data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given that a ship travels in one direction and Emily walks parallel to the riverbank in the opposite direction, counting 210 steps from back to front and 42 steps from front to back, determine the length of the ship in terms of Emily's equal steps. Please output the final answer within \\boxed{}."... | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "36500",
"split": "train"
} |
math | [
{
"content": "One angle of a triangle is twice another, and the sides opposite these angles have lengths 12 and 18. Compute the length of the third side of the triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "36501",
"split": "train"
} |
math | [
{
"content": "Given a regular hexagon \\( A_6 \\) where each of its 6 vertices is colored with either red or blue, determine the number of type II colorings of the vertices of the hexagon. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "36502",
"split": "train"
} |
math | [
{
"content": "A triangular region is bounded by the two coordinate axes and the line given by the equation $2x + y = 6$. What is the area of the region, in square units? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "36503",
"split": "train"
} |
math | [
{
"content": "Simplify first, then evaluate: $({\\frac{{x-1}}{x}-\\frac{{x-2}}{{x+1}}})÷\\frac{{2{x^2}-x}}{{{x^2}+2x+1}}$, where $x$ satisfies $x^{2}-2x-2=0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "36504",
"split": "train"
} |
math | [
{
"content": "A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$... | {
"ground_truth": "683",
"style": "rule"
} | {
"index": "36505",
"split": "train"
} |
math | [
{
"content": "The integer $N$ is the smallest positive integer that is a multiple of 2024, has more than 100 positive divisors (including 1 and $N$), and has fewer than 110 positive divisors (including 1 and $N$). What is the sum of the digits of $N$? Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "36506",
"split": "train"
} |
math | [
{
"content": "Using the six digits 0, 1, 2, 3, 4, 5 to form a six-digit number without repeating any digit, how many such numbers are there where the unit digit is less than the ten's digit? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "36507",
"split": "train"
} |
math | [
{
"content": "Three real numbers $a, b,$ and $c$ have a sum of 114 and a product of 46656. If $b=ar$ and $c=ar^2$ for some real number $r$, what is the value of $a+c$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "78",
"style": "rule"
} | {
"index": "36508",
"split": "train"
} |
math | [
{
"content": "For real numbers \\( x \\) and \\( y \\), define the operation \\( \\star \\) as follows: \\( x \\star y = xy + 4y - 3x \\).\n\nCompute the value of the expression\n\n$$\n((\\ldots)(((2022 \\star 2021) \\star 2020) \\star 2019) \\star \\ldots) \\star 2) \\star 1\n$$ Please output the final answer ... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "36509",
"split": "train"
} |
math | [
{
"content": "Given \\( A, B, C, D \\in\\{1, 2, \\cdots, 6\\} \\), and each is distinct from the others. If the curves \\( y = Ax^{2} + B \\) and \\( y = Cx^{2} + D \\) intersect, then there are \\(\\quad\\) different ways to choose \\( A, B, C, D \\) (the intersection of the curves is independent of the order ... | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "36510",
"split": "train"
} |
math | [
{
"content": "For what value of $a$ does the equation $3(2x-a) = 2(3x+12)$ have infinitely many solutions $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-8",
"style": "rule"
} | {
"index": "36511",
"split": "train"
} |
math | [
{
"content": "Given a square \\(ABCD\\) on the plane, find the minimum of the ratio \\(\\frac{OA+OC}{OB+OD}\\), where \\(O\\) is an arbitrary point on the plane. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{\\sqrt{2}}",
"style": "rule"
} | {
"index": "36512",
"split": "train"
} |
math | [
{
"content": "Suppose $d\\not=0$. We can write $\\left(12d+13+14d^2\\right)+\\left(2d+1\\right)$, in the form $ad+b+cd^2$, where $a$, $b$, and $c$ are integers. Find $a+b+c$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "36513",
