data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "(1) Let the function $f(x)=|x+2|+|x-a|$. If the inequality $f(x) \\geqslant 3$ always holds for $x$ in $\\mathbb{R}$, find the range of the real number $a$.\n\n(2) Given positive numbers $x$, $y$, $z$ satisfying $x+2y+3z=1$, find the minimum value of $\\frac{3}{x}+\\frac{2}{y}+\\frac{1}{z}$. Pleas... | {
"ground_truth": "16+8\\sqrt{3}",
"style": "rule"
} | {
"index": "2000",
"split": "train"
} |
math | [
{
"content": "In $\\Delta ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, where $b=6$, and the area of $\\Delta ABC$ is $15$. The radius of the circumcircle is $5$.\n\n(Ⅰ) Find the value of $\\sin 2B$;\n\n(Ⅱ) Find the perimeter of $\\Delta ABC$. Please out... | {
"ground_truth": "6+6 \\sqrt {6}",
"style": "rule"
} | {
"index": "2001",
"split": "train"
} |
math | [
{
"content": "If $x@y=xy-2x$, what is the value of $(5@3)-(3@5)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "2002",
"split": "train"
} |
math | [
{
"content": "Given that $O$ is a regular octahedron, that $C$ is the cube whose vertices are the centers of the faces of $O,$ and that the ratio of the volume of $O$ to that of $C$ is $\\frac mn,$ where $m$ and $n$ are relatively prime integers, find $m+n.$ Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "2003",
"split": "train"
} |
math | [
{
"content": "If $y= \\frac{x^2+2x+8}{x-4}$, at what value of $x$ will there be a vertical asymptote? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "2004",
"split": "train"
} |
math | [
{
"content": "A geometric sequence starts $16$, $-24$, $36$, $-54$. What is the common ratio of this sequence? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{2}",
"style": "rule"
} | {
"index": "2005",
"split": "train"
} |
math | [
{
"content": "Let the function \n$$\nf(x) = A \\sin(\\omega x + \\varphi) \\quad (A>0, \\omega>0).\n$$\n\nIf \\( f(x) \\) is monotonic on the interval \\(\\left[\\frac{\\pi}{6}, \\frac{\\pi}{2}\\right]\\) and \n$$\nf\\left(\\frac{\\pi}{2}\\right) = f\\left(\\frac{2\\pi}{3}\\right) = -f\\left(\\frac{\\pi}{6}\\ri... | {
"ground_truth": "\\pi",
"style": "rule"
} | {
"index": "2006",
"split": "train"
} |
math | [
{
"content": "Monica decides to tile the floor of her 15-foot by 20-foot dining room. She plans to create a two-foot-wide border using one-foot by one-foot square tiles around the edges of the room and fill in the rest of the floor with three-foot by three-foot square tiles. Calculate the total number of tiles ... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "2007",
"split": "train"
} |
math | [
{
"content": "In a WeChat group, five members simultaneously grab for four red envelopes, each person can grab at most one, and all red envelopes are claimed. Among the four red envelopes, there are two containing 2 yuan and two containing 3 yuan. Determine the number of scenarios in which both members A and B ... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "2008",
"split": "train"
} |
math | [
{
"content": "The consecutive angles of a particular trapezoid form an arithmetic sequence. If the largest angle measures $120^{\\circ}$, what is the measure of the smallest angle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60^\\circ",
"style": "rule"
} | {
"index": "2009",
"split": "train"
} |
math | [
{
"content": "A pyramid with a square base has all edges of 1 unit in length. What is the radius of the sphere that can be inscribed in the pyramid? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{6} - \\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "2010",
"split": "train"
} |
math | [
{
"content": "For $-1<s<1$, let $T(s)$ denote the sum of the geometric series \\[20 + 20s + 20s^2 + 20s^3+\\cdots .\\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b) = 4800$. Find $T(b) + T(-b)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "480",
"style": "rule"
} | {
"index": "2011",
"split": "train"
} |
math | [
{
"content": "The lengths of the six edges of a tetrahedron \\(ABCD\\) are 7, 13, 18, 27, 36, and 41, respectively. If \\(AB = 41\\), what is the length of \\(CD\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "2012",
