data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Each segment whose ends are vertices of a regular 100-sided polygon is colored - in red if there are an even number of vertices between its ends, and in blue otherwise (in particular, all sides of the 100-sided polygon are red). Numbers are placed at the vertices, the sum of the squares of which i... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "33300",
"split": "train"
} |
math | [
{
"content": "When $8000^{50}$ is expanded out, the result is $1$ followed by how many zeros? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "33301",
"split": "train"
} |
math | [
{
"content": "Let $A B C D$ be an isosceles trapezoid with $A D=B C=255$ and $A B=128$. Let $M$ be the midpoint of $C D$ and let $N$ be the foot of the perpendicular from $A$ to $C D$. If $\\angle M B C=90^{\\circ}$, compute $\\tan \\angle N B M$. Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "\\frac{120}{353}",
"style": "rule"
} | {
"index": "33302",
"split": "train"
} |
math | [
{
"content": "Point $M$ lies on the diagonal $BD$ of parallelogram $ABCD$ such that $MD = 3BM$ . Lines $AM$ and $BC$ intersect in point $N$ . What is the ratio of the area of triangle $MND$ to the area of parallelogram $ABCD$ ? Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "3/8",
"style": "rule"
} | {
"index": "33303",
"split": "train"
} |
math | [
{
"content": "The wavelength of red light that the human eye can see is $0.000077$ cm. Please round the data $0.000077$ to $0.00001$ and express it in scientific notation as ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8 \\times 10^{-5}",
"style": "rule"
} | {
"index": "33304",
"split": "train"
} |
math | [
{
"content": "When manufacturing a steel cable, it was found that the cable has the same length as the curve given by the system of equations:\n\n$$\n\\left\\{\\begin{array}{l}\nx+y+z=10 \\\\\nx y+y z+x z=-22\n\\end{array}\\right.\n$$\n\nFind the length of the cable. Please output the final answer within \\boxe... | {
"ground_truth": "4 \\pi \\sqrt{\\frac{83}{3}}",
"style": "rule"
} | {
"index": "33305",
"split": "train"
} |
math | [
{
"content": "A right cylinder with a height of 5 inches has a radius of 2 inches. What is the area of the curved surface of the cylinder, in square inches? Express your answer in terms of $\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20\\pi",
"style": "rule"
} | {
"index": "33306",
"split": "train"
} |
math | [
{
"content": "There are $10$ girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "512",
"style": "rule"
} | {
"index": "33307",
"split": "train"
} |
math | [
{
"content": "Given the function $y=\\cos (x+\\frac{π}{3})$, determine the horizontal shift of the graph of the function $y=\\sin x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5\\pi}{6}",
"style": "rule"
} | {
"index": "33308",
"split": "train"
} |
math | [
{
"content": "At the mountain hut, the coach said, \"If we continue at this comfortable pace of $4 \\mathrm{~km}$ per hour, we will arrive at the station 45 minutes after the departure of our train.\"\n\nThen he pointed to a group that had just passed us: \"They are using poles, and therefore achieve an average... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "33309",
"split": "train"
} |
math | [
{
"content": "Find the number of ordered pairs of integers $(a, b)$ such that the sequence\\[3, 4, 5, a, b, 30, 40, 50\\]is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "228",
"style": "rule"
} | {
"index": "33310",
"split": "train"
} |
math | [
{
"content": "In a small town, the police are looking for a wanderer. There is a four in five chance that he is in one of the eight bars in the town, with no preference for any particular one. Two officers visited seven bars but did not find the wanderer. What are the chances of finding him in the eighth bar? P... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "33311",
"split": "train"
} |
math | [
{