"split": "train"
} |
math | [
{
"content": "Given that $\\cos (α-β)= \\frac{3}{5}$, $\\sin β=- \\frac{5}{13}$, and $α∈\\left( \\left. 0, \\frac{π}{2} \\right. \\right)$, $β∈\\left( \\left. - \\frac{π}{2},0 \\right. \\right)$, find the value of $\\sin α$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{33}{65}",
"style": "rule"
} | {
"index": "36514",
"split": "train"
} |
math | [
{
"content": "Can three persons, having one double motorcycle, overcome the distance of $70$ km in $3$ hours? Pedestrian speed is $5$ km / h and motorcycle speed is $50$ km / h. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\text{No}",
"style": "rule"
} | {
"index": "36515",
"split": "train"
} |
math | [
{
"content": "In the diagram, \\(ABCD\\) is a right trapezoid with \\(AD = 2\\) as the upper base, \\(BC = 6\\) as the lower base. Point \\(E\\) is on \\(DC\\). The area of triangle \\(ABE\\) is 15.6 and the area of triangle \\(AED\\) is 4.8. Find the area of trapezoid \\(ABCD\\). Please output the final answer... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "36516",
"split": "train"
} |
math | [
{
"content": "For a natural number $n$, if $n+6$ divides $n^3+1996$, then $n$ is called a lucky number of 1996. For example, since $4+6$ divides $4^3+1996$, 4 is a lucky number of 1996. Find the sum of all lucky numbers of 1996. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3720",
"style": "rule"
} | {
"index": "36517",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=2\\sin (wx+\\varphi+ \\frac {\\pi}{3})+1$ ($|\\varphi| < \\frac {\\pi}{2},w > 0$) is an even function, and the distance between two adjacent axes of symmetry of the function $f(x)$ is $\\frac {\\pi}{2}$.\n$(1)$ Find the value of $f( \\frac {\\pi}{8})$.\n$(2)$ When $x\\in(-... | {
"ground_truth": "2\\pi",
"style": "rule"
} | {
"index": "36518",
"split": "train"
} |
math | [
{
"content": "The total GDP of the capital city in 2022 is 41600 billion yuan, express this number in scientific notation. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4.16 \\times 10^{4}",
"style": "rule"
} | {
"index": "36519",
"split": "train"
} |
math | [
{
"content": "A rectangle with dimensions \\(24 \\times 60\\) is divided into unit squares by lines parallel to its sides. Into how many parts will this rectangle be divided if its diagonal is also drawn? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1512",
"style": "rule"
} | {
"index": "36520",
"split": "train"
} |
math | [
{
"content": "The region $G$ is bounded by the ellipsoid $\\frac{x^{2}}{16}+\\frac{y^{2}}{9}+\\frac{z^{2}}{4}=1$, and the region $g$ is bounded by this ellipsoid and the sphere $x^{2}+y^{2}+z^{2}=4$. A point is randomly chosen within the region $G$. What is the probability that it belongs to region $g$ (event $... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "36521",
"split": "train"
} |
math | [
{
"content": "A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k, 1 \\le k \\le 8.$ A tower is to be built using all 8 cubes according to the rules:\nAny cube may be the bottom cube in the tower.\nThe cube immediately on top of a cube with edge-length $k$ must have edge-length... | {
"ground_truth": "458",
"style": "rule"
} | {
"index": "36522",
"split": "train"
} |
math | [
{
"content": "Five squares and two right-angled triangles are positioned as shown. The areas of three squares are \\(3 \\, \\mathrm{m}^{2}, 7 \\, \\mathrm{m}^{2}\\), and \\(22 \\, \\mathrm{m}^{2}\\). What is the area, in \\(\\mathrm{m}^{2}\\), of the square with the question mark?\nA) 18\nB) 19\nC) 20\nD) 21\nE... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "36523",
"split": "train"
} |
math | [
{
"content": "A particular convex polygon with seven sides has exactly one right angle. How many diagonals does this seven-sided polygon have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "36524",
"split": "train"
} |
math | [
{
