problem stringlengths 10 5.15k | answer dict |
|---|---|
Find one fourth of 12.8, expressed as a simplified improper fraction and also as a mixed number. | {
"answer": "3 \\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, $sin(A+\frac{π}{4})sin(B+\frac{π}{4})=cosAcosB$. Find:<br/>
$(1)$ the value of angle $C$;<br/>
$(2)$ if $AB=\sqrt{2}$, find the minimum value of $\overrightarrow{CA}•\overrightarrow{CB}$. | {
"answer": "-\\sqrt{2}+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A local government intends to encourage entrepreneurship by rewarding newly established small and micro enterprises with an annual output value between 500,000 and 5,000,000 RMB. The reward scheme follows these principles: The bonus amount $y$ (in ten thousand RMB) increases with the yearly output value $x$ (in ten thousand RMB), the bonus is no less than 700,000 RMB, and the bonus does not exceed 15% of the annual output value.
1. If an enterprise has an output value of 1,000,000 RMB and is eligible for a 90,000 RMB bonus, analyze whether the function $y=\log x + kx + 5$ (where $k$ is a constant) is in line with the government's reward requirements, and explain why (given $\log 2 \approx 0.3, \log 5 \approx 0.7$).
2. If the function $f(x) = \frac{15x - a}{x + 8}$ is adopted as the reward model, determine the minimum value of the positive integer $a$. | {
"answer": "315",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Convert -630° to radians. | {
"answer": "-\\frac{7\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the value of $x$ if $x=\frac{2023^2 - 2023 + 1}{2023}$? | {
"answer": "2022 + \\frac{1}{2023}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the relationship between \(\arcsin \cos \arcsin x\) and \(\arccos \sin \arccos x\). | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A chess piece called "the four-liner" attacks two vertical and two horizontal squares adjacent to the square on which it stands. What is the maximum number of non-attacking four-liners that can be placed on a $10 \times 10$ board? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a right rectangular prism \(B\) with edge lengths \(2,\ 5,\) and \(6\), including its interior. For any real \(r \geq 0\), let \(T(r)\) be the set of points in 3D space within a distance \(r\) from some point in \(B\). The volume of \(T(r)\) is expressed as \(ar^{3} + br^{2} + cr + d\), where \(a,\) \(b,\) \(c,\) and \(d\) are positive real numbers. Calculate \(\frac{bc}{ad}\).
A) $\frac{1560}{60}$
B) $\frac{8112}{240}$
C) $\frac{3792}{120}$
D) $\frac{6084}{180}$ | {
"answer": "\\frac{8112}{240}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In acute triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $b=2$, $B= \frac{π}{3}$, and $c\sin A= \sqrt{3}a\cos C$, find the area of $\triangle ABC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The value of \(\frac{1^{2}-3^{2}+5^{2}-7^{2}+\cdots+97^{2}-99^{2}}{1-3+5-7+\cdots+97-99}\) is: | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle PQR,$ where $PQ=PR=17$ and $QR=15.$ Points $G,H,$ and $I$ are on sides $\overline{PQ},$ $\overline{QR},$ and $\overline{PR},$ respectively, such that $\overline{GH}$ and $\overline{HI}$ are parallel to $\overline{PR}$ and $\overline{PQ},$ respectively. What is the perimeter of parallelogram $PGHI$? | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $a \in \{0,1,2\}, b \in \{-1,1,3,5\}$, determine the probability that the function $f(x)=ax^{2}-2bx$ is increasing in the interval $(1,+\infty)$. | {
"answer": "\\dfrac{5}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Use the method of random simulation experiments to estimate the probability of having exactly two days of rain in three days: First, use a calculator to generate random integers between \\(0\\) and \\(9\\), with \\(1\\), \\(2\\), \\(3\\), \\(4\\) representing rain, and \\(5\\), \\(6\\), \\(7\\), \\(8\\), \\(9\\), \\(0\\) representing no rain; then, take every three random numbers as a group, representing the rain situation for these three days. Through random simulation experiments, the following \\(20\\) groups of random numbers were generated: Based on this, estimate the probability of having exactly two days of rain in these three days to be approximately
\\(907\\) \\(966\\) \\(191\\) \\(925\\) \\(271\\) \\(932\\) \\(812\\) \\(458\\) \\(569\\) \\(683\\)
\\(431\\) \\(257\\) \\(393\\) \\(027\\) \\(556\\) \\(488\\) \\(730\\) \\(113\\) \\(537\\) \\(989\\) | {
"answer": "0.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: $(-2)^{2}+\sqrt{(-3)^{2}}-\sqrt[3]{27}+|\sqrt{3}-2|$. | {
"answer": "6 - \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $0 < \beta < \frac{\pi}{2} < \alpha < \pi$ and $\cos \left(\alpha- \frac{\beta}{2}\right)=- \frac{1}{9}, \sin \left( \frac{\alpha}{2}-\beta\right)= \frac{2}{3}$, calculate the value of $\cos (\alpha+\beta)$. | {
"answer": "-\\frac{239}{729}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a moving circle $E$ that passes through the point $F(1,0)$, and is tangent to the line $l: x=-1$.
