data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Lucia needs to save 35 files onto disks, each with a capacity of 1.6 MB. 5 of the files are 0.9 MB each, 10 of the files are 0.8 MB each, and the rest are 0.5 MB each. Calculate the smallest number of disks needed to store all 35 files. Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "39000",
"split": "train"
} |
math | [
{
"content": "Given the parametric equation of curve $C_1$ as $$\\begin{cases} x=2t-1 \\\\ y=-4t-2 \\end{cases}$$ (where $t$ is the parameter), and establishing a polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C_2$ is $$\... | {
"ground_truth": "\\frac {3 \\sqrt {5}}{10}",
"style": "rule"
} | {
"index": "39001",
"split": "train"
} |
math | [
{
"content": "The graph of the function $y= \\sqrt {3}\\cos x+\\sin x, (x\\in \\mathbb{R})$ is translated to the right by $\\theta$ ($\\theta>0$) units. Determine the minimum value of $\\theta$ such that the resulting graph is symmetric about the y-axis. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "\\frac{5\\pi}{6}",
"style": "rule"
} | {
"index": "39002",
"split": "train"
} |
math | [
{
"content": "Let $N$ be the greatest integer multiple of $36$ all of whose digits are even and no two of whose digits are the same. Find the remainder when $N$ is divided by $1000$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "640",
"style": "rule"
} | {
"index": "39003",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{v}$ be a vector such that\n\\[\\left\\| \\mathbf{v} + \\begin{pmatrix} 4 \\\\ 2 \\end{pmatrix} \\right\\| = 10.\\]\nFind the smallest possible value of $\\|\\mathbf{v}\\|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10 - 2\\sqrt{5}",
"style": "rule"
} | {
"index": "39004",
"split": "train"
} |
math | [
{
"content": "Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations: \\begin{align*} abc&=70,\\\\ cde&=71,\\\\ efg&=72. \\end{align*} Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "39005",
"split": "train"
} |
math | [
{
"content": "What is the sum of the different prime factors of $150280?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "39006",
"split": "train"
} |
math | [
{
"content": "The volume of a prism is equal to the area of its base times its depth. If the prism has identical bases with area $400 \\mathrm{~cm}^{2}$ and depth 8 cm, what is its volume? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3200 \\mathrm{~cm}^{3}",
"style": "rule"
} | {
"index": "39007",
"split": "train"
} |
math | [
{
"content": "For real numbers \\(x\\), \\(y\\), and \\(z\\), consider the matrix\n\\[\n\\begin{pmatrix} x+y & x & y \\\\ x & y+z & y \\\\ y & x & x+z \\end{pmatrix}\n\\]\nDetermine whether this matrix is invertible. If not, list all possible values of\n\\[\n\\frac{x}{y + z} + \\frac{y}{x + z} + \\frac{z}{x + y... | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "39008",
"split": "train"
} |
math | [
{
"content": "In a convex polygon, all its diagonals are drawn. These diagonals divide the polygon into several smaller polygons. What is the maximum number of sides that a polygon in the subdivision can have if the original polygon has:\n\na) 13 sides;\n\nb) 1950 sides? Please output the final answer within \\... | {
"ground_truth": "1950",
"style": "rule"
} | {
"index": "39009",
"split": "train"
} |
math | [
{
"content": "Xinyi is a beautiful city with abundant tourism resources. The Malingshan Scenic Area is a $4A$-rated tourist attraction in our city. The scenic area has developed a souvenir that costs $30$ yuan to produce and is sold. The selling price is set not to be lower than the cost and not higher than $54... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "39010",
"split": "train"
} |
math | [
{
"content": "There is a certain regularity in the operation between rational numbers and irrational numbers. For example, if $a$ and $b$ are rational numbers, and $a(\\pi +3)+b=0$, then $a=0$, $b=0$. Given that $m$ and $n$ are rational numbers:<br/>$(1)$ If $(m-3)×\\sqrt{6}+n-3=0$, then the square root of $mn$... | {
"ground_truth": "\\frac{25}{9}",
"style": "rule"
} | {