"content": "For a real number \\( x \\), \\([x]\\) denotes the greatest integer less than or equal to \\( x \\). Given a sequence of positive numbers \\( \\{a_n\\} \\) such that \\( a_1 = 1 \\) and \\( S_n = \\frac{1}{2} \\left( a_n + \\frac{1}{a_n} \\right) \\), where \\( S_n \\) is the sum of the first \\( ... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "33312",
"split": "train"
} |
math | [
{
"content": "Jerry cuts a wedge from a 6-cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "603",
"style": "rule"
} | {
"index": "33313",
"split": "train"
} |
math | [
{
"content": "The sum of three numbers $a$, $b$, and $c$ is 99. If we increase $a$ by 6, decrease $b$ by 6 and multiply $c$ by 5, the three resulting numbers are equal. What is the value of $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "33314",
"split": "train"
} |
math | [
{
"content": "Find maximal positive integer $p$ such that $5^7$ is sum of $p$ consecutive positive integers Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "33315",
"split": "train"
} |
math | [
{
"content": "A regular decagon $A_{0} A_{1} A_{2} \\cdots A_{9}$ is given in the plane. Compute $\\angle A_{0} A_{3} A_{7}$ in degrees. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "54^{\\circ}",
"style": "rule"
} | {
"index": "33316",
"split": "train"
} |
math | [
{
"content": "The largest whole number such that seven times the number is less than 100 is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "33317",
"split": "train"
} |
math | [
{
"content": "Given that $f(x+6) + f(x-6) = f(x)$ for all real $x$, determine the least positive period $p$ for these functions. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "33318",
"split": "train"
} |
math | [
{
"content": "In a new diagram, $A$ is the center of a circle with radii $AB=AC=8$. The sector $BOC$ is shaded except for a triangle $ABC$ within it, where $B$ and $C$ lie on the circle. If the central angle of $BOC$ is $240^\\circ$, what is the perimeter of the shaded region? Please output the final answer wit... | {
"ground_truth": "16 + \\frac{32}{3}\\pi",
"style": "rule"
} | {
"index": "33319",
"split": "train"
} |
math | [
{
"content": "Suppose that \\( a^3 \\) varies inversely with \\( b^2 \\). If \\( a = 5 \\) when \\( b = 2 \\), find the value of \\( a \\) when \\( b = 8 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2.5",
"style": "rule"
} | {
"index": "33320",
"split": "train"
} |
math | [
{
"content": "Given a triangle $ABC$ with internal angles $A$, $B$, $C$ opposite to sides $a$, $b$, $c$ respectively, and $A=2C$.\n\n(Ⅰ) If $\\triangle ABC$ is an acute triangle, find the range of $\\frac{a}{c}$.\n\n(Ⅱ) If $b=1, c=3$, find the area of $\\triangle ABC$. Please output the final answer within \\bo... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "33321",
"split": "train"
} |
math | [
{
"content": "Express $0.5\\overline{023}$ as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1045}{1998}",
"style": "rule"
} | {
"index": "33322",
"split": "train"
} |
math | [
{
"content": "In the $3 \times 3$ grid shown, the central square contains the integer 5. The remaining eight squares contain $a, b, c, d, e, f, g, h$, which are each to be replaced with an integer from 1 to 9, inclusive. Integers can be repeated. There are $N$ ways to complete the grid so that the sums of the i... | {
"ground_truth": "73",
"style": "rule"
} | {
"index": "33323",
"split": "train"
} |
math | [
{
"content": "Expand the following expression: $(9x+4)\\cdot 2x^2$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18x^3+8x^2",
"style": "rule"
} | {
"index": "33324",
"split": "train"
} |
math | [
{
"content": "Liquid $X$ does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film $0.1$cm thick. A rectangular box measuring $6$cm by $3$cm by $12$cm is filled with liquid $X$. Its contents are poured onto a large body of water. What will be the radius, in centi... | {
"ground_truth": "\\sqrt{\\frac{2160}{\\pi}}",
"style": "rule"
} | {
"index": "33325",
"split": "train"
} |
math | [
{
"content": "Find the common ratio of the infinite geometric series: $$\\frac{-4}{7}+\\frac{14}{3}+\\frac{-98}{9} + \\dots$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{49}{6}",
"style": "rule"
} | {
"index": "33326",
"split": "train"