(Ⅰ) Find the equation of the trajectory $G$ for the center $E$ of the moving circle.
(Ⅱ) Given the point $A(3,0)$, if a line with a slope of $1$ intersects with the line segment $OA$ (not passing through the origin $O$ and point $A$) and intersects the curve $G$ at points $B$ and $C$, find the maximum value of the area of $\triangle ABC$. | {
"answer": "\\frac{32 \\sqrt{3}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a\cos B - b\cos A = c$, and $C = \frac{π}{5}$, then find the measure of angle $B$. | {
"answer": "\\frac{3\\pi}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$2018$ people (call them $A, B, C, \ldots$ ) stand in a line with each permutation equally likely. Given that $A$ stands before $B$ , what is the probability that $C$ stands after $B$ ? | {
"answer": "1/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and it is given that $b^{2}=ac$ and $a^{2}+bc=c^{2}+ac$. Calculate the value of $\dfrac {c}{b\sin B}$. | {
"answer": "\\dfrac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the circle \\({{x}^{2}}+{{y}^{2}}-6x-2y+3=0\\) and the distance from its center to the line \\(x+ay-1=0\\) is \\(1,\\) then the real number \\(a=\\)_______ | {
"answer": "- \\dfrac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $y=\frac{1}{4}x^2$ and the circle $C: (x-1)^2+(y-2)^2=r^2$ $(r > 0)$ share a common point $P$. If the tangent line to the parabola at point $P$ also touches circle $C$, find the value of $r$. | {
"answer": "r = \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point $P$ lies in the same plane as a given square of side $2$. Let the vertices of the square, taken counterclockwise, be $A, B, C$ and $D$. Also, let the distances from $P$ to $B, C$ and $D$, respectively, be $v, w$ and $t$. What is the greatest distance that $P$ can be from $A$ if $v^2 + w^2 = t^2$?
A) $\sqrt{8}$
B) $\sqrt{10}$
C) $\sqrt{12}$
D) $\sqrt{14}$ | {
"answer": "\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum distance between point $P$ on the line $y = 2x + 1$ and point $Q$ on the curve $y = x + \ln{x}$. | {
"answer": "\\frac{2\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a workshop, each participant has a 1 in 40 chance of being late. What is the probability that out of any three participants chosen at random, exactly one will be late? Express your answer as a percent rounded to the nearest tenth. | {
"answer": "7.1\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | {
"answer": "2\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Amelia has a coin that lands heads with a probability of $\frac{3}{7}$, and Blaine has a coin that lands on heads with a probability of $\frac{1}{4}$. Initially, they simultaneously toss their coins once. If both get heads, they stop; otherwise, if either gets a head and the other a tail, that person wins. If both get tails, they start tossing their coins alternately as in the original scenario until one gets a head and wins. Amelia goes first in the alternate rounds. What is the probability that Amelia wins the game?