"split": "train"
} |
math | [
{
"content": "Once in a restaurant ***Dr. Strange*** found out that there were 12 types of food items from 1 to 12 on the menu. He decided to visit the restaurant 12 days in a row and try a different food everyday. 1st day, he tries one of the items from the first two. On the 2nd day, he eats either item 3 or t... | {
"ground_truth": "2048",
"style": "rule"
} | {
"index": "2013",
"split": "train"
} |
math | [
{
"content": "What is the probability that Claudia gets at least 2 more heads than tails if she flips 12 coins? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{793}{2048}",
"style": "rule"
} | {
"index": "2014",
"split": "train"
} |
math | [
{
"content": "A cone with a base radius of $15$ cm and a height of $30$ cm has a sphere inscribed within it. The radius of the sphere can be expressed in the form $b\\sqrt{d} - b$ cm. Determine the values of $b$ and $d$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12.5",
"style": "rule"
} | {
"index": "2015",
"split": "train"
} |
math | [
{
"content": "Find the area of the circle defined by \\(x^2 + 4x + y^2 + 10y + 13 = 0\\) that lies above the line \\(y = -2\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\pi",
"style": "rule"
} | {
"index": "2016",
"split": "train"
} |
math | [
{
"content": "If $x$ is an even number, then find the largest integer that always divides the expression \\[(15x+3)(15x+9)(5x+10).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "2017",
"split": "train"
} |
math | [
{
"content": "Find the curve defined by the equation\n\\[r = \\frac{1}{1 - \\cos \\theta}.\\](A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\text{(C)}",
"style": "rule"
} | {
"index": "2018",
"split": "train"
} |
math | [
{
"content": "Sara baked 60 pies, of which one third contained chocolate, three-fifths contained berries, half contained cinnamon, and one-fifth contained poppy seeds. What is the largest possible number of pies that had none of these ingredients? Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "2019",
"split": "train"
} |
math | [
{
"content": "Let $a_1, a_2, \\dots, a_k$ be a finite arithmetic sequence with $a_4 + a_7 + a_{10} = 17$ and $a_4 + a_5 + \\dots + a_{13} + a_{14} = 77$. If $a_k = 13$, then $k = $ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "2020",
"split": "train"
} |
math | [
{
"content": "The first three terms of a geometric progression are $\\sqrt{2}, \\sqrt[3]{2}, \\sqrt[6]{2}$. Find the fourth term. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "2021",
"split": "train"
} |
math | [
{
"content": "The oranges are placed in a pyramid-like stack with the base being a rectangle that is 5 oranges wide and 7 oranges long. Each orange above the first layer rests in a pocket formed by four oranges in the layer below. Determine the total number of oranges in the stack. Please output the final answe... | {
"ground_truth": "85",
"style": "rule"
} | {
"index": "2022",
"split": "train"
} |
math | [
{
"content": "In an isosceles trapezoid, the height is 10, and the diagonals are mutually perpendicular. Find the midsegment (the line connecting the midpoints of the non-parallel sides) of the trapezoid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "2023",
"split": "train"
} |
math | [
{
"content": "To enhance students' physical fitness, our school has set up sports interest classes for seventh graders. Among them, the basketball interest class has $x$ students, the number of students in the soccer interest class is $2$ less than twice the number of students in the basketball interest class, ... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "2024",
"split": "train"
} |
math | [
{
"content": "Simplify $(2^8 + 4^5)(2^3 - (-2)^3)^7$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1280 \\cdot 16^7",
"style": "rule"
} | {
"index": "2025",
"split": "train"
} |
math | [
{
"content": "Given that $θ$ is an angle in the third quadrant and $\\sin (θ- \\frac {π}{4})= \\frac {3}{5}$, find $\\tan (θ+ \\frac {π}{4})=$____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {4}{3}",