} |
math | [
{
"content": "Given that $\\sin\\alpha = \\frac{3}{5}$, and $\\alpha \\in \\left(\\frac{\\pi}{2}, \\pi \\right)$. \n(1) Find the value of $\\tan\\left(\\alpha+\\frac{\\pi}{4}\\right)$; \n(2) If $\\beta \\in (0, \\frac{\\pi}{2})$, and $\\cos(\\alpha-\\beta) = \\frac{1}{3}$, find the value of $\\cos\\beta$. Pleas... | {
"ground_truth": "\\frac{6\\sqrt{2} - 4}{15}",
"style": "rule"
} | {
"index": "33327",
"split": "train"
} |
math | [
{
"content": "A point moving in the positive direction of the $OX$ axis has its horizontal coordinate given by $x(t) = 5(t + 1)^2 + \\frac{a}{(t + 1)^5}$, where $a$ is a positive constant. Find the minimum value of $a$ such that $x(t) \\geqslant 24$ for all $t \\geqslant 0$. Please output the final answer withi... | {
"ground_truth": "2 \\sqrt{\\left( \\frac{24}{7} \\right)^{7}}",
"style": "rule"
} | {
"index": "33328",
"split": "train"
} |
math | [
{
"content": "Complex numbers $a,$ $b,$ and $c$ are zeros of a polynomial $P(z) = z^3 + qz + r,$ and $|a|^2 + |b|^2 + |c|^2 = 250.$ The points corresponding to $a,$ $b,$ and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h.$ Find $h^2.$ Please output the final answer within \\box... | {
"ground_truth": "375",
"style": "rule"
} | {
"index": "33329",
"split": "train"
} |
math | [
{
"content": "If $x, y, z \\in \\mathbb{R}$ are solutions to the system of equations $$ \\begin{cases}\nx - y + z - 1 = 0\nxy + 2z^2 - 6z + 1 = 0\n\\end{cases} $$ what is the greatest value of $(x - 1)^2 + (y + 1)^2$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "33330",
"split": "train"
} |
math | [
{
"content": "$-2^{3}+|2-3|-2\\times \\left(-1\\right)^{2023}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-5",
"style": "rule"
} | {
"index": "33331",
"split": "train"
} |
math | [
{
"content": "If \\( x \\) and \\( y \\) are real numbers such that \\( x + y = 4 \\) and \\( xy = -2 \\), then the value of \\( x + \\frac{x^3}{y^2} + \\frac{y^3}{x^2} + y \\) is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "440",
"style": "rule"
} | {
"index": "33332",
"split": "train"
} |
math | [
{
"content": "The Houson Association of Mathematics Educators decides to hold a grand forum on mathematics education and invites a number of politicians from the United States to participate. Around lunch time the politicians decide to play a game. In this game, players can score 19 points for pegging the coord... | {
"ground_truth": "1209",
"style": "rule"
} | {
"index": "33333",
"split": "train"
} |
math | [
{
"content": "For an integer $n \\geq 0$, let $f(n)$ be the smallest possible value of $|x+y|$, where $x$ and $y$ are integers such that $3 x-2 y=n$. Evaluate $f(0)+f(1)+f(2)+\\cdots+f(2013)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2416",
"style": "rule"
} | {
"index": "33334",
"split": "train"
} |
math | [
{
"content": "In an isosceles triangle $ABC$ with $AB = AC = 6$ units and $BC = 5$ units, a point $P$ is randomly selected inside the triangle $ABC$. What is the probability that $P$ is closer to vertex $C$ than to either vertex $A$ or vertex $B$? Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "33335",
"split": "train"
} |
math | [
{
"content": "Place parentheses and operation signs in the sequence 22222 so that the result is 24. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(2+2+2) \\times (2+2)",
"style": "rule"
} | {
"index": "33336",
"split": "train"
} |
math | [
{
"content": "Compute the number of distinct pairs of the form (first three digits of $x$, first three digits of $x^{4}$ ) over all integers $x>10^{10}$. For example, one such pair is $(100,100)$ when $x=10^{10^{10}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4495",
"style": "rule"
} | {
"index": "33337",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be a triangle with area $5$ and $BC = 10.$ Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G.$ Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2+AC^2.$ *Proposed by... | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "33338",
"split": "train"
} |
math | [
{