"content": "Mindy made three purchases for $1.98, $5.04, and $9.89. What was her total, to the nearest dollar? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "36525",
"split": "train"
} |
math | [
{
"content": "Let $n$ be a positive integer. $n$ people take part in a certain party. For any pair of the participants, either the two are acquainted with each other or they are not. What is the maximum possible number of the pairs for which the two are not acquainted but have a common acquaintance among the pa... | {
"ground_truth": "\\binom{n-1}{2}",
"style": "rule"
} | {
"index": "36526",
"split": "train"
} |
math | [
{
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.5 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (i = -5; i <= 5; ++i)... | {
"ground_truth": "\\text{A}",
"style": "rule"
} | {
"index": "36527",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\frac{1}{2}\\sin 2x\\sin φ+\\cos^2x\\cos φ+\\frac{1}{2}\\sin (\\frac{3π}{2}-φ)(0 < φ < π)$, whose graph passes through the point $(\\frac{π}{6},\\frac{1}{2})$.\n\n(I) Find the interval(s) where the function $f(x)$ is monotonically decreasing on $[0,π]$;\n\n(II) If ${x}_{0... | {
"ground_truth": "\\frac{7-24\\sqrt{3}}{100}",
"style": "rule"
} | {
"index": "36528",
"split": "train"
} |
math | [
{
"content": "If the numbers $x$ and $y$ are inversely proportional and when the sum of $x$ and $y$ is 54, $x$ is three times $y$, find the value of $y$ when $x = 5$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "109.35",
"style": "rule"
} | {
"index": "36529",
"split": "train"
} |
math | [
{
"content": "Given the numbers 252 and 630, find the ratio of the least common multiple to the greatest common factor. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "36530",
"split": "train"
} |
math | [
{
"content": "Aquatic plants require a specific type of nutrient solution. Given that each time $a (1 \\leqslant a \\leqslant 4$ and $a \\in R)$ units of the nutrient solution are released, its concentration $y (\\text{g}/\\text{L})$ changes over time $x (\\text{days})$ according to the function $y = af(x)$, wh... | {
"ground_truth": "24-16\\sqrt{2}",
"style": "rule"
} | {
"index": "36531",
"split": "train"
} |
math | [
{
"content": "What is $(5^{-2})^0 + (5^0)^3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "36532",
"split": "train"
} |
math | [
{
"content": "What is $(3+3+5)\\div2 - 1\\div2$ in simplest terms? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "36533",
"split": "train"
} |
math | [
{
"content": "Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\\frac{12}5\\sqrt2$. What is the volume of the tetrahedron? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\frac{24}5$",
"style": "rule"
} | {
"index": "36534",
"split": "train"
} |
math | [
{
"content": "Let $f(n)$ be the largest prime factor of $n$. Estimate $$N=\\left\\lfloor 10^{4} \\cdot \\frac{\\sum_{n=2}^{10^{6}} f\\left(n^{2}-1\\right)}{\\sum_{n=2}^{10^{6}} f(n)}\\right\\rfloor$$ An estimate of $E$ will receive $\\max \\left(0,\\left\\lfloor 20-20\\left(\\frac{|E-N|}{10^{3}}\\right)^{1 / 3}... | {
"ground_truth": "18215",
"style": "rule"
} | {
"index": "36535",
"split": "train"
} |
math | [
{
"content": "Find the number of real solutions of the equation\n\\[\\frac{x}{100} = \\sin x.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "63",
"style": "rule"
} | {
"index": "36536",
"split": "train"
} |
math | [
{
"content": "Given the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ (where $a > 0$, $b > 0$), a line with an angle of $60^\\circ$ passes through one of the foci and intersects the y-axis and the right branch of the hyperbola. Find the eccentricity of the hyperbola if the point where the line intersects ... | {
"ground_truth": "2 + \\sqrt{3}",
"style": "rule"
} | {
"index": "36537",
"split": "train"
} |
math | [
{
"content": "A point $(x, y)$ is randomly selected such that $0 \\leq x \\leq 3$ and $0 \\leq y \\leq 3$. What is the probability that $x + 2y \\leq 6$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "36538",
"split": "train"
} |
math | [
{