"style": "rule"
} | {
"index": "2026",
"split": "train"
} |
math | [
{
"content": "Alice picks a number uniformly at random from the first $5$ even positive integers, and Palice picks a number uniformly at random from the first $5$ odd positive integers. If Alice picks a larger number than Palice with probability $\\frac{m}{n}$ for relatively prime positive integers $m,n$... | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "2027",
"split": "train"
} |
math | [
{
"content": "The integer \\( n \\) is the smallest positive number that satisfies the following conditions:\n1. \\( n \\) is a multiple of 75.\n2. \\( n \\) has exactly 75 positive divisors (including 1 and itself).\n\nFind the value of \\( \\frac{n}{75} \\). Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "432",
"style": "rule"
} | {
"index": "2028",
"split": "train"
} |
math | [
{
"content": "Find the largest real number $k$ such that there exists a sequence of positive reals $\\left\\{a_{i}\\right\\}$ for which $\\sum_{n=1}^{\\infty} a_{n}$ converges but $\\sum_{n=1}^{\\infty} \\frac{\\sqrt{a_{n}}}{n^{k}}$ does not. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "2029",
"split": "train"
} |
math | [
{
"content": "A calculator has digits from 0 to 9 and signs of two operations. Initially, the display shows the number 0. You can press any keys. The calculator performs operations in the sequence of keystrokes. If an operation sign is pressed several times in a row, the calculator remembers only the last press... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "2030",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ and $AEFG$ be two faces of a cube with edge $AB=10$. A beam of light emanates from vertex $A$ and reflects off face $AEFG$ at point $Q$, which is 3 units from $\\overline{EF}$ and 6 units from $\\overline{AG}$. The beam continues to be reflected off the faces of the cube. The length of ... | {
"ground_truth": "155",
"style": "rule"
} | {
"index": "2031",
"split": "train"
} |
math | [
{
"content": "A circle with center $C$ is tangent to the positive $x$ and $y$-axes and externally tangent to the circle centered at $(3,0)$ with radius $1$. What is the sum of all possible radii of the circle with center $C$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "2032",
"split": "train"
} |
math | [
{
"content": "Two arithmetic sequences $\\{a_{n}\\}$ and $\\{b_{n}\\}$ have the sums of the first $n$ terms as $S_{n}$ and $T_{n}$, respectively. It is known that $\\frac{{S}_{n}}{{T}_{n}}=\\frac{7n+2}{n+3}$. Find $\\frac{{a}_{7}}{{b}_{7}}$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac{93}{16}",
"style": "rule"
} | {
"index": "2033",
"split": "train"
} |
math | [
{
"content": "Li Yun is sitting by the window in a train moving at a speed of 60 km/h. He sees a freight train with 30 cars approaching from the opposite direction. When the head of the freight train passes the window, he starts timing, and he stops timing when the last car passes the window. The recorded time ... | {
"ground_truth": "44",
"style": "rule"
} | {
"index": "2034",
"split": "train"
} |
math | [
{
"content": "Find the greatest $a$ such that $\\frac{7\\sqrt{(2a)^2+(1)^2}-4a^2-1}{\\sqrt{1+4a^2}+3}=2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "2035",
"split": "train"
} |
math | [
{
"content": "How many nine-digit integers of the form 'pqrpqrpqr' are multiples of 24? (Note that p, q, and r need not be different.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "112",
"style": "rule"
} | {
"index": "2036",
"split": "train"
} |
math | [
{
"content": "Kimothy starts in the bottom-left square of a 4 by 4 chessboard. In one step, he can move up, down, left, or right to an adjacent square. Kimothy takes 16 steps and ends up where he started, visiting each square exactly once (except for his starting/ending square). How many paths could he have tak... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "2037",
"split": "train"
} |
math | [
{