"index": "39011",
"split": "train"
} |
math | [
{
"content": "A rectangle has a perimeter of 80 cm and each side has an integer length. How many non-congruent rectangles meet these criteria? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "39012",
"split": "train"
} |
math | [
{
"content": "Let $a_0 = 2,$ $b_0 = 3,$ and\n\\[a_{n + 1} = \\frac{a_n^2}{b_n} \\quad \\text{and} \\quad b_{n + 1} = \\frac{b_n^2}{a_n}\\]for all $n \\ge 0.$ Then $b_8 = \\frac{3^m}{2^n}$ for some integers $m$ and $n.$ Enter the ordered pair $(m,n).$ Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "(3281,3280)",
"style": "rule"
} | {
"index": "39013",
"split": "train"
} |
math | [
{
"content": "Line $\\ell$ passes through $A$ and into the interior of the equilateral triangle $ABC$ . $D$ and $E$ are the orthogonal projections of $B$ and $C$ onto $\\ell$ respectively. If $DE=1$ and $2BD=CE$ , then the area of $ABC$ can be expressed as $m\\sqrt n$ , where $m$ and $... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "39014",
"split": "train"
} |
math | [
{
"content": "Find the largest positive integer \\( n \\) for which we can find a set of distinct positive integers such that each integer is at most 2002 and if \\( a \\) and \\( b \\) are in the set, then \\( a^2 \\) and \\( ab \\) are not in the set. Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "1958",
"style": "rule"
} | {
"index": "39015",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=e^{x}-ax-1$ ($a$ is a real number), and $g(x)=\\ln x-x$.\n\n(I) Discuss the monotonic intervals of the function $f(x)$.\n\n(II) Find the extreme values of the function $g(x)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "39016",
"split": "train"
} |
math | [
{
"content": "A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the letter that is $1$ place to its right in the alphabet (asumming that the letter $A$ is one place to the right of the letter $Z$). The second time this same letter appears in the gi... | {
"ground_truth": "s",
"style": "rule"
} | {
"index": "39017",
"split": "train"
} |
math | [
{
"content": "From the natural numbers 1 to 2008, the maximum number of numbers that can be selected such that the sum of any two selected numbers is not divisible by 3 is ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "671",
"style": "rule"
} | {
"index": "39018",
"split": "train"
} |
math | [
{
"content": "Four complex numbers lie at the vertices of a square in the complex plane. Three of the numbers are $1+2i, -2+i$, and $-1-2i$. The fourth number is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$2-i$",
"style": "rule"
} | {
"index": "39019",
"split": "train"
} |
math | [
{
"content": "Let $a$, $b$, and $c$ be the three roots of the equation $$\n5x^3 + 505x + 1010 = 0.\n$$ Find $(a + b)^3 + (b + c)^3 + (c + a)^3.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "606",
"style": "rule"
} | {
"index": "39020",
"split": "train"
} |
math | [
{
"content": "Two sides of a triangle measure 4 cm and 9 cm. What is the greatest integer number of centimeters that could be the third side of this triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "39021",
"split": "train"
} |
math | [
{
"content": "Calculate the value of $\\frac{17!}{7!10!}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "408408",
"style": "rule"
} | {
"index": "39022",
"split": "train"
} |
math | [
{
"content": "Simplify $(2x)^4 + (3x)(x^3)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19x^4",
"style": "rule"
} | {
"index": "39023",
"split": "train"
} |
math | [
{
"content": "Find the volume of the region given by the inequality\n\\[\n|x + y + z| + |x + y - z| + |x - y + z| + |-x + y + z| \\le 6.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "39024",
"split": "train"
} |
math | [
{
"content": "Calculate the sum of the first 4 terms of the geometric sequence $\\{a_n\\}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "39025",
"split": "train"
} |
math | [
{
"content": "Alicia earns 20 dollars per hour, of which $1.45\\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "39026",
"split": "train"
} |
math | [
{