"content": "Determine by how many times the number \\((2014)^{2^{2014}} - 1\\) is greater than the number written in the following form:\n\\[\n\\left(\\left((2014)^{2^0} + 1\\right) \\cdot \\left((2014)^{2^1} + 1\\right) \\cdot \\left((2014)^{2^2} + 1\\right) \\ldots \\cdot \\left((2014)^{2^{2013}} + 1\\right... | {
"ground_truth": "2013",
"style": "rule"
} | {
"index": "33339",
"split": "train"
} |
math | [
{
"content": "Given a positive integer \\( A \\) whose prime factorization can be written as \\( A = 2^{\\alpha} \\times 3^{\\beta} \\times 5^{\\gamma} \\), where \\(\\alpha, \\beta, \\gamma\\) are natural numbers. If one-half of \\( A \\) is a perfect square, one-third of \\( A \\) is a perfect cube, and one-f... | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "33340",
"split": "train"
} |
math | [
{
"content": "How many two-digit numbers have digits whose sum is either a perfect square or a prime number up to 25? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "41",
"style": "rule"
} | {
"index": "33341",
"split": "train"
} |
math | [
{
"content": "Let $n$ be a positive integer and let $d_{1},d_{2},,\\ldots ,d_{k}$ be its divisors, such that $1=d_{1}<d_{2}<\\ldots <d_{k}=n$ . Find all values of $n$ for which $k\\geq 4$ and $n=d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}$ . Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "130",
"style": "rule"
} | {
"index": "33342",
"split": "train"
} |
math | [
{
"content": "If\n\\[\n(1 + \\tan 1^\\circ)(1 + \\tan 2^\\circ)(1 + \\tan 3^\\circ) \\dotsm (1 + \\tan 89^\\circ) = 2^m,\n\\]\nthen find $m.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "33343",
"split": "train"
} |
math | [
{
"content": "In a regular hexagon divided into 6 regions, plant ornamental plants such that the same type of plant is planted within one region, and different types of plants are planted in adjacent regions. There are 4 different types of plants available. How many planting schemes are possible? Please output ... | {
"ground_truth": "732",
"style": "rule"
} | {
"index": "33344",
"split": "train"
} |
math | [
{
"content": "We call a pair of natural numbers \\((a, p)\\) good if the number \\(a^3 + p^3\\) is divisible by \\(a^2 - p^2\\), with \\(a > p\\).\n\n(a) (1 point) Specify any possible value of \\(a\\) for which the pair \\((a, 13)\\) is good.\n\n(b) (3 points) Find the number of good pairs for which \\(p\\) is... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "33345",
"split": "train"
} |
math | [
{
"content": "Compute and simplify the following expressions:\n1. $(1)(1 \\frac{1}{2})^{0}-(1-0.5^{-2})÷(\\frac{27}{8})^{\\frac{2}{3}}$\n2. $\\sqrt{2 \\sqrt{2 \\sqrt{2}}}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2^{\\frac{7}{8}}",
"style": "rule"
} | {
"index": "33346",
"split": "train"
} |
math | [
{
"content": "Let \\( x \\) be a non-zero real number such that \n\n\\[ \\sqrt[5]{x^{3}+20 x}=\\sqrt[3]{x^{5}-20 x} \\]. \n\nFind the product of all possible values of \\( x \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-5",
"style": "rule"
} | {
"index": "33347",
"split": "train"
} |
math | [
{
"content": "Among the numbers $1, 2, 3, \\cdots, 50$, if 10 consecutive numbers are selected, what is the probability that exactly 3 of them are prime numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22/41",
"style": "rule"
} | {
"index": "33348",
"split": "train"
} |
math | [
{
"content": "The number of trailing zeros in 2006! is to be calculated. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "500",
"style": "rule"
} | {
"index": "33349",
"split": "train"
} |
math | [
{
"content": "What is $a-2b$, where $a=4-2i$ and $b=3+2i$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2-6i",
"style": "rule"
} | {
"index": "33350",
"split": "train"
} |
math | [
{
"content": "In a trapezoid \\(ABCD\\) with bases \\(AD=12\\) and \\(BC=8\\), circles constructed on the sides \\(AB\\), \\(BC\\), and \\(CD\\) as diameters intersect at one point. The length of diagonal \\(AC\\) is 12. Find the length of \\(BD\\). Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "33351",
"split": "train"
} |
math | [
{