"content": "Six pepperoni circles will exactly fit across the diameter of a $12$-inch pizza when placed. If a total of $24$ circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered by pepperoni? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "2038",
"split": "train"
} |
math | [
{
"content": "What is the value of the expression $(37 + 12)^2 - (37^2 +12^2)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "888",
"style": "rule"
} | {
"index": "2039",
"split": "train"
} |
math | [
{
"content": "Let \\( g(n) = (n^2 - 2n + 1)^{1/3} + (n^2 - 1)^{1/3} + (n^2 + 2n + 1)^{1/3} \\). Find \\( \\frac{1}{g(1)} + \\frac{1}{g(3)} + \\frac{1}{g(5)} + \\ldots + \\frac{1}{g(999999)} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "2040",
"split": "train"
} |
math | [
{
"content": "Compute $\\cos 150^\\circ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "2041",
"split": "train"
} |
math | [
{
"content": "Before the soccer match between the \"North\" and \"South\" teams, five predictions were made:\n\na) There will be no draw;\n\nb) \"South\" will concede goals;\n\nc) \"North\" will win;\n\nd) \"North\" will not lose;\n\ne) Exactly 3 goals will be scored in the match.\n\nAfter the match, it was fou... | {
"ground_truth": "2-1",
"style": "rule"
} | {
"index": "2042",
"split": "train"
} |
math | [
{
"content": "Solve for $R$ if $\\sqrt[4]{R^3} = 64\\sqrt[16]{4}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "256 \\cdot 2^{1/6}",
"style": "rule"
} | {
"index": "2043",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $AB=BC=29$, and $AC=42$. What is the area of $\\triangle ABC$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "420",
"style": "rule"
} | {
"index": "2044",
"split": "train"
} |
math | [
{
"content": "In a new configuration, six circles with a radius of 5 units intersect at a single point. What is the number of square units in the area of the shaded region? The region is formed similarly to the original problem where the intersections create smaller sector-like areas. Express your answer in ter... | {
"ground_truth": "75\\pi - 25\\sqrt{3}",
"style": "rule"
} | {
"index": "2045",
"split": "train"
} |
math | [
{
"content": "Using the side lengths 2, 3, 5, 7, and 11, how many different triangles with exactly two equal sides can be formed? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "2046",
"split": "train"
} |
math | [
{
"content": "Given that $AC$ and $CE$ are two diagonals of a regular hexagon $ABCDEF$, and points $M$ and $N$ divide $AC$ and $CE$ internally such that $\\frac{AM}{AC}=\\frac{CN}{CE}=r$. If points $B$, $M$, and $N$ are collinear, find the value of $r$. Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "\\frac{1}{\\sqrt{3}}",
"style": "rule"
} | {
"index": "2047",
"split": "train"
} |
math | [
{
"content": "Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)\n[asy] import thre... | {
"ground_truth": "1680",
"style": "rule"
} | {
"index": "2048",
"split": "train"
} |
math | [
{
"content": "During a break, a fly entered the math classroom and began crawling on a poster depicting the graph of a quadratic function \\( y = f(x) \\) with a leading coefficient of -1. Initially, the fly moved exactly along the parabola to the point with an abscissa of 2, but then started moving along a str... | {
"ground_truth": "6070",
"style": "rule"
} | {
"index": "2049",
"split": "train"
} |
math | [
{
"content": "There are two circles: one with center at point \\( A \\) and radius 6, and the other with center at point \\( B \\) and radius 3. Their common internal tangent touches the circles respectively at points \\( C \\) and \\( D \\). The lines \\( AB \\) and \\( CD \\) intersect at point \\( E \\). Fin... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "2050",
"split": "train"
} |
math | [
{
"content": "The equation of the line joining the complex numbers $-1 + 2i$ and $2 + 3i$ can be expressed in the form\n\\[az + b \\overline{z} = d\\]for some complex numbers $a$, $b$, and real number $d$. Find the product $ab$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "2051",
"split": "train"
} |
math | [
{