"content": "The tail of a 1-mile long train exits a tunnel exactly 3 minutes after the front of the train entered the tunnel. If the train is moving 60 miles per hour, how many miles long is the tunnel? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\text{ miles}",
"style": "rule"
} | {
"index": "36539",
"split": "train"
} |
math | [
{
"content": "Let \\( D \\) be a point inside \\( \\triangle ABC \\) such that \\( \\angle BAD = \\angle BCD \\) and \\( \\angle BDC = 90^\\circ \\). If \\( AB = 5 \\), \\( BC = 6 \\), and \\( M \\) is the midpoint of \\( AC \\), find the length of \\( DM \\). Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{\\sqrt{11}}{2}",
"style": "rule"
} | {
"index": "36540",
"split": "train"
} |
math | [
{
"content": "How many ordered triples of positive integers $(a, b, c)$ are there for which $a^{4} b^{2} c=54000$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "36541",
"split": "train"
} |
math | [
{
"content": "In the game of Frood, dropping $n$ froods gives a score of the sum of the first $n$ positive integers. For example, dropping five froods scores $1 + 2 + 3 + 4 + 5 = 15$ points. Eating $n$ froods earns $10n$ points. For example, eating five froods earns $10(5) = 50$ points. What is the least number... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "36542",
"split": "train"
} |
math | [
{
"content": "The positive integers $x$ and $y$ are the two smallest positive integers for which the product of $360$ and $x$ is a square and the product of $360$ and $y$ is a cube. What is the sum of $x$ and $y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "85",
"style": "rule"
} | {
"index": "36543",
"split": "train"
} |
math | [
{
"content": "The formula expressing the relationship between $x$ and $y$ in the table is:\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline x & 2 & 3 & 4 & 5 & 6\\\n\\hline y & 0 & 2 & 6 & 12 & 20\\\n\\hline\n\\end{tabular} Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$y = x^{2}-3x+2$",
"style": "rule"
} | {
"index": "36544",
"split": "train"
} |
math | [
{
"content": "Points $( \\sqrt{\\pi} , a)$ and $( \\sqrt{\\pi} , b)$ are distinct points on the graph of $y^2 + x^4 = 2x^2 y + 1$. What is $|a-b|$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "36545",
"split": "train"
} |
math | [
{
"content": "Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.472",
"style": "rule"
} | {
"index": "36546",
"split": "train"
} |
math | [
{
"content": "A right circular cone has base radius $r$ and height $h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The c... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "36547",
"split": "train"
} |
math | [
{
"content": "Determine the sum of all integer values $n$ for which $\\binom{25}{n} + \\binom{25}{12} = \\binom{26}{13}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "36548",
"split": "train"
} |
math | [
{
"content": "Point $A$ , $B$ , $C$ , and $D$ form a rectangle in that order. Point $X$ lies on $CD$ , and segments $\\overline{BX}$ and $\\overline{AC}$ intersect at $P$ . If the area of triangle $BCP$ is 3 and the area of triangle $PXC$ is 2, what is the area of the entire rectangle? Please o... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "36549",
"split": "train"
} |
math | [
{
"content": "Given that in acute triangle ABC, the sides opposite to angles A, B, C are a, b, c respectively, and bsinA = acos(B - $ \\frac{\\pi}{6}$).\n(1) Find the value of angle B.\n(2) If b = $\\sqrt{13}$, a = 4, and D is a point on AC such that S<sub>△ABD</sub> = 2$\\sqrt{3}$, find the length of AD. Pleas... | {
"ground_truth": "\\frac{2 \\sqrt{13}}{3}",
"style": "rule"
} | {
"index": "36550",
"split": "train"
} |
math | [
{
"content": "Compute the square of 1033 without a calculator. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1067089",
"style": "rule"
} | {
"index": "36551",
"split": "train"
} |
math | [
{
"content": "In the arithmetic sequence $\\{a\\_n\\}$, the common difference $d=\\frac{1}{2}$, and the sum of the first $100$ terms $S\\_{100}=45$. Find the value of $a\\_1+a\\_3+a\\_5+...+a\\_{99}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-69",
