-2)^{2}}$;\n2. $\\frac{1}{2}\\log\\frac{32}{49} - \\frac{4}{3}\\log\\sqrt{8} + \\log\\sqrt{245}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "4270",
"split": "train"
} |
math | [
{
"content": "$ABCDE$ is a regular pentagon. $AP$, $AQ$ and $AR$ are the perpendiculars dropped from $A$ onto $CD$, $CB$ extended and $DE$ extended, respectively. Let $O$ be the center of the pentagon. If $OP = 1$, then find $AO + AQ + AR$.\n\n[asy]\n\nunitsize(2 cm);\n\npair A, B, C, D, E, O, P, Q, R;\n\nA ... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "4271",
"split": "train"
} |
math | [
{
"content": "A vertex-induced subgraph is a subset of the vertices of a graph together with any edges whose endpoints are both in this subset. An undirected graph contains 10 nodes and $m$ edges, with no loops or multiple edges. What is the minimum possible value of $m$ such that this graph must contain a none... | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "4272",
"split": "train"
} |
math | [
{
"content": "Assume that $y_1, y_2, ... , y_8$ are real numbers such that\n\\[\n\\begin{aligned}\ny_1 + 4y_2 + 9y_3 + 16y_4 + 25y_5 + 36y_6 + 49y_7 + 64y_8 &= 3, \\\\\n4y_1 + 9y_2 + 16y_3 + 25y_4 + 36y_5 + 49y_6 + 64y_7 + 81y_8 &= 15, \\\\\n9y_1 + 16y_2 + 25y_3 + 36y_4 + 49y_5 + 64y_6 + 81y_7 + 100y_8 &= 140.\... | {
"ground_truth": "472",
"style": "rule"
} | {
"index": "4273",
"split": "train"
} |
math | [
{
"content": "The sequences of positive integers $1,a_2, a_3,...$ and $1,b_2, b_3,...$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$. Please output the final answer withi... | {
"ground_truth": "262",
"style": "rule"
} | {
"index": "4274",
"split": "train"
} |
math | [
{
"content": "Given \\(x\\) and \\(y\\) are real numbers, and:\n\\[\nz_1 = x + (y + 2)i,\n\\]\n\\[\nz_2 = (x - 2) + yi,\n\\]\nwith the condition \\(\\left|z_1\\right| + \\left|z_2\\right| = 4\\),\n\nFind the maximum value of \\(|x + y|\\). Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "4275",
"split": "train"
} |
math | [
{
"content": "Right triangle $PQR$ has one leg of length 9 cm, one leg of length 12 cm and a right angle at $P$. A square has one side on the hypotenuse of triangle $PQR$ and a vertex on each of the two legs of triangle $PQR$. What is the length of one side of the square, in cm? Express your answer as a common ... | {
"ground_truth": "\\frac{45}{8}",
"style": "rule"
} | {
"index": "4276",
"split": "train"
} |
math | [
{
"content": "Tyler rolls two $ 4025 $ sided fair dice with sides numbered $ 1, \\dots , 4025 $ . Given that the number on the first die is greater than or equal to the number on the second die, what is the probability that the number on the first die is less than or equal to $ 2012 $ ? Please output the fina... | {
"ground_truth": "1006/4025",
"style": "rule"
} | {
"index": "4277",
"split": "train"
} |
math | [
{
"content": "How many integers $n$ satisfy $(n+2)(n-8) \\le 0$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "4278",
"split": "train"
} |
math | [
{
"content": "Calculate the smallest root \\(x_0\\) of the equation\n\n$$\nx^{2}-\\sqrt{\\lg x +100}=0\n$$\n\n(with a relative error of no more than \\(10^{-390} \\%\\)). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10^{-100}",
"style": "rule"
} | {
"index": "4279",
"split": "train"
} |
math | [
{
"content": "Given a point $P(x,y)$ moving on the ellipse $\\frac{x^{2}}{4} + \\frac{y^{2}}{3} = 1$, let $d = \\sqrt{x^{2} + y^{2} + 4y + 4} - \\frac{x}{2}$. Find the minimum value of $d$.\nA) $\\sqrt{5} - 2$\nB) $2\\sqrt{2} - 1$\nC) $\\sqrt{5} - 1$\nD) $\\sqrt{6} - 1$ Please output the final answer within \\b... | {
"ground_truth": "2\\sqrt{2} - 1",
"style": "rule"
} | {
"index": "4280",
"split": "train"
} |
math | [
{
"content": "Given that the quadratic equation \\( (5a + 2b)x^2 + ax + b = 0 \\) has a unique solution for \\( x \\), find the value of \\( x \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "4281",
"split": "train"
} |
math | [
{
"content": "In the middle of a vast prairie, a firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at $50$ miles per hour along the highways and at $14$ miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within si... | {
"ground_truth": "731",
"style": "rule"
} | {
"index": "4282",
"split": "train"
} |
math | [
{
"content": "Every 1 kilogram of soybeans can produce 0.8 kilograms of soybean oil. With 20 kilograms of soybeans, you can produce \\_\\_\\_\\_\\_\\_ kilograms of soybean oil. To obtain 20 kilograms of soybean oil, you need \\_\\_\\_\\_\\_\\_ kilograms of soybeans. Please output the final answer within \\boxed... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "4283",
"split": "train"
} |
math | [
{