"content": "The operation $\\Diamond$ is defined by $a\\Diamond b=ab^2-b+1$. What is the value of $(-1)\\Diamond 6$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-41",
"style": "rule"
} | {
"index": "2052",
"split": "train"
} |
math | [
{
"content": "Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. What is the probability that the four chords form a convex quadrilateral? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{91}",
"style": "rule"
} | {
"index": "2053",
"split": "train"
} |
math | [
{
"content": "Compute the argument of the sum:\n\\[ e^{2\\pi i/40} + e^{6\\pi i/40} + e^{10\\pi i/40} + e^{14\\pi i/40} + e^{18\\pi i/40} + e^{22\\pi i/40} + e^{26\\pi i/40} + e^{30\\pi i/40} + e^{34\\pi i/40} + e^{38\\pi i/40} \\]\nand express it in the form \\( r e^{i \\theta} \\), where \\( 0 \\le \\theta < ... | {
"ground_truth": "\\frac{\\pi}{2}",
"style": "rule"
} | {
"index": "2054",
"split": "train"
} |
math | [
{
"content": "From the $8$ vertices of a cube, select any $4$ vertices. The probability that these $4$ points lie in the same plane is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{6}{35}",
"style": "rule"
} | {
"index": "2055",
"split": "train"
} |
math | [
{
"content": "The repeating decimals $0.abab\\overline{ab}$ and $0.abcabc\\overline{abc}$ satisfy\n\\[0.abab\\overline{ab}+0.abcabc\\overline{abc}=\\frac{33}{37},\\]\nwhere $a$, $b$, and $c$ are (not necessarily distinct) digits. Find the three digit number $abc$. Please output the final answer within \\boxed{}... | {
"ground_truth": "447",
"style": "rule"
} | {
"index": "2056",
"split": "train"
} |
math | [
{
"content": "Form five-digit numbers without repeating digits using the numbers \\\\(0\\\\), \\\\(1\\\\), \\\\(2\\\\), \\\\(3\\\\), and \\\\(4\\\\).\n\n\\\\((\\\\)I\\\\()\\\\) How many of these five-digit numbers are even?\n\n\\\\((\\\\)II\\\\()\\\\) How many of these five-digit numbers are less than \\\\(3200... | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "2057",
"split": "train"
} |
math | [
{
"content": "Given the curve $f(x)=x^2-2x$, find the slope angle of the tangent line at the point $(\\frac{3}{2},f(\\frac{3}{2}))$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "2058",
"split": "train"
} |
math | [
{
"content": "For what values of $x$ is $\\frac{\\log{(3-x)}}{\\sqrt{x-1}}$ defined? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(1,3)",
"style": "rule"
} | {
"index": "2059",
"split": "train"
} |
math | [
{
"content": "An isosceles trapezoid has legs of length 30 cm each, two diagonals of length 40 cm each and the longer base is 50 cm. What is the trapezoid's area in sq cm? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "768",
"style": "rule"
} | {
"index": "2060",
"split": "train"
} |
math | [
{
"content": "The even throws belong to B. Bons. Therefore, Bons wins only when the total number of throws, including the last successful one, is even. The probability of rolling a six is $\\frac{1}{6}$. The probability of the opposite event is $\\frac{5}{6}$. Hence, the probability that the total number of thr... | {
"ground_truth": "\\frac{5}{11}",
"style": "rule"
} | {
"index": "2061",
"split": "train"
} |
math | [
{
"content": "Given that $f(x)$ is a function defined on $\\mathbb{R}$, and for any $x \\in \\mathbb{R}$, it holds that $f(x+2) = f(2-x) + 4f(2)$, if the graph of the function $y=f(x+1)$ is symmetric about the point $(-1,0)$ and $f(1)=3$, then find $f(2015)$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "2062",
"split": "train"
} |
math | [
{
"content": "It is known that \\( x = 2a^{5} = 5b^{2} > 0 \\), where \\( a \\) and \\( b \\) are integers. What is the smallest possible value of \\( x \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "200000",
"style": "rule"
} | {
"index": "2063",
"split": "train"
} |
math | [
{
"content": "The six-digit number $20210A$ is prime for only one digit $A.$ What is $A?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "2064",
"split": "train"
} |
math | [
{