"style": "rule"
} | {
"index": "36552",
"split": "train"
} |
math | [
{
"content": "In the sequence $\\{a_n\\}$, $a_1=2$, $a_2=5$, $a_{n+1}=a_{n+2}+a_{n}$, calculate the value of $a_6$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "36553",
"split": "train"
} |
math | [
{
"content": "Given that point $P$ is a moving point on the curve $y= \\frac {3-e^{x}}{e^{x}+1}$, find the minimum value of the slant angle $\\alpha$ of the tangent line at point $P$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3\\pi}{4}",
"style": "rule"
} | {
"index": "36554",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=a\\ln(x+1)+bx+1$ \n$(1)$ If the function $y=f(x)$ has an extremum at $x=1$, and the tangent line to the curve $y=f(x)$ at the point $(0,f(0))$ is parallel to the line $2x+y-3=0$, find the value of $a$; \n$(2)$ If $b= \\frac{1}{2}$, discuss the monotonicity of the functio... | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "36555",
"split": "train"
} |
math | [
{
"content": "Given real numbers $x$, $y$, and $z$ are chosen independently and at random from the interval $[0, m]$ for some positive integer $m$. The probability that no two of $x$, $y$, and $z$ are within 2 units of each other is greater than $\\frac{1}{2}$. Determine the smallest possible value of $m$. Plea... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "36556",
"split": "train"
} |
math | [
{
"content": "Regular hexagon $A B C D E F$ has side length 2. A laser beam is fired inside the hexagon from point $A$ and hits $\\overline{B C}$ at point $G$. The laser then reflects off $\\overline{B C}$ and hits the midpoint of $\\overline{D E}$. Find $B G$. Please output the final answer within \\boxed{}.",... | {
"ground_truth": "\\frac{2}{5}",
"style": "rule"
} | {
"index": "36557",
"split": "train"
} |
math | [
{
"content": "Given that $|\\vec{a}|=4$, and $\\vec{e}$ is a unit vector. When the angle between $\\vec{a}$ and $\\vec{e}$ is $\\frac{2\\pi}{3}$, the projection of $\\vec{a} + \\vec{e}$ on $\\vec{a} - \\vec{e}$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5\\sqrt{21}}{7}",
"style": "rule"
} | {
"index": "36558",
"split": "train"
} |
math | [
{
"content": "Select 5 elements from the set $\\{x|1\\leq x \\leq 11, \\text{ and } x \\in \\mathbb{N}^*\\}$ to form a subset of this set, and any two elements in this subset do not sum up to 12. How many different subsets like this are there? (Answer with a number). Please output the final answer within \\boxe... | {
"ground_truth": "112",
"style": "rule"
} | {
"index": "36559",
"split": "train"
} |
math | [
{
"content": "Parabola C is defined by the equation y²=2px (p>0). A line l with slope k passes through point P(-4,0) and intersects with parabola C at points A and B. When k=$\\frac{1}{2}$, points A and B coincide.\n\n1. Find the equation of parabola C.\n2. If A is the midpoint of PB, find the length of |AB|. P... | {
"ground_truth": "2\\sqrt{11}",
"style": "rule"
} | {
"index": "36560",
"split": "train"
} |
math | [
{
"content": "In an institute, there are truth-tellers, who always tell the truth, and liars, who always lie. One day, each of the employees made two statements:\n\n1) There are fewer than ten people in the institute who work more than I do.\n2) In the institute, at least one hundred people have a salary greate... | {
"ground_truth": "110",
"style": "rule"
} | {
"index": "36561",
"split": "train"
} |
math | [
{
"content": "Given an arithmetic sequence $\\{a_{n}\\}$ with the sum of the first $n$ terms as $S_{n}$, if $a_{2}+a_{4}+3a_{7}+a_{9}=24$, calculate the value of $S_{11}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "44",
"style": "rule"
} | {
"index": "36562",
"split": "train"
} |
math | [
{
"content": "How many positive divisors does the number $360$ have? Also, calculate the sum of all positive divisors of $360$ that are greater than $30$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1003",
"style": "rule"
} | {
"index": "36563",
"split": "train"