A) $\frac{5}{14}$
B) $\frac{9}{14}$
C) $\frac{11}{14}$
D) $\frac{13}{14}$
E) $\frac{1}{2}$ | {
"answer": "\\frac{9}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The greatest common divisor of two positive integers is $(x+7)$ and their least common multiple is $x(x+7)$, where $x$ is a positive integer. If one of the integers is 56, what is the smallest possible value of the other one? | {
"answer": "294",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In March of this year, the Municipal Bureau of Industry and Commerce conducted a quality supervision and random inspection of beverages in the circulation field within the city. The results showed that the qualification rate of a newly introduced X beverage in the market was 80%. Now, three people, A, B, and C, gather and choose 6 bottles of this beverage, with each person drinking two bottles. Calculate:
(Ⅰ) The probability that A drinks two bottles of X beverage and both are qualified;
(Ⅱ) The probability that A, B, and C each drink two bottles, and exactly one person drinks unqualified beverages (rounded to 0.01). | {
"answer": "0.44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the total number of digits used when the first 3002 positive even integers are written? | {
"answer": "11456",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\cos\left(\frac {\pi}{4} + \theta\right) = -\frac {3}{5}$, and $\frac {11\pi}{12} < \theta < \frac {5\pi}{4}$, find the value of $\frac {\sin{2\theta} + 2\sin^{2}{\theta}}{1 - \tan{\theta}}$. | {
"answer": "\\frac {28}{75}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$, $b$, $c$ be positive numbers, and $a+b+9c^2=1$. The maximum value of $\sqrt{a} + \sqrt{b} + \sqrt{3}c$ is \_\_\_\_\_\_. | {
"answer": "\\frac{\\sqrt{21}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $\left(\frac{x}{x-1}-1\right) \div \frac{{x}^{2}-1}{{x}^{2}-2x+1}$, where $x=\sqrt{5}-1$. | {
"answer": "\\frac{\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function \( f(x) \) is an even function and has a period of 4, if the equation \( f(x) = 0 \) has exactly one root, which is 1, in the interval \([0,2]\), what is the sum of all the roots of \( f(x) = 0 \) in the interval \([0,17]\)? | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $sin2θ=-\frac{1}{3}$, if $\frac{π}{4}＜θ＜\frac{3π}{4}$, then $\tan \theta =$____. | {
"answer": "-3-2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\cos \alpha =\dfrac{4}{5}$ and $\cos (\alpha +\beta )=\dfrac{5}{13}$, where $\alpha$ and $\beta$ are acute angles.
1. Find the value of $\sin 2\alpha$.
2. Find the value of $\sin \beta$. | {
"answer": "\\dfrac{33}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For lines $l_1: x + ay + 3 = 0$ and $l_2: (a-2)x + 3y + a = 0$ to be parallel, determine the values of $a$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For an arithmetic sequence $a_1, a_2, a_3, \dots,$ let
\[ S_n = a_1 + a_2 + a_3 + \dots + a_n, \]
and let
\[ T_n = S_1 + S_2 + S_3 + \dots + S_n. \]
Given the value of $S_{2023}$, determine the smallest integer $n$ for which you can uniquely determine the value of $T_n$. | {
"answer": "3034",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, $\alpha$ is an angle in the fourth quadrant. The terminal side of angle $\alpha$ intersects the unit circle $O$ at point $P(x_{0}, y_{0})$. If $\cos(\alpha - \frac{\pi}{3}) = -\frac{\sqrt{3}}{3}$, determine the $y_{0}$ coordinate. | {
"answer": "\\frac{-\\sqrt{6}-3}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The positive integer equal to the expression
\[ \sum_{i=0}^{9} \left(i+(-9)^i\right)8^{9-i} \binom{9}{i}\]
is divisible by exactly six distinct primes. Find the sum of these six distinct prime factors.
*Team #7* | {
"answer": "835",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular prism has 4 green faces, 2 yellow faces, and 6 blue faces. What's the probability that when it is rolled, a blue face will be facing up? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A robot has 6 modules, each needing a base and a cap attached sequentially. In how many different orders can the robot attach its bases and caps, assuming that, on each module, the base must be attached before the cap?
A) $6! \cdot 2^6$
B) $(6!)^2$
C) $12!$
D) $\frac{12!}{2^6}$
E) $2^{12} \cdot 6!$ | {
"answer": "\\frac{12!}{2^6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a geometric sequence $\{a_n\}$ with a common ratio $q$, and the product of its first $n$ terms is $T_n$, where the first term $a_1 > 1$, and $a_{2014}a_{2015} - 1 > 0$, $\frac{a_{2014} - 1}{a_{2015} - 1} < 0$, find the largest natural number $n$ such that $T_n > 1$. | {
"answer": "4028",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The roots of the equation $x^2 + kx + 8 = 0$ differ by $\sqrt{72}$. Find the greatest possible value of $k$. | {
"answer": "2\\sqrt{26}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the circle $C: x^2 + y^2 - (6 - 2m)x - 4my + 5m^2 - 6m = 0$, and a fixed line $l$ passing through the point $A(1, 0)$, for any real number $m$, the chord intercepted by circle $C$ on line $l$ always has a constant length $A$. Find the constant value of $A$. | {
"answer": "\\frac{2\\sqrt{145}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A parking lot has 20 spaces in a row. Fifteen cars arrive, each of which requires one parking space, and their drivers choose spaces at random from among the available spaces. Uncle Ben arrives in his RV, which requires 3 adjacent spaces. What is the probability that he is able to park?