"content": "A certain high school encourages students to spontaneously organize various sports competitions. Two students, A and B, play table tennis matches in their spare time. The competition adopts a \"best of seven games\" format (i.e., the first player to win four games wins the match and the competitio... | {
"ground_truth": "\\frac{19}{9}",
"style": "rule"
} | {
"index": "4284",
"split": "train"
} |
math | [
{
"content": "The arithmetic mean of 12 scores is 82. When the highest and lowest scores are removed, the new mean becomes 84. If the highest of the 12 scores is 98, what is the lowest score? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "46",
"style": "rule"
} | {
"index": "4285",
"split": "train"
} |
math | [
{
"content": "In three-digit numbers, if the digit in the tens place is smaller than the digits in both the hundreds and ones places, the number is called a \"concave number,\" such as 304, 968, etc. How many distinct three-digit concave numbers are there without any repeated digits? Please output the final ans... | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "4286",
"split": "train"
} |
math | [
{
"content": "Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is an ace? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{52}",
"style": "rule"
} | {
"index": "4287",
"split": "train"
} |
math | [
{
"content": "Compute the limit of the function:\n\n$$\n\\lim _{x \\rightarrow \\pi} \\frac{\\ln (2+\\cos x)}{\\left(3^{\\sin x}-1\\right)^{2}}\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2 \\ln^2 3}",
"style": "rule"
} | {
"index": "4288",
"split": "train"
} |
math | [
{
"content": "The unit's digit (one's digit) of the product of any six consecutive positive whole numbers is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "4289",
"split": "train"
} |
math | [
{
"content": "Given a family of sets \\(\\{A_{1}, A_{2}, \\ldots, A_{n}\\}\\) that satisfies the following conditions:\n(1) Each set \\(A_{i}\\) contains exactly 30 elements;\n(2) For any \\(1 \\leq i < j \\leq n\\), the intersection \\(A_{i} \\cap A_{j}\\) contains exactly 1 element;\n(3) The intersection \\(A... | {
"ground_truth": "871",
"style": "rule"
} | {
"index": "4290",
"split": "train"
} |
math | [
{
"content": "$k, a_2, a_3$ and $k, b_2, b_3$ are both nonconstant geometric sequences with different common ratios. We have $$a_3-b_3=3(a_2-b_2).$$Find the sum of the common ratios of the two sequences. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "4291",
"split": "train"
} |
math | [
{
"content": "Three fair six-sided dice are rolled (meaning all outcomes are equally likely). What is the probability that the numbers on the top faces form an arithmetic sequence with a common difference of 1? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\frac{1}{9}$",
"style": "rule"
} | {
"index": "4292",
"split": "train"
} |
math | [
{
"content": "The time right now is 9 o'clock. On a 12-hour clock, what time will it be 2004 hours from now? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "4293",
"split": "train"
} |
math | [
{
"content": "Evaluate the expression $\\frac{3^2 + 3^0 + 3^{-1} + 3^{-2}}{3^{-1} + 3^{-2} + 3^{-3} + 3^{-4}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3807}{180}",
"style": "rule"
} | {
"index": "4294",
"split": "train"
} |
math | [
{
"content": "Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between \\(12^2\\) and \\(13^2\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "165",
"style": "rule"
} | {
"index": "4295",
"split": "train"
} |
math | [
{
"content": "Let \\(A\\) and \\(B\\) be two moving points on the ellipse \\(x^2 + 3y^2 = 1\\), and \\(OA \\perp OB\\) (where \\(O\\) is the origin). Find the product of the maximum and minimum values of \\( |AB| \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2 \\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "4296",
"split": "train"
} |
math | [
{
"content": "What is the maximum number of checkers that can be placed on a $6 \\times 6$ board such that no three checkers (specifically, the centers of the cells they occupy) lie on the same straight line (at any angle)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "4297",
"split": "train"
} |
math | [
{
"content": "A room is 24 feet long and 14 feet wide. Find the ratio of the length to its perimeter and the ratio of the width to its perimeter. Express each ratio in the form $a:b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7:38",
"style": "rule"
} | {
"index": "4298",
"split": "train"
} |
math | [
{
"content": "Five consecutive two-digit positive integers, each less than 40, are not prime. What is the largest of these five integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "4299",
"split": "train"
} |
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