} |
math | [
{
"content": "Find the three-digit integer in the decimal system that satisfies the following properties:\n1. When the digits in the tens and units places are swapped, the resulting number can be represented in the octal system as the original number.\n2. When the digits in the hundreds and tens places are swap... | {
"ground_truth": "139",
"style": "rule"
} | {
"index": "36564",
"split": "train"
} |
math | [
{
"content": "A line with negative slope passing through the point $(18,8)$ intersects the $x$ and $y$ axes at $(a,0)$ and $(0,b)$ , respectively. What is the smallest possible value of $a+b$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "36565",
"split": "train"
} |
math | [
{
"content": "If $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^3 + bx^2 + 1,$ find $b.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "36566",
"split": "train"
} |
math | [
{
"content": "Let $C$ and $C^{\\prime}$ be two externally tangent circles with centers $O$ and $O^{\\prime}$ and radii 1 and 2, respectively. From $O$, a tangent is drawn to $C^{\\prime}$ with the point of tangency at $P^{\\prime}$, and from $O^{\\prime}$, a tangent is drawn to $C$ with the point of tangency at... | {
"ground_truth": "\\frac{4\\sqrt{2} - \\sqrt{5}}{3}",
"style": "rule"
} | {
"index": "36567",
"split": "train"
} |
math | [
{
"content": "Find the number of functions $f : \\mathbb{R} \\to \\mathbb{R}$ such that\n\\[f(x + y) f(x - y) = (f(x) + f(y))^2 - 4x^2 f(y)\\]for all real numbers $x$ and $y.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "36568",
"split": "train"
} |
math | [
{
"content": "A right circular cone has base of radius 1 and height 3. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{6}{2 + 3\\sqrt{2}}",
"style": "rule"
} | {
"index": "36569",
"split": "train"
} |
math | [
{
"content": "In a regular tetrahedron \\( P-ABCD \\), where each face is an equilateral triangle with side length 1, points \\( M \\) and \\( N \\) are the midpoints of edges \\( AB \\) and \\( BC \\), respectively. Find the distance between the skew lines \\( MN \\) and \\( PC \\). Please output the final ans... | {
"ground_truth": "\\frac{\\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "36570",
"split": "train"
} |
math | [
{
"content": "Given that $\\alpha$ and $\\beta$ are two interior angles of an oblique triangle, if $\\frac{{\\cos \\alpha - \\sin \\alpha}}{{\\cos \\alpha + \\sin \\alpha}} = \\cos 2\\beta$, then the minimum value of $\\tan \\alpha + \\tan \\beta$ is ______. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "-\\frac{1}{4}",
"style": "rule"
} | {
"index": "36571",
"split": "train"
} |
math | [
{
"content": "Calculate \\(3^3 \\cdot 4^3\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1728",
"style": "rule"
} | {
"index": "36572",
"split": "train"
} |
math | [
{
"content": "Find the area of a triangle with side lengths 8, 9, and 9. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{65}",
"style": "rule"
} | {
"index": "36573",
"split": "train"
} |
math | [
{
"content": "Let $BCDE$ be a trapezoid with $BE\\parallel CD$ , $BE = 20$ , $BC = 2\\sqrt{34}$ , $CD = 8$ , $DE = 2\\sqrt{10}$ . Draw a line through $E$ parallel to $BD$ and a line through $B$ perpendicular to $BE$ , and let $A$ be the intersection of these two lines. Let $M$ be the intersect... | {
"ground_truth": "203",
"style": "rule"
} | {
"index": "36574",
"split": "train"
} |
math | [
{
"content": "Given an ellipse C: $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}=1(a＞b＞0)$$ with left and right foci $F_1$ and $F_2$, respectively. Point A is the upper vertex of the ellipse, $|F_{1}A|= \\sqrt {2}$, and the area of △$F_{1}AF_{2}$ is 1.\n(1) Find the standard equation of the ellipse.\n(2) Let M ... | {
"ground_truth": "y=- \\frac {1}{3}",
"style": "rule"
} | {
"index": "36575",
"split": "train"
} |
math | [
{
"content": "While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments $AB = 10$ and $CD = 7$ on a plane. Across all possible configurations of those two segments, determine the maximum possible area of quadrilateral $ACBD$ . Please output the final answer within \\... | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "36576",