"content": "Each of the cells of a $7 \\times 7$ grid is painted with a color chosen randomly and independently from a set of $N$ fixed colors. Call an edge hidden if it is shared by two adjacent cells in the grid that are painted the same color. Determine the least $N$ such that the expected number of ... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "33352",
"split": "train"
} |
math | [
{
"content": "For a function $f(x)$ with domain $I$, if there exists an interval $\\left[m,n\\right]\\subseteq I$ such that $f(x)$ is a monotonic function on the interval $\\left[m,n\\right]$, and the range of the function $y=f(x)$ for $x\\in \\left[m,n\\right]$ is $\\left[m,n\\right]$, then the interval $\\lef... | {
"ground_truth": "\\frac{2\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "33353",
"split": "train"
} |
math | [
{
"content": "A conservatory houses five pairs of different animals, one male and one female of each type. The feeder must alternate between feeding a male and a female each time. If the feeder begins by feeding a female hippopotamus, how many ways can the feeder complete feeding all the animals? Please output ... | {
"ground_truth": "2880",
"style": "rule"
} | {
"index": "33354",
"split": "train"
} |
math | [
{
"content": "Using the data presented in the chart, what was the average daily high temperature in Brixton from September 15th, 2008 to September 21st, 2008, inclusive? The daily high temperatures in degrees Fahrenheit were recorded as 51, 64, 61, 59, 48, 63, and 55. Please output the final answer within \\box... | {
"ground_truth": "57.3",
"style": "rule"
} | {
"index": "33355",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = \\begin{cases} x-5, & x\\geq 2000 \\\\ f[f(x+8)], & x<2000 \\end{cases}$, calculate $f(1996)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2002",
"style": "rule"
} | {
"index": "33356",
"split": "train"
} |
math | [
{
"content": "Compute the product\n\\[\n\\prod_{n = 1}^{15} \\frac{n^2 + 5n + 6}{n+2}.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{18!}{6}",
"style": "rule"
} | {
"index": "33357",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, the angles $\\angle B = 30^\\circ$ and $\\angle A = 90^\\circ$ are known. Point $K$ is marked on side $AC$, and points $L$ and $M$ are marked on side $BC$ such that $KL = KM$ (point $L$ is on segment $BM$).\n\nFind the length of segment $LM$, given that $AK = 4$, $BL = 31$, and ... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "33358",
"split": "train"
} |
math | [
{
"content": "Two spheres touch the plane of triangle \\(ABC\\) at points \\(B\\) and \\(C\\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 12, and the distance between their centers is \\(4 \\sqrt{29}\\). The center of a third sphere with radius 8 is at point \\(A\\)... | {
"ground_truth": "4\\sqrt{5}",
"style": "rule"
} | {
"index": "33359",
"split": "train"
} |
math | [
{
"content": "As usual, let \\( n! \\) denote the product of the integers from 1 to \\( n \\) inclusive. Determine the largest integer \\( m \\) such that \\( m! \\) divides \\( 100! + 99! + 98! \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "98",
"style": "rule"
} | {
"index": "33360",
"split": "train"
} |
math | [
{
"content": "Convert from kilometers to miles. In the problem 3.125, the Fibonacci numeral system was introduced as being useful when converting distances from kilometers to miles or vice versa.\n\nSuppose we want to find out how many miles are in 30 kilometers. For this, we represent the number 30 in the Fibo... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "33361",
"split": "train"
} |
math | [
{
"content": "Given that line $l$ passes through point $P(-1,2)$ with a slope angle of $\\frac{2\\pi}{3}$, and the circle's equation is $\\rho=2\\cos (\\theta+\\frac{\\pi}{3})$:\n(1) Find the parametric equation of line $l$;\n(2) Let line $l$ intersect the circle at points $M$ and $N$, find the value of $|PM|\\... | {
"ground_truth": "6+2\\sqrt{3}",
"style": "rule"
} | {
"index": "33362",
"split": "train"
} |
math | [
{