A) $\frac{273}{969}$
B) $\frac{232}{323}$
C) $\frac{143}{364}$
D) $\frac{17}{28}$
E) $\frac{58}{81}$ | {
"answer": "\\frac{232}{323}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | {
"answer": "\\sqrt{40}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Eliminate every second number in a clockwise direction from numbers $1, 2, 3, \cdots, 2001$ that have been placed on a circle, starting with the number 2, until only one number remains. What is the last remaining number? | {
"answer": "1955",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, $c$ are the opposite sides of the acute angles $A$, $B$, $C$ of triangle $\triangle ABC$, $\overrightarrow{m}=(3a,3)$, $\overrightarrow{n}=(-2\sin B,b)$, and $\overrightarrow{m} \cdot \overrightarrow{n}=0$.
$(1)$ Find $A$;
$(2)$ If $a=2$ and the perimeter of $\triangle ABC$ is $6$, find the area of $\triangle ABC$. | {
"answer": "6 - 3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
After the year 2002, which is a palindrome, identify the next year where the sum of the product of its digits is greater than 15. Find the sum of the product of the digits of that year. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, its left and right foci are $F_1$ and $F_2$ respectively. Point $P(1, \frac{\sqrt{2}}{2})$ is on the ellipse, and $|PF_1| + |PF_2| = 2\sqrt{2}$.
$(1)$ Find the standard equation of ellipse $C$;
$(2)$ A line $l$ passing through $F_2$ intersects the ellipse at points $A$ and $B$. Find the maximum area of $\triangle AOB$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ is 120°, and the magnitude of $\overrightarrow {a}$ is 2. If $(\overrightarrow {a} + \overrightarrow {b}) \cdot (\overrightarrow {a} - 2\overrightarrow {b}) = 0$, find the projection of $\overrightarrow {b}$ on $\overrightarrow {a}$. | {
"answer": "-\\frac{\\sqrt{33} + 1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of real solutions to the equation:
\[
\frac{1^2}{x - 1} + \frac{2^2}{x - 2} + \frac{3^2}{x - 3} + \dots + \frac{120^2}{x - 120} = x.
\] | {
"answer": "121",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, it is given that $\cos(2X-Z) + \sin(X+Y) = 2$ and $XY = 6$. Determine the length of $YZ$. | {
"answer": "6\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, $XY = 5$, $XZ = 7$, and $YZ = 8$. The medians $XM$, $YN$, and $ZO$ of triangle $XYZ$ intersect at the centroid $G$. Let the projections of $G$ onto $YZ$, $XZ$, and $XY$ be $P$, $Q$, and $R$ respectively. Find $GP + GQ + GR$. | {
"answer": "\\frac{131\\sqrt{3}}{42}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $min|a, b|$ denote the minimum value between $a$ and $b$. When positive numbers $x$ and $y$ vary, let $t = min|2x+y, \frac{2y}{x^2+2y^2}|$, then the maximum value of $t$ is ______. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distance between two parallel lines $x-y=1$ and $2x-2y+3=0$ is ______. | {
"answer": "\\frac{5\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a large circle with a radius of 11 and small circles with a radius of 1, find the maximum number of small circles that can be tangentially inscribed in the large circle without overlapping. | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ with its right focus $F$, draw a line perpendicular to the $x$-axis passing through $F$, intersecting the two asymptotes at points $A$ and $B$, and intersecting the hyperbola at point $P$ in the first quadrant. Denote $O$ as the origin. If $\overrightarrow{OP} = λ\overrightarrow{OA} + μ\overrightarrow{OB} (λ, μ \in \mathbb{R})$, and $λ^2 + μ^2 = \frac{5}{8}$, find the eccentricity of the hyperbola. | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The centers of the faces of the right rectangular prism shown below are joined to create an octahedron, What is the volume of the octahedron?