"split": "train"
} |
math | [
{
"content": "A natural number \\( 1 \\leq n \\leq 221 \\) is called lucky if, when dividing 221 by \\( n \\), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "115",
"style": "rule"
} | {
"index": "36577",
"split": "train"
} |
math | [
{
"content": "Choose six out of the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to fill in the blanks below, so that the equation is true. Each blank is filled with a single digit, and no two digits are the same. \n\n$\\square + \\square \\square = \\square \\square \\square$. \n\nWhat is the largest possible three... | {
"ground_truth": "105",
"style": "rule"
} | {
"index": "36578",
"split": "train"
} |
math | [
{
"content": "A certain chemical reaction requires a catalyst to accelerate the reaction, but using too much of this catalyst affects the purity of the product. If the amount of this catalyst added is between 500g and 1500g, and the 0.618 method is used to arrange the experiment, then the amount of catalyst add... | {
"ground_truth": "882",
"style": "rule"
} | {
"index": "36579",
"split": "train"
} |
math | [
{
"content": "Calculate: $99\\times 101=\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9999",
"style": "rule"
} | {
"index": "36580",
"split": "train"
} |
math | [
{
"content": "Determine the number of prime dates in a non-leap year where the day and the month are both prime numbers, and the year has one fewer prime month than usual. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "41",
"style": "rule"
} | {
"index": "36581",
"split": "train"
} |
math | [
{
"content": "A new model car travels 4.2 kilometers more per liter of gasoline than an old model car. Additionally, the fuel consumption for the new model is 2 liters less per 100 km. How many liters of gasoline per 100 km does the new car consume? Round your answer to the nearest hundredth if necessary. Pleas... | {
"ground_truth": "5.97",
"style": "rule"
} | {
"index": "36582",
"split": "train"
} |
math | [
{
"content": "Let the product $(12)(15)(16)$, each factor written in base $b$, equals $3146$ in base $b$. Let $s=12+15+16$, each term expressed in base $b$. Then $s$, in base $b$, is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "44",
"style": "rule"
} | {
"index": "36583",
"split": "train"
} |
math | [
{
"content": "Given that $f(x) = x^k$ where $k < 0$, what is the range of $f(x)$ on the interval $[1, \\infty)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(0,1]",
"style": "rule"
} | {
"index": "36584",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $A > B$, $\\cos C= \\frac {5}{13}$, and $\\cos (A-B)= \\frac {3}{5}$.\n\n(1) Find the value of $\\cos 2A$;\n\n(2) If $c=15$, find the value of $a$. Please output the final answer wit... | {
"ground_truth": "2 \\sqrt {65}",
"style": "rule"
} | {
"index": "36585",
"split": "train"
} |
math | [
{
"content": "Let $f(x) = e^x - ax + 3$ where $a \\in \\mathbb{R}$.\n1. Discuss the monotonicity of the function $f(x)$.\n2. If the minimum value of the function $f(x)$ on the interval $[1,2]$ is $4$, find the value of $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "e - 1",
"style": "rule"
} | {
"index": "36586",
"split": "train"
} |
math | [
{
"content": "Compute $1-2+3-4+\\dots+100-101$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-151",
"style": "rule"
} | {
"index": "36587",
"split": "train"
} |
math | [
{
"content": "Kelvin the Frog has a pair of standard fair 8-sided dice (each labelled from 1 to 8). Alex the sketchy Kat also has a pair of fair 8-sided dice, but whose faces are labelled differently (the integers on each Alex's dice need not be distinct). To Alex's dismay, when both Kelvin and Alex roll their ... | {
"ground_truth": "24, 28, 32",
"style": "rule"
} | {
"index": "36588",
"split": "train"
} |
math | [
{