"content": "Consider the hyperbola $x^{2}-y^{2}=8$ with left and right foci denoted as $F_{1}$ and $F_{2}$, respectively. Let $P_{n}(x_{n},y_{n})$ be a sequence of points on its right branch such that $|P_{n+1}F_{2}|=|P_{n}F_{1}|$ and $P_{1}F_{2} \\perp F_{1}F_{2}$. Determine the value of $x_{2016}$. Please o... | {
"ground_truth": "8064",
"style": "rule"
} | {
"index": "33363",
"split": "train"
} |
math | [
{
"content": "The coefficients of the polynomial\n\\[ a_{12} x^{12} + a_{11} x^{11} + \\dots + a_2 x^2 + a_1 x + a_0 = 0 \\]\nare all integers, and its roots $s_1, s_2, \\dots, s_{12}$ are all integers. Furthermore, the roots of the polynomial\n\\[ a_0 x^{12} + a_1 x^{11} + a_2 x^{10} + \\dots + a_{11} x + a_{1... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "33364",
"split": "train"
} |
math | [
{
"content": "Kimberly borrows $1000$ dollars from Lucy, who charged interest of $5\\%$ per month (which compounds monthly). What is the least integer number of months after which Kimberly will owe more than twice as much as she borrowed? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "33365",
"split": "train"
} |
math | [
{
"content": "Find the area of the region described by $x \\ge 0,$ $y \\ge 0,$ and\n\\[50 \\{x\\} \\ge \\lfloor x \\rfloor - \\lfloor y \\rfloor.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25.5",
"style": "rule"
} | {
"index": "33366",
"split": "train"
} |
math | [
{
"content": "Given the line $y=a (0 < a < 1)$ and the function $f(x)=\\sin \\omega x$ intersect at 12 points on the right side of the $y$-axis. These points are denoted as $(x\\_1)$, $(x\\_2)$, $(x\\_3)$, ..., $(x\\_{12})$ in order. It is known that $x\\_1= \\dfrac {\\pi}{4}$, $x\\_2= \\dfrac {3\\pi}{4}$, and ... | {
"ground_truth": "66\\pi",
"style": "rule"
} | {
"index": "33367",
"split": "train"
} |
math | [
{
"content": "Given that both $α$ and $β$ are acute angles, $\\cos α= \\frac {1}{7}$, and $\\cos (α+β)=- \\frac {11}{14}$, find the value of $\\cos β$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {1}{2}",
"style": "rule"
} | {
"index": "33368",
"split": "train"
} |
math | [
{
"content": "For a real number \\( x \\), let \\( [x] \\) denote the greatest integer that does not exceed \\( x \\). For a certain integer \\( k \\), there exist exactly 2008 positive integers \\( n_{1}, n_{2}, \\cdots, n_{2008} \\), such that \\( k=\\left[\\sqrt[3]{n_{1}}\\right]=\\left[\\sqrt[3]{n_{2}}\\rig... | {
"ground_truth": "668",
"style": "rule"
} | {
"index": "33369",
"split": "train"
} |
math | [
{
"content": "Find all functions $f:\\mathbb{R}^+ \\rightarrow \\mathbb{R}^+$, such that $$f(x^{2023}+f(x)f(y))=x^{2023}+yf(x)$$ for all $x, y>0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "f(x) = x",
"style": "rule"
} | {
"index": "33370",
"split": "train"
} |
math | [
{
"content": "If $5^x=100$, what is the value of $5^{x+2}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2500",
"style": "rule"
} | {
"index": "33371",
"split": "train"
} |
math | [
{
"content": "Three different integers are randomly chosen from the set $$\\{ -6, -3, 0, 2, 5, 7 \\}$$. What is the probability that their sum is even? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{19}{20}",
"style": "rule"
} | {
"index": "33372",
"split": "train"
} |
math | [
{
"content": "A circle $U$ has a circumference of $18\\pi$ inches, and segment $AB$ is a diameter. If the measure of angle $UAV$ is $45^{\\circ}$, what is the length, in inches, of segment $AV$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9\\sqrt{2 - \\sqrt{2}}",
"style": "rule"
} | {
"index": "33373",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ with vertices $A(1, -3)$, $B(-2, 0)$ and $C(4, 3)$ is reflected over the $y$-axis to form triangle $A'B'C'$. What is the length of a segment drawn from $A$ to $A'$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "33374",
"split": "train"
} |
math | [
{
"content": "A note contains three two-digit numbers that are said to form a sequence with a fourth number under a cryptic condition. The numbers provided are 46, 19, and 63, but the fourth number is unreadable. You know that the sum of the digits of all four numbers is $\\frac{1}{4}$ of the total sum of these... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "33375",