"content": "At $2:15$ o'clock, the hour and minute hands of a clock form an angle of: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22\\frac {1}{2}^{\\circ}",
"style": "rule"
} | {
"index": "39027",
"split": "train"
} |
math | [
{
"content": "If the integer solutions to the system of inequalities\n\\[\n\\begin{cases}\n9x - a \\geq 0, \\\\\n8x - b < 0\n\\end{cases}\n\\]\nare only 1, 2, and 3, find the number of ordered pairs \\((a, b)\\) of integers that satisfy this system. Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "39028",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = x^2 + ax + b$ ($a, b \\in \\mathbb{R}$).\n\n(Ⅰ) Given $x \\in [0, 1]$,\n\n(i) If $a = b = 1$, find the range of the function $f(x)$;\n\n(ii) If the range of the function $f(x)$ is $[0, 1]$, find the values of $a$ and $b$;\n\n(Ⅱ) When $|x| \\geq 2$, it always holds that $... | {
"ground_truth": "74",
"style": "rule"
} | {
"index": "39029",
"split": "train"
} |
math | [
{
"content": "Let \\( n \\) be a fixed integer with \\( n \\geq 2 \\). Determine the minimal constant \\( c \\) such that the inequality\n\n\\[ \\sum_{1 \\leq i < j \\leq n} x_i x_j (x_i^2 + x_j^2) \\leq c \\left( \\sum_{i=1}^{n} x_i \\right)^4 \\]\n\nholds for all non-negative real numbers \\( x_1, x_2, \\ldot... | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "39030",
"split": "train"
} |
math | [
{
"content": "In a triangle $ABC$ , let $I$ denote the incenter. Let the lines $AI,BI$ and $CI$ intersect the incircle at $P,Q$ and $R$ , respectively. If $\\angle BAC = 40^o$ , what is the value of $\\angle QPR$ in degrees ? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "39031",
"split": "train"
} |
math | [
{
"content": "A sign painter paints individual numerals for a row of 100 houses. The houses are numbered with consecutive integers from 1 to 100. How many 9s are painted, and what is the total count of all digits used in painting? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "192",
"style": "rule"
} | {
"index": "39032",
"split": "train"
} |
math | [
{
"content": "One endpoint of a line segment is $(4,3)$ and its midpoint is $(2,9)$. What is the sum of the coordinates of the other endpoint? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "39033",
"split": "train"
} |
math | [
{
"content": "Given the numbers $12534, 25341, 53412, 34125$, calculate their sum. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "125412",
"style": "rule"
} | {
"index": "39034",
"split": "train"
} |
math | [
{
"content": "Let $\\triangle ABC$ be a right triangle at $A$ with circumcircle $\\omega$. The tangents to $\\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppose $BT = CT = 25$, $BC = 34$, and $TX^2 + TY^2 + XY^2 = 1975$. Find $XY^2$.... | {
"ground_truth": "987.5",
"style": "rule"
} | {
"index": "39035",
"split": "train"
} |
math | [
{
"content": "Given \\( AB \\) as the diameter of the smallest radius circle centered at \\( C(0,1) \\) that intersects the graph of \\( y = \\frac{1}{|x|-1} \\), where \\( O \\) is the origin. Find the value of \\( \\overrightarrow{OA} \\cdot \\overrightarrow{OB} \\). Please output the final answer within \\bo... | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "39036",
"split": "train"
} |
math | [
{
"content": "If $g(2x - 5) = 3x + 9$ for all real $x$, then what is $g(1)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "39037",
"split": "train"
} |
math | [
{
"content": "How many non-empty subsets $T$ of $\\{1,2,3,\\ldots,17\\}$ have the following two properties?\n$(1)$ No two consecutive integers belong to $T$.\n$(2)$ If $T$ contains $k$ elements, then $T$ contains no number less than $k+1$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "594",
"style": "rule"
} | {
"index": "39038",
"split": "train"
} |
math | [
{
"content": "In the adjoining figure $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, then the area of $CMN$ in square inches is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{3}-3",
"style": "rule"
} | {
"index": "39039",
"split": "train"
} |
math | [
{