"content": "A part of a book has fallen out. The number of the first fallen page is 387, and the number of the last page consists of the same digits but in a different order. How many sheets fell out of the book? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "176",
"style": "rule"
} | {
"index": "36589",
"split": "train"
} |
math | [
{
"content": "Given the assumption that smoking is unrelated to lung disease, calculate the confidence level that can be concluded from the chi-square statistic $K^2=5.231$, with $P(K^2 \\geq 3.841) = 0.05$ and $P(K^2 \\geq 6.635) = 0.01$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "95\\%",
"style": "rule"
} | {
"index": "36590",
"split": "train"
} |
math | [
{
"content": "A room is shaped like an 'L'. One part is a rectangle that is 23 feet long and 15 feet wide, and a square with a side of 8 feet is attached at one end of the rectangle, extending its width. Calculate the ratio of the total length of the room to its perimeter. Please output the final answer within ... | {
"ground_truth": "1:4",
"style": "rule"
} | {
"index": "36591",
"split": "train"
} |
math | [
{
"content": "Given a positive integer \\( n \\geq 3 \\), for an \\( n \\)-element real array \\(\\left(x_{1}, x_{2}, \\cdots, x_{n}\\right)\\), if every permutation \\( y_{1}, y_{2}, \\cdots, y_{n} \\) of it satisfies \\(\\sum_{i=1}^{n-1} y_{i} y_{i+1} \\geq -1\\), then the real array \\(\\left(x_{1}, x_{2}, \... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "36592",
"split": "train"
} |
math | [
{
"content": "In triangle \\(ABC\\), side \\(AC = 42\\). The angle bisector \\(CL\\) is divided by the point of intersection of the angle bisectors of the triangle in the ratio \\(2:1\\) from the vertex. Find the length of side \\(AB\\) if the radius of the circle inscribed in triangle \\(ABC\\) is 14. Please o... | {
"ground_truth": "56",
"style": "rule"
} | {
"index": "36593",
"split": "train"
} |
math | [
{
"content": "In the two regular tetrahedra \\(A-OBC\\) and \\(D-OBC\\) with coinciding bases, \\(M\\) and \\(N\\) are the centroids of \\(\\triangle ADC\\) and \\(\\triangle BDC\\) respectively. Let \\(\\overrightarrow{OA}=\\boldsymbol{a}, \\overrightarrow{OB}=\\boldsymbol{b}, \\overrightarrow{OC}=\\boldsymbol... | {
"ground_truth": "439",
"style": "rule"
} | {
"index": "36594",
"split": "train"
} |
math | [
{
"content": "Let the hyperbola $\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1(a>b>0)$ have its right focus at $F$ and eccentricity $e$. A line passing through $F$ with a slope of 1 intersects the asymptotes of the hyperbola at points $A$ and $B$. If the midpoint of $A$ and $B$ is $M$ and $|FM|=c$, find $e$. Plea... | {
"ground_truth": "\\sqrt[4]{2}",
"style": "rule"
} | {
"index": "36595",
"split": "train"
} |
math | [
{
"content": "Triangle $\\triangle ABC$ has a right angle at $C$, $\\angle A = 60^\\circ$, and $AC=10$. Find the radius of the incircle of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5(\\sqrt{3}-1)",
"style": "rule"
} | {
"index": "36596",
"split": "train"
} |
math | [
{
"content": "Let $a,$ $b,$ $c,$ $d$ be distinct integers, and let $\\omega$ be a complex number such that $\\omega^4 = 1$ and $\\omega \\neq 1.$ Find the smallest possible value of\n\\[|a + b \\omega + c \\omega^2 + d \\omega^3|.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{4.5}",
"style": "rule"
} | {
"index": "36597",
"split": "train"
} |
math | [
{
"content": "At the end of $1997$, Jason was one-third as old as his grandmother. The sum of the years in which they were born was $3852$. How old will Jason be at the end of $2004$?\nA) $41.5$\nB) $42.5$\nC) $43.5$\nD) $44.5$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42.5",
"style": "rule"
} | {
"index": "36598",
"split": "train"
} |
math | [
{
"content": "A grocer arranges a display of cans where the top row has three cans, and each lower row has two more cans than the row above it. The total number of cans in the display is 169. How many rows does the display contain? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "36599",
"split": "train"
} |
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