[asy]
import three; size(2inch);
currentprojection=orthographic(4,2,2);
draw((0,0,0)--(0,0,3),dashed);
draw((0,0,0)--(0,4,0),dashed);
draw((0,0,0)--(5,0,0),dashed);
draw((5,4,3)--(5,0,3)--(5,0,0)--(5,4,0)--(0,4,0)--(0,4,3)--(0,0,3)--(5,0,3));
draw((0,4,3)--(5,4,3)--(5,4,0));
label("3",(5,0,3)--(5,0,0),W);
label("4",(5,0,0)--(5,4,0),S);
label("5",(5,4,0)--(0,4,0),SE);
[/asy] | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A hexagonal prism has a regular hexagonal base, and its lateral edges are perpendicular to the base. It is known that all the vertices of the hexagonal prism are on the same spherical surface, and the volume of the hexagonal prism is $\frac{9}{8}$, with a base perimeter of 3. The volume of this sphere is ______. | {
"answer": "\\frac{4\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the final 5 contestants of "The Voice" season 4 must sign with one of the three companies A, B, and C, with each company signing at least 1 person and at most 2 people, calculate the total number of possible different signing schemes. | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$ , find the smallest possible value of $$ |(\cot A + \cot B)(\cot B +\cot C)(\cot C + \cot A)| $$ | {
"answer": "\\frac{8\\sqrt{3}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circular paper with a radius of 6 inches has a section removed to form a \(240^\circ\) sector. This sector is then used to form a right circular cone by bringing the two radii together. Find the circumference of the base of the cone in terms of \(\pi\). | {
"answer": "8\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \sin\left(\frac{5\pi}{3}x + \frac{\pi}{6}\right) + \frac{3x}{2x-1}$, then the value of $f\left(\frac{1}{2016}\right) + f\left(\frac{3}{2016}\right) + f\left(\frac{5}{2016}\right) + f\left(\frac{7}{2016}\right) + \ldots + f\left(\frac{2015}{2016}\right) = \_\_\_\_\_\_$. | {
"answer": "1512",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\Delta ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $a\sin C=\sqrt{3}c\cos A$.
1. Find the size of angle $A$.
2. If $a=\sqrt{13}$ and $c=3$, find the area of $\Delta ABC$. | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the product of the numerator and the denominator when $0.\overline{0012}$ is expressed as a fraction in lowest terms? | {
"answer": "13332",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the difference between the sum of the first 1000 even counting numbers including 0, and the sum of the first 1000 odd counting numbers? | {
"answer": "-1000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number of games won by five basketball teams is shown in a bar chart. The teams' names are not displayed. The following clues provide information about the teams:
1. The Hawks won more games than the Falcons.
2. The Warriors won more games than the Knights, but fewer games than the Royals.
3. The Knights won more than 22 games.
How many games did the Warriors win? The win numbers given in the bar chart are 23, 28, 33, 38, and 43 games respectively. | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers $a$ and $b$ satisfying $a^{2}-4\ln a-b=0$, find the minimum value of $\left(a-c\right)^{2}+\left(b+2c\right)^{2}$. | {
"answer": "\\frac{9}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A granary has collected 1536 shi of rice, and upon inspection, it is found that out of 224 grains, 28 are weeds. Determine the approximate amount of weeds in this batch of rice. | {
"answer": "192",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a pocket, there are several balls of three different colors (enough in quantity), and each time 2 balls are drawn. To ensure that the result of drawing is the same 5 times, at least how many times must one draw? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The inclination angle of the line given by the parametric equations \[
\begin{cases}
x=1+t \\
y=1-t
\end{cases}
\]
calculate the inclination angle. | {
"answer": "\\frac {3\\pi }{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The hyperbola $tx^{2}-y^{2}-1=0$ has asymptotes that are perpendicular to the line $2x+y+1=0$. Find the eccentricity of this hyperbola. | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\cos ( \sqrt {3}x+\phi)- \sqrt {3}\sin ( \sqrt {3}x+\phi)$, find the smallest positive value of $\phi$ such that $f(x)$ is an even function. | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sphere intersects the $xy$-plane in a circle centered at $(3, 5, 0)$ with radius 2. The sphere also intersects the $yz$-plane in a circle centered at $(0, 5, -8),$ with radius $r.$ Find $r.$ | {
"answer": "\\sqrt{59}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A shooter fires 5 shots in succession, hitting the target with scores of: $9.7$, $9.9$, $10.1$, $10.2$, $10.1$. The variance of this set of data is __________. | {
"answer": "0.032",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a matrix $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$ where $a_{11}, a_{12}, a_{21}, a_{22} \in \{0, 1\}$, and the determinant of $A$ is 0. Determine the number of distinct matrices $A$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that five students from Maplewood school worked for 6 days, six students from Oakdale school worked for 4 days, and eight students from Pinecrest school worked for 7 days, and the total amount paid for the students' work was 1240 dollars, determine the total amount earned by the students from Oakdale school, ignoring additional fees. | {
"answer": "270.55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among 50 school teams participating in the HKMO, no team answered all four questions correctly. The first