"content": "Given that $F_1$ and $F_2$ are the left and right foci of a hyperbola $E$, and point $P$ is on $E$, with $\\angle F_1 P F_2 = \\frac{\\pi}{6}$ and $(\\overrightarrow{F_2 F_1} + \\overrightarrow{F_2 P}) \\cdot \\overrightarrow{F_1 P} = 0$, determine the eccentricity $e$ of hyperbola $E$. Please out... | {
"ground_truth": "\\frac{\\sqrt{3} + 1}{2}",
"style": "rule"
} | {
"index": "39040",
"split": "train"
} |
math | [
{
"content": "Let point $P$ be a point on the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$. Let $F_1$ and $F_2$ respectively be the left and right foci of the ellipse, and let $I$ be the incenter of $\\triangle PF_1F_2$. If $S_{\\triangle IPF_1} + S_{\\triangle IPF_2} = 2S_{\\triangle IF_1F_2}$... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "39041",
"split": "train"
} |
math | [
{
"content": "The Smith family has 4 sons and 3 daughters. In how many ways can they be seated in a row of 7 chairs such that at least 2 boys are next to each other? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4896",
"style": "rule"
} | {
"index": "39042",
"split": "train"
} |
math | [
{
"content": "Find the minimum value of the sum of the distances from a point in space to the vertices of a regular tetrahedron with edge length 1. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{6}",
"style": "rule"
} | {
"index": "39043",
"split": "train"
} |
math | [
{
"content": "The bending resistance of a beam with a rectangular cross-section is proportional to the product of its width and the square of its height. What should be the width of the beam section, cut from a round log with a diameter of \\( 15 \\sqrt{3} \\), to maximize its bending resistance? Please output ... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "39044",
"split": "train"
} |
math | [
{
"content": "Four consecutive positive prime numbers have a sum that is divisible by three. What is the smallest possible value of this sum? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "39045",
"split": "train"
} |
math | [
{
"content": "In the diagram, square $ABCD$ has sides of length $4,$ and $\\triangle ABE$ is equilateral. Line segments $BE$ and $AC$ intersect at $P.$ Point $Q$ is on $BC$ so that $PQ$ is perpendicular to $BC$ and $PQ=x.$ [asy]\npair A, B, C, D, E, P, Q;\nA=(0,0);\nB=(4,0);\nC=(4,-4);\nD=(0,-4);\nE=(2,-3.464)... | {
"ground_truth": "105^\\circ",
"style": "rule"
} | {
"index": "39046",
"split": "train"
} |
math | [
{
"content": "How many positive integer divisors of $201^9$ are perfect squares or perfect cubes (or both)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "39047",
"split": "train"
} |
math | [
{
"content": "Fill the letters a, b, and c into a 3×3 grid such that each letter does not repeat in any row or column. The number of different filling methods is ___. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "39048",
"split": "train"
} |
math | [
{
"content": "Suppose $A_1,A_2,\\cdots ,A_n \\subseteq \\left \\{ 1,2,\\cdots ,2018 \\right \\}$ and $\\left | A_i \\right |=2, i=1,2,\\cdots ,n$, satisfying that $$A_i + A_j, \\; 1 \\le i \\le j \\le n ,$$ are distinct from each other. $A + B = \\left \\{ a+b|a\\in A,\\,b\\in B \\right \\}$. Determine the maxi... | {
"ground_truth": "4033",
"style": "rule"
} | {
"index": "39049",
"split": "train"
} |
math | [
{
"content": "What is the smallest number of people that can be broken up both into 15 groups of equal membership and into 48 groups of equal membership? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "39050",
"split": "train"
} |
math | [
{
"content": "Given in triangle $\\triangle ABC$ the internal angles are $A$, $B$, and $C$, and the centroid is $G$. If $2\\sin A \\cdot \\overrightarrow{GA} + \\sqrt{3}\\sin B \\cdot \\overrightarrow{GB} + 3\\sin C \\cdot \\overrightarrow{GC} = \\overrightarrow{0}$, then $\\cos B = \\_\\_\\_\\_\\_\\_$. Please ... | {
"ground_truth": "\\frac{1}{12}",
"style": "rule"
} | {
"index": "39051",
"split": "train"
} |
math | [
{
"content": "The Smith family has four girls aged 5, 5, 5, and 12, and two boys aged 13 and 16. What is the mean (average) of the ages of the children? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9.33",