"content": "A cylindrical can has a circumference of 24 inches and a height of 7 inches. A spiral strip is painted on the can such that it winds around the can precisely once, reaching from the bottom to the top. However, instead of reaching directly above where it started, it ends 3 inches horizontally to th... | {
"ground_truth": "\\sqrt{778}",
"style": "rule"
} | {
"index": "2065",
"split": "train"
} |
math | [
{
"content": "In how many ways can the number 210 be factored into a product of four natural numbers? The order of the factors does not matter. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "2066",
"split": "train"
} |
math | [
{
"content": "On the altitude \\( BH \\) of triangle \\( ABC \\) a certain point \\( D \\) is marked. Line \\( AD \\) intersects side \\( BC \\) at point \\( E \\), and line \\( CD \\) intersects side \\( AB \\) at point \\( F \\). Points \\( G \\) and \\( J \\) are the projections of points \\( F \\) and \\( E... | {
"ground_truth": "\\sqrt{2} : 1",
"style": "rule"
} | {
"index": "2067",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be an acute triangle with circumcircle $\\omega$ . Let $D$ and $E$ be the feet of the altitudes from $B$ and $C$ onto sides $AC$ and $AB$ , respectively. Lines $BD$ and $CE$ intersect $\\omega$ again at points $P \\neq B$ and $Q \\neq C$ . Suppose that $PD=3$ , $... | {
"ground_truth": "\\sqrt{23}",
"style": "rule"
} | {
"index": "2068",
"split": "train"
} |
math | [
{
"content": "It is known that $F_1$ and $F_2$ are the upper and lower foci of the ellipse $C: \\frac {y^{2}}{a^{2}}+ \\frac {x^{2}}{b^{2}}=1$ ($a>b>0$), where $F_1$ is also the focus of the parabola $C_1: x^{2}=4y$. Point $M$ is the intersection of $C_1$ and $C_2$ in the second quadrant, and $|MF_{1}|= \\frac ... | {
"ground_truth": "2 \\sqrt {6}",
"style": "rule"
} | {
"index": "2069",
"split": "train"
} |
math | [
{
"content": "Jamie King invested some money in real estate and mutual funds. The total amount he invested was $\\$200,\\!000$. If he invested 5.5 times as much in real estate as he did in mutual funds, what was his total investment in real estate? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "169,230.77",
"style": "rule"
} | {
"index": "2070",
"split": "train"
} |
math | [
{
"content": "The function $f(x)$ satisfies\n\\[f(xy) = \\frac{f(x)}{y}\\]for all positive real numbers $x$ and $y.$ If $f(30) = 20,$ find $f(40).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "2071",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}=(\\cos α,\\sin α)$, $\\overrightarrow{b}=(\\cos β,\\sin β)$, and $|\\overrightarrow{a}- \\overrightarrow{b}|= \\frac {4 \\sqrt {13}}{13}$.\n(1) Find the value of $\\cos (α-β)$;\n(2) If $0 < α < \\frac {π}{2}$, $- \\frac {π}{2} < β < 0$, and $\\sin β=- \\frac {4}{... | {
"ground_truth": "\\frac {16}{65}",
"style": "rule"
} | {
"index": "2072",
"split": "train"
} |
math | [
{
"content": "To qualify for the competition, wrestler Vladimir must have three matches and win at least two of them consecutively. His opponents are Andrei (A) and Boris (B). Vladimir can choose to schedule the matches in the order ABA or BAB. The probability of Vladimir losing one match to Boris is 0.3, and t... | {
"ground_truth": "0.742",
"style": "rule"
} | {
"index": "2073",
"split": "train"
} |
math | [
{
"content": "The common ratio of the geometric sequence \\( a + \\log_{2} 3, a + \\log_{4} 3, a + \\log_{8} 3 \\) is? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "2074",
"split": "train"
} |
math | [
{
"content": "A group of 50 girls sit in a circle. They throw a ball around the circle. Each girl who gets the ball throws it to the girl sitting six places anticlockwise from her, who catches it. Freda catches the ball 100 times. In that time, how many girls never get to catch the ball? Please output the final... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "2075",
"split": "train"
} |
math | [
{
"content": "Given a triangle \\(ABC\\) with sides opposite to the angles \\(A\\), \\(B\\), and \\(C\\) being \\(a\\), \\(b\\), and \\(c\\) respectively, and it is known that \\( \\sqrt {3}\\sin A-\\cos (B+C)=1\\) and \\( \\sin B+\\sin C= \\dfrac {8}{7}\\sin A\\) with \\(a=7\\):\n(Ⅰ) Find the value of angle \\... | {