"split": "train"
} |
math | [
{
"content": "The quadrilateral $ABCD$ has the following equality $\\angle ABC=\\angle BCD=150^{\\circ}$. Moreover, $AB=18$ and $BC=24$, the equilateral triangles $\\triangle APB,\\triangle BQC,\\triangle CRD$ are drawn outside the quadrilateral. If $P(X)$ is the perimeter of the polygon $X$, then the following... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "33376",
"split": "train"
} |
math | [
{
"content": "The recruits were standing in a row, one behind the other, facing the same direction. Among them were three brothers: Peter, Nicholas, and Denis. There were 50 people ahead of Peter, 100 ahead of Nicholas, and 170 ahead of Denis. Upon the command \"About face!\", everyone turned to face the opposi... | {
"ground_truth": "211",
"style": "rule"
} | {
"index": "33377",
"split": "train"
} |
math | [
{
"content": "\\(a_n\\) is the last digit of \\(1 + 2 + \\ldots + n\\). Find \\(a_1 + a_2 + \\ldots + a_{1992}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6984",
"style": "rule"
} | {
"index": "33378",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be constants. Suppose that the equation \\[\\frac{(x+a)(x+b)(x+12)}{(x+3)^2} = 0\\]has exactly $3$ distinct roots, while the equation \\[\\frac{(x+2a)(x+3)(x+6)}{(x+b)(x+12)} = 0\\]has exactly $1$ distinct root. Compute $100a + b.$ Please output the final answer within \\boxed{}.",... | {
"ground_truth": "156",
"style": "rule"
} | {
"index": "33379",
"split": "train"
} |
math | [
{
"content": "Define the lengths of intervals $(m, n)$, $[m, n)$, $(m, n]$, and $[m, n]$ to be $n - m$ ($n, m \\in \\mathbf{R}$ and $n > m$). Find the sum of the lengths of the intervals for real numbers $x$ that satisfy the inequality\n\\[\n\\frac{1}{x-20}+\\frac{1}{x-17} \\geqslant \\frac{1}{512}\n\\] Please ... | {
"ground_truth": "1024",
"style": "rule"
} | {
"index": "33380",
"split": "train"
} |
math | [
{
"content": "Calculate the definite integral:\n\n$$\n\\int_{0}^{2 \\pi} \\sin ^{4} 3 x \\cos ^{4} 3 x \\, d x\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3\\pi}{64}",
"style": "rule"
} | {
"index": "33381",
"split": "train"
} |
math | [
{
"content": "If the complex number $z$ satisfies $z(1-i)=|1-i|+i$, then the imaginary part of $\\overline{z}$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\dfrac{\\sqrt{2}+1}{2}",
"style": "rule"
} | {
"index": "33382",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ and $BCFG$ be two faces of a cube with $AB=12$. A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P$, which is 7 units from $\\overline{BG}$ and 5 units from $\\overline{BC}$. The beam continues to be reflected off the faces of the cube. The length of the l... | {
"ground_truth": "230",
"style": "rule"
} | {
"index": "33383",
"split": "train"
} |
math | [
{
"content": "Evaluate the infinite geometric series:\n$$\\frac{5}{3} - \\frac{5}{4} + \\frac{25}{48} - \\frac{125}{384} + \\dots$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{20}{21}",
"style": "rule"
} | {
"index": "33384",
"split": "train"
} |
math | [
{
"content": "At the beginning of every period of British Literature, Mrs. Crabapple picks a random student to receive a crabapple as a gift, but really, as you might imagine, they are quite bitter and nasty. Given that there are 11 students in her class and her class meets four times a week, how many different... | {
"ground_truth": "14,\\!641",
"style": "rule"
} | {
"index": "33385",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, $c$, and the vectors $\\overrightarrow{m}=({\\cos C, \\cos({\\frac{\\pi}{2}-B})})$, $\\overrightarrow{n}=({\\cos({-4\\pi+B}), -\\sin C})$, and $\\overrightarrow{m} \\cdot \\overrightarrow{n}=-\\frac{\\sqrt{2... | {
"ground_truth": "5 + 2\\sqrt{2} + \\sqrt{5}",
"style": "rule"
} | {
"index": "33386",
"split": "train"
} |
math | [
{
"content": "Determine the value of \\( n \\) if we know that\n\n$$\n\\binom{n}{5}=\\frac{n(n-1)(n-2)(n-3)(n-4)}{2 \\cdot 3 \\cdot 4 \\cdot 5}\n$$\n\n(which, as we know, is an integer) in the decimal system is of the form \\(\\overline{ababa}\\), where \\( a \\) and \\( b \\) represent digits. Please output th... | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "33387",