"style": "rule"
} | {
"index": "39052",
"split": "train"
} |
math | [
{
"content": "Palindromic primes are two-digit prime numbers such that the number formed when the digits are reversed is also prime. What is the sum of all palindromic primes less than 50? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "109",
"style": "rule"
} | {
"index": "39053",
"split": "train"
} |
math | [
{
"content": "Cameron writes down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. How many integers are in Cameron's list? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "871",
"style": "rule"
} | {
"index": "39054",
"split": "train"
} |
math | [
{
"content": "The sequence $\\left(z_{n}\\right)$ of complex numbers satisfies the following properties: $z_{1}$ and $z_{2}$ are not real. $z_{n+2}=z_{n+1}^{2} z_{n}$ for all integers $n \\geq 1$. $\\frac{z_{n+3}}{z_{n}^{2}}$ is real for all integers $n \\geq 1$. $\\left|\\frac{z_{3}}{z_{4}}\\right|=\\left|\\fr... | {
"ground_truth": "65536",
"style": "rule"
} | {
"index": "39055",
"split": "train"
} |
math | [
{
"content": "Given in the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\\begin{cases} x=3+5\\cos \\alpha \\\\ y=4+5\\sin \\alpha \\end{cases}$, ($\\alpha$ is the parameter), points $A$ and $B$ are on curve $C$. With the origin $O$ as the pole and the positive half-axis of $x$ as ... | {
"ground_truth": "\\frac{25\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "39056",
"split": "train"
} |
math | [
{
"content": "Given four points \\( A, B, C, \\) and \\( D \\) in space that are not coplanar, each pair of points is connected by an edge with a probability of \\( \\frac{1}{2} \\). The events of whether any two pairs of points are connected are mutually independent. What is the probability that \\( A \\) and ... | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "39057",
"split": "train"
} |
math | [
{
"content": "Points $D$ and $E$ are chosen on side $BC$ of triangle $ABC$ such that $E$ is between $B$ and $D$ and $BE=1$ , $ED=DC=3$ . If $\\angle BAD=\\angle EAC=90^\\circ$ , the area of $ABC$ can be expressed as $\\tfrac{p\\sqrt q}r$ , where $p$ and $r$ are relatively prime positive... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "39058",
"split": "train"
} |
math | [
{
"content": "A speaker talked for sixty minutes to a full auditorium. Twenty percent of the audience heard the entire talk and ten percent slept through the entire talk. Half of the remainder heard one third of the talk and the other half heard two thirds of the talk. What was the average number of minutes of ... | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "39059",
"split": "train"
} |
math | [
{
"content": "Given the inequality $ax^{2}+bx+c \\gt 0$ with the solution set $\\{x\\left|\\right.1 \\lt x \\lt 2\\}$, find the solution set of the inequality $cx^{2}+bx+a \\gt 0$ in terms of $x$. When studying the above problem, Xiaoming and Xiaoning respectively came up with the following Solution 1 and Solut... | {
"ground_truth": "-490",
"style": "rule"
} | {
"index": "39060",
"split": "train"
} |
math | [
{
"content": "If the inequality system $\\left\\{\\begin{array}{l}{x-m＞0}\\\\{x-2＜0}\\end{array}\\right.$ has only one positive integer solution, then write down a value of $m$ that satisfies the condition: ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.5",
"style": "rule"
} | {
"index": "39061",
"split": "train"
} |
math | [
{
"content": "The polynomial $x^3 - ax^2 + bx - 2310$ has three positive integer roots. What is the smallest possible value of $a$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "88",
"style": "rule"
} | {
"index": "39062",
"split": "train"
} |
math | [
{
"content": "Let $f(x) = \\frac{3}{9^x + 3}.$ Find\n\\[f \\left( \\frac{1}{1001} \\right) + f \\left( \\frac{2}{1001} \\right) + f \\left( \\frac{3}{1001} \\right) + \\dots + f \\left( \\frac{1000}{1001} \\right).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "500",
"style": "rule"
} | {
"index": "39063",
"split": "train"
} |
math | [
{