"ground_truth": "\\dfrac {15 \\sqrt {3}}{4}",
"style": "rule"
} | {
"index": "2076",
"split": "train"
} |
math | [
{
"content": "Solve the equation:\n\n$$\n\\begin{gathered}\n\\frac{10}{x+10}+\\frac{10 \\cdot 9}{(x+10)(x+9)}+\\frac{10 \\cdot 9 \\cdot 8}{(x+10)(x+9)(x+8)}+\\cdots+ \\\\\n+\\frac{10 \\cdot 9 \\ldots 2 \\cdot 1}{(x+10)(x+9) \\ldots(x+1)}=11\n\\end{gathered}\n$$ Please output the final answer within \\boxed{}.",... | {
"ground_truth": "-\\frac{1}{11}",
"style": "rule"
} | {
"index": "2077",
"split": "train"
} |
math | [
{
"content": "Let $\\varphi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Let $S$ be the set of positive integers $n$ such that $\\frac{2 n}{\\varphi(n)}$ is an integer. Compute the sum $\\sum_{n \\in S} \\frac{1}{n}$. Please output the final answer with... | {
"ground_truth": "\\frac{10}{3}",
"style": "rule"
} | {
"index": "2078",
"split": "train"
} |
math | [
{
"content": "In the center of a circular field, there is a house of geologists. Eight straight roads emanate from the house, dividing the field into 8 equal sectors. Two geologists set off on a journey from their house at a speed of 4 km/h choosing any road randomly. Determine the probability that the distance... | {
"ground_truth": "0.375",
"style": "rule"
} | {
"index": "2079",
"split": "train"
} |
math | [
{
"content": "A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is $\\frac{1}{3}$. When $4$ black cards are added to the deck, the probability of choosing red becomes $\\frac{1}{4}$. How many cards were in the deck originally? Please output the final answer ... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "2080",
"split": "train"
} |
math | [
{
"content": "How many non-empty subsets $S$ of $\\{1, 2, 3, \\ldots, 10\\}$ satisfy the following two properties?\n1. No two consecutive integers belong to $S$.\n2. If $S$ contains $k$ elements, then $S$ contains no number less than $k$. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "143",
"style": "rule"
} | {
"index": "2081",
"split": "train"
} |
math | [
{
"content": "An ellipse C is defined by the equation $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}=1$$ (a > b > 0), and its four vertices form a rhombus with side length $$\\sqrt {3}$$ and area $$2 \\sqrt {2}$$.\n(1) Find the equation of ellipse C.\n(2) A line l passes through point Q(0, -2) and intersects el... | {
"ground_truth": "\\frac {4 \\sqrt {21}}{11}",
"style": "rule"
} | {
"index": "2082",
"split": "train"
} |
math | [
{
"content": "Given that the angle between the generating line and the axis of a cone is $\\frac{\\pi}{3}$, and the length of the generating line is $3$, find the maximum value of the cross-sectional area through the vertex. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{2}",
"style": "rule"
} | {
"index": "2083",
"split": "train"
} |
math | [
{
"content": "A square 10cm on each side has four quarter circles drawn with centers at the four corners. How many square centimeters are in the area of the shaded region? Express your answer in terms of $\\pi$.\n\n[asy]\nunitsize (1.5 cm);\n\ndraw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle);\nfilldraw(arc((1,1),1,2... | {
"ground_truth": "100-25\\pi",
"style": "rule"
} | {
"index": "2084",
"split": "train"
} |
math | [
{
"content": "Count the number of permutations $a_{1} a_{2} \\ldots a_{7}$ of 1234567 with longest decreasing subsequence of length at most two (i.e. there does not exist $i<j<k$ such that $a_{i}>a_{j}>a_{k}$ ). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "429",
"style": "rule"
} | {
"index": "2085",
"split": "train"
} |
math | [
{
"content": "The quadratic $4x^2 - 40x + 100$ can be written in the form $(ax+b)^2 + c$, where $a$, $b$, and $c$ are constants. What is $2b-3c$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-20",
"style": "rule"
} | {
"index": "2086",
"split": "train"
} |
math | [
{