"split": "train"
} |
math | [
{
"content": "Let $a, b, c, d$ be real numbers. Suppose that all the roots of $z^4+az^3+bz^2+cz+d=0$ are complex numbers lying on a circle in the complex plane centered at $0+0i$ and having radius $1$. The sum of the reciprocals of the roots is necessarily Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "$-a$",
"style": "rule"
} | {
"index": "33388",
"split": "train"
} |
math | [
{
"content": "For how many $n$ with $1 \\leq n \\leq 100$ can a unit square be divided into $n$ congruent figures? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "33389",
"split": "train"
} |
math | [
{
"content": "Ten families have an average of 2 children per family. If exactly two of these families are childless, what is the average number of children in the families with children? Express your answer as a decimal to the nearest tenth. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "2.5",
"style": "rule"
} | {
"index": "33390",
"split": "train"
} |
math | [
{
"content": "In a certain sequence the first term is $a_1=2007$ and the second term is $a_2=2008$. Furthermore, the values of the remaining terms are chosen so that $a_n+a_{n+1}+a_{n+2}=n$ for all $n\\ge 1$. Determine $a_{1000}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\mathbf{2340}",
"style": "rule"
} | {
"index": "33391",
"split": "train"
} |
math | [
{
"content": "For each positive integer $n$, let $k(n)$ be the number of ones in the binary representation of $2023 \\cdot n$. What is the minimum value of $k(n)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "33392",
"split": "train"
} |
math | [
{
"content": "In triangle \\( \\triangle ABC \\), the sides opposite to angles \\( A \\), \\( B \\), and \\( C \\) are \\( a \\), \\( b \\), and \\( c \\) respectively. If the sizes of angles \\( A \\), \\( B \\), and \\( C \\) form a geometric progression, and \\( b^2 - a^2 = ac \\), what is the radian measure... | {
"ground_truth": "\\frac{2 \\pi}{7}",
"style": "rule"
} | {
"index": "33393",
"split": "train"
} |
math | [
{
"content": "In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "33394",
"split": "train"
} |
math | [
{
"content": "Altitudes $\\overline{AP}$ and $\\overline{BQ}$ of an acute triangle $\\triangle ABC$ intersect at point $H$. If $HP=8$ and $HQ=3$, then calculate $(BP)(PC)-(AQ)(QC)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "55",
"style": "rule"
} | {
"index": "33395",
"split": "train"
} |
math | [
{
"content": "(1) Given $\\sin \\left( \\frac{\\pi }{3}-\\alpha \\right)=\\frac{1}{2}$, find the value of $\\cos \\left( \\frac{\\pi }{6}+\\alpha \\right)$;\n\n(2) Given $\\cos \\left( \\frac{5\\pi }{12}+\\alpha \\right)=\\frac{1}{3}$ and $-\\pi < \\alpha < -\\frac{\\pi }{2}$, find the value of $\\cos \\left( \... | {
"ground_truth": "- \\frac{1}{3}+ \\frac{2 \\sqrt{2}}{3}",
"style": "rule"
} | {
"index": "33396",
"split": "train"
} |
math | [
{
"content": "How many whole numbers between 1 and 500 do not contain the digit 2? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "323",
"style": "rule"
} | {
"index": "33397",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, $A'$, $B'$, and $C'$ are on the sides $BC$, $AC$, and $AB$, respectively. Given that $AA'$, $BB'$, and $CC'$ are concurrent at the point $O$, and that $\\frac{AO}{OA'}+\\frac{BO}{OB'}+\\frac{CO}{OC'}=92$, find $\\frac{AO}{OA'}\\cdot \\frac{BO}{OB'}\\cdot \\frac{CO}{OC'}$. Please... | {
"ground_truth": "94",
"style": "rule"
} | {
"index": "33398",
"split": "train"
} |
math | [
{
"content": "Let $\\{a_n\\}$ be a decreasing geometric sequence, where $q$ is the common ratio, and $S_n$ is the sum of the first $n$ terms. Given that $\\{a_1, a_2, a_3\\} \\subseteq \\{-4, -3, -2, 0, 1, 2, 3, 4\\}$, find the value of $$\\frac {S_{8}}{1-q^{4}}$$. Please output the final answer within \\boxed{... | {
"ground_truth": "\\frac {17}{2}",
"style": "rule"
} | {
"index": "33399",
"split": "train"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.