"content": "Define a function $f(x)$ on $\\mathbb{R}$ that satisfies $f(x+6)=f(x)$. When $x \\in [-3,-1)$, $f(x)=-(x+2)^{2}$, and when $x \\in [-1,3)$, $f(x)=x$. Find the value of $f(1)+f(2)+f(3)+\\ldots+f(2016)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "336",
"style": "rule"
} | {
"index": "39064",
"split": "train"
} |
math | [
{
"content": "A real number $a$ is chosen randomly and uniformly from the interval $[-10, 15]$. Find the probability that the roots of the polynomial\n\\[ x^4 + 3ax^3 + (3a - 3)x^2 + (-5a + 4)x - 3 \\]\nare all real. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{23}{25}",
"style": "rule"
} | {
"index": "39065",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 180 = 0$ has integral solutions? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "39066",
"split": "train"
} |
math | [
{
"content": "Given an ellipse M: $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}=1$$ (a>0, b>0) with two vertices A(-a, 0) and B(a, 0). Point P is a point on the ellipse distinct from A and B. The slopes of lines PA and PB are k₁ and k₂, respectively, and $$k_{1}k_{2}=- \\frac {1}{2}$$.\n(1) Find the eccentrici... | {
"ground_truth": "\\frac { \\sqrt {2}}{2}",
"style": "rule"
} | {
"index": "39067",
"split": "train"
} |
math | [
{
"content": "There are $2^{10} = 1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "548",
"style": "rule"
} | {
"index": "39068",
"split": "train"
} |
math | [
{
"content": "The KML airline operates shuttle services between some cities such that from any city, you cannot directly reach more than three other cities. However, with at most one transfer, you can travel from any city to any other city. What is the maximum number of cities between which the planes operate? ... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "39069",
"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}(1, 1, 2)$\n\n$A_{2}(-1, 1, 3)$\n\n$A_{3}(2, -2, 4)$\n\n$A_{4}(-1, 0, -2)$ Please output the final answer within \\box... | {
"ground_truth": "\\sqrt{\\frac{35}{2}}",
"style": "rule"
} | {
"index": "39070",
"split": "train"
} |
math | [
{
"content": "An ellipse whose axes are parallel to the coordinate axes is tangent to the $x$-axis at $(6, 0)$ and tangent to the $y$-axis at $(0, 2)$. Find the distance between the foci of the ellipse. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{2}",
"style": "rule"
} | {
"index": "39071",
"split": "train"
} |
math | [
{
"content": "Three of the four vertices of a square are $(2, 8)$, $(13, 8)$, and $(13, -3)$. What is the area of the intersection of this square region and the region inside the graph of the equation $(x - 2)^2 + (y + 3)^2 = 16$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\pi",
"style": "rule"
} | {
"index": "39072",
"split": "train"
} |
math | [
{
"content": "Calculate the definite integral:\n\n$$\n\\int_{\\pi / 4}^{\\arcsin \\sqrt{2 / 3}} \\frac{8 \\tan x \\, dx}{3 \\cos ^{2} x+8 \\sin 2 x-7}\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{21} \\ln \\left| \\frac{7\\sqrt{2} - 2}{5} \\right| - \\frac{4}{3} \\ln |2 - \\sqrt{2}|",
"style": "rule"
} | {
"index": "39073",
"split": "train"
} |
math | [
{
"content": "Two of the vertices of a regular octahedron are to be chosen at random. What is the probability that they will be the endpoints of an edge of the octahedron? Express your answer as a common fraction. [asy]\nsize(150);\npair A, B, C, D, E, F;\nA=(1,1);\nB=(-1,-1);\nC=(0,6);\nD=(0,-6);\nE=(6, 0);\nF... | {
"ground_truth": "\\frac{4}{5}",
"style": "rule"
} | {
"index": "39074",
"split": "train"
} |
math | [
{
"content": "Let $A,$ $B,$ and $C$ be points on a circle of radius $18.$ If $\\angle ACB = 70^\\circ,$ what is the circumference of the minor arc ${AB}$? Express your answer in terms of $\\pi.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14\\pi",
"style": "rule"
} | {
"index": "39075",
"split": "train"
} |
math | [
{
"content": "If the ratio of the legs of a right triangle is $1: 2$, then the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1: 4",
"style": "rule"
} | {
"index": "39076",
"split": "train"
} |
math | [
{