"content": "Chandra has four bowls. Each one is a different color (red, blue, yellow, green). She also has exactly one glass the same color as each bowl. If she chooses a bowl and a glass from the cupboard, how many pairings are possible? One such pairing is a blue bowl and a yellow glass. Please output t... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "2087",
"split": "train"
} |
math | [
{
"content": "Spinners $A$ and $B$ are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even? Express your answer as a common fraction. [asy]\n/* AMC8 2004 #21 Problem */\ndraw(circle((0,0), 2));\ndraw(circle((6,... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "2088",
"split": "train"
} |
math | [
{
"content": "In the rhombus \\(ABCD\\), point \\(Q\\) divides side \\(BC\\) in the ratio \\(1:3\\) starting from vertex \\(B\\), and point \\(E\\) is the midpoint of side \\(AB\\). It is known that the median \\(CF\\) of triangle \\(CEQ\\) is equal to \\(2\\sqrt{2}\\), and \\(EQ = \\sqrt{2}\\). Find the radius... | {
"ground_truth": "\\frac{\\sqrt{7}}{2}",
"style": "rule"
} | {
"index": "2089",
"split": "train"
} |
math | [
{
"content": "Let $\\{a_k\\}$ be a sequence of integers such that $a_1=1$ and $a_{m+n}=a_m+a_n+mn,$ for all positive integers $m$ and $n.$ Find $a_{12}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "78",
"style": "rule"
} | {
"index": "2090",
"split": "train"
} |
math | [
{
"content": "Given a biased coin with probabilities of $\\frac{3}{4}$ for heads and $\\frac{1}{4}$ for tails, and outcomes of tosses being independent, calculate the probabilities of winning Game A and Game B. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "2091",
"split": "train"
} |
math | [
{
"content": "Suppose that $\\triangle{ABC}$ is an equilateral triangle of side length $s$, with the property that there is a unique point $P$ inside the triangle such that $AP=1$, $BP=\\sqrt{3}$, and $CP=2$. What is $s$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{7}",
"style": "rule"
} | {
"index": "2092",
"split": "train"
} |
math | [
{
"content": "Write the number in the form of a fraction (if possible):\n\n$$\nx=0.5123412341234123412341234123412341234 \\ldots\n$$\n\nCan you generalize this method to all real numbers with a periodic decimal expansion? And conversely? Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "\\frac{51229}{99990}",
"style": "rule"
} | {
"index": "2093",
"split": "train"
} |
math | [
{
"content": "Compute $7\\left(\\frac{1}{5}\\right)^3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{125}",
"style": "rule"
} | {
"index": "2094",
"split": "train"
} |
math | [
{
"content": "If the acute angle $\\theta$ satisfies $\\sin (\\pi \\cos \\theta) = \\cos (\\pi \\sin \\theta)$, then what is $\\sin 2\\theta$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "2095",
"split": "train"
} |
math | [
{
"content": "Suppose that the graph of \\[2x^2 + y^2 + 8x - 10y + c = 0\\]consists of a single point. (In this case, we call the graph a degenerate ellipse.) Find $c.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "2096",
"split": "train"
} |
math | [
{
"content": "Let $\\triangle ABC$ be an acute isosceles triangle with circumcircle $\\omega$. The tangents to $\\omega$ at vertices $B$ and $C$ intersect at point $T$. Let $Z$ be the projection of $T$ onto $BC$. Assume $BT = CT = 20$, $BC = 24$, and $TZ^2 + 2BZ \\cdot CZ = 478$. Find $BZ \\cdot CZ$. Please out... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "2097",
"split": "train"
} |
math | [
{
"content": "Given that the random variable $X$ follows a normal distribution $N(1,4)$, and $P(0 \\leq X \\leq 2) = 0.68$, find $P(X > 2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.16",
"style": "rule"
} | {
"index": "2098",
"split": "train"
} |
math | [
{
"content": "One of the three cards had the number 18, another had the number 75, and the third had some two-digit number. If you sum all the distinct six-digit numbers that can be obtained by arranging these cards in a row, you get the number 2606058. What number is written on the third card? Please output th... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "2099",
"split": "train"
} |
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