"content": "A bag contains $5$ small balls of the same shape and size, with $2$ red balls and $3$ white balls. Three balls are randomly drawn from the bag.<br/>$(1)$ Find the probability of drawing exactly one red ball.<br/>$(2)$ If the random variable $X$ represents the number of red balls drawn, find the di... | {
"ground_truth": "\\frac{3}{10}",
"style": "rule"
} | {
"index": "39077",
"split": "train"
} |
math | [
{
"content": "Let \\( a \\) and \\( b \\) be real numbers, and consider the function \\( f(x) = x^3 + ax^2 + bx \\). Suppose there exist three real numbers \\( x_1, x_2, x_3 \\) such that \\( x_1 + 1 \\leq x_2 \\leq x_3 - 1 \\) and \\( f(x_1) = f(x_2) = f(x_3) \\). Find the minimum value of \\( |a| + 2|b| \\). ... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "39078",
"split": "train"
} |
math | [
{
"content": "I had been planning to work for 20 hours a week for 12 weeks this summer to earn $\\$3000$ to buy a used car. Unfortunately, I got sick for the first two weeks of the summer and didn't work any hours. How many hours a week will I have to work for the rest of the summer if I still want to buy the c... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "39079",
"split": "train"
} |
math | [
{
"content": "How many zeros are at the end of the product 25 $\\times$ 240? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "39080",
"split": "train"
} |
math | [
{
"content": "A hyperbola has its center at the origin O, with its foci on the x-axis and two asymptotes denoted as l₁ and l₂. A line perpendicular to l₁ passes through the right focus F intersecting l₁ and l₂ at points A and B, respectively. It is known that the magnitudes of vectors |OA|, |AB|, and |OB| form ... | {
"ground_truth": "\\frac{\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "39081",
"split": "train"
} |
math | [
{
"content": "A tank is to be filled with water. When the tank is one-sixth full, 130 gallons of water are added, making the tank three-fifths full. How many gallons does the tank contain when it is completely full? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "39082",
"split": "train"
} |
math | [
{
"content": "Given that \\( I \\) is the incenter of \\( \\triangle ABC \\), and\n\\[ 9 \\overrightarrow{CI} = 4 \\overrightarrow{CA} + 3 \\overrightarrow{CB}. \\]\n\nLet \\( R \\) and \\( r \\) be the circumradius and inradius of \\( \\triangle ABC \\), respectively. Find \\(\\frac{r}{R} = \\). Please output ... | {
"ground_truth": "5/16",
"style": "rule"
} | {
"index": "39083",
"split": "train"
} |
math | [
{
"content": "What is the remainder when $2x^2-17x+47$ is divided by $x-5$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "39084",
"split": "train"
} |
math | [
{
"content": "Given that \\( a \\) is a real number, and for any \\( k \\in [-1,1] \\), when \\( x \\in (0,6] \\), the following inequality is always satisfied:\n\\[ 6 \\ln x + x^2 - 8 x + a \\leq k x. \\]\n\nFind the maximum value of \\( a \\). Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "6 - 6 \\ln 6",
"style": "rule"
} | {
"index": "39085",
"split": "train"
} |
math | [
{
"content": "Find the largest real $ T$ such that for each non-negative real numbers $ a,b,c,d,e$ such that $ a\\plus{}b\\equal{}c\\plus{}d\\plus{}e$ : \\[ \\sqrt{a^{2}\\plus{}b^{2}\\plus{}c^{2}\\plus{}d^{2}\\plus{}e^{2}}\\geq T(\\sqrt a\\plus{}\\sqrt b\\plus{}\\sqrt c\\plus{}\\sqrt d\\plus{}\\sqrt e)^{2}... | {
"ground_truth": "\\frac{\\sqrt{30}}{30 + 12\\sqrt{6}}",
"style": "rule"
} | {
"index": "39086",
"split": "train"
} |
math | [
{
"content": "How many non-empty subsets of $\\{1,2,3,4,5,6,7,8\\}$ have exactly $k$ elements and do not contain the element $k$ for some $k=1,2, \\ldots, 8$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "127",
"style": "rule"
} | {
"index": "39087",
"split": "train"
} |
math | [
{
"content": "Determine the smallest positive integer \\( a \\) for which \\( 47^n + a \\cdot 15^n \\) is divisible by 1984 for all odd \\( n \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1055",
"style": "rule"
} | {
"index": "39088",
"split": "train"
} |
math | [
{
"content": "Given point $P(2,-1)$,\n(1) Find the general equation of the line that passes through point $P$ and has a distance of 2 units from the origin.\n(2) Find the general equation of the line that passes through point $P$ and has the maximum distance from the origin. Calculate the maximum distance. Plea... | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "39089",
"split": "train"
} |
math | [
{
"content": "Consider the set \\(S\\) of all complex numbers \\(z\\) with nonnegative real and imaginary part such that \\(\\left|z^{2}+2\\right| \\leq|z|\\). Across all \\(z \\in S\\), compute the minimum possible value of \\(\\tan \\theta\\), where \\(\\theta\\) is the angle formed between \\(z\\) and the re... | {
"ground_truth": "\\sqrt{7}",
"style": "rule"
} | {
"index": "39090",
"split": "train"
} |
math | [
{
"content": "In the polar coordinate system, the curve $C\\_1$: $ρ=2\\cos θ$, and the curve $C\\_2$: $ρ\\sin ^{2}θ=4\\cos θ$. Establish a rectangular coordinate system $(xOy)$ with the pole as the coordinate origin and the polar axis as the positive semi-axis $x$. The parametric equation of the curve $C$ is $\... | {
"ground_truth": "\\frac {11}{3}",
"style": "rule"
} | {
"index": "39091",
"split": "train"
} |
math | [
{
"content": "Using equal-length toothpicks to form a rectangular diagram as shown, if the length of the rectangle is 20 toothpicks long and the width is 10 toothpicks long, how many toothpicks are used? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "430",
"style": "rule"
} | {
"index": "39092",
"split": "train"
} |
math | [
{
"content": "In a math competition consisting of problems $A$, $B$, and $C$, among the 39 participants, each person answered at least one problem correctly. Among the people who answered $A$ correctly, the number of people who only answered $A$ is 5 more than the number of people who answered other problems as... | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "39093",
"split": "train"
} |
math | [
{
"content": "Given the line $y=x+\\sqrt{6}$, the circle $(O)$: $x^2+y^2=5$, and the ellipse $(E)$: $\\frac{y^2}{a^2}+\\frac{x^2}{b^2}=1$ $(b > 0)$ with an eccentricity of $e=\\frac{\\sqrt{3}}{3}$. The length of the chord intercepted by line $(l)$ on circle $(O)$ is equal to the length of the major axis of the ... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "39094",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be a right triangle with $m(\\widehat{A})=90^\\circ$ . Let $APQR$ be a square with area $9$ such that $P\\in [AC]$ , $Q\\in [BC]$ , $R\\in [AB]$ . Let $KLMN$ be a square with area $8$ such that $N,K\\in [BC]$ , $M\\in [AB]$ , and $L\\in [AC]$ . What is $|AB|+|AC|$ ? Ple... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "39095",
"split": "train"
} |
math | [
{
"content": "Ms. Hamilton's eighth-grade class wants to participate in the annual three-person-team basketball tournament. The losing team of each game is eliminated from the tournament. If sixteen teams compete, how many games will be played to determine the winner? Please output the final answer within \\b... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "39096",
"split": "train"
} |
math | [
{
"content": "The sum of the digits of the year 2004 is 6. What is the first year after 2000 for which the sum of the digits is 12? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2019",
"style": "rule"
} | {
"index": "39097",
"split": "train"
} |
math | [
{
"content": "Given the singing scores 9.4, 8.4, 9.4, 9.9, 9.6, 9.4, 9.7, calculate the average and variance of the remaining data after removing the highest and lowest scores. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.016",
"style": "rule"
} | {
"index": "39098",
"split": "train"
} |
math | [
{
"content": "Katrine has a bag containing 4 buttons with distinct letters M, P, F, G on them (one letter per button). She picks buttons randomly, one at a time, without replacement, until she picks the button with letter G. What is the probability that she has at least three picks and her third pick is the b... | {
"ground_truth": "1/12",
"style": "rule"
} | {
"index": "39099",
"split": "train"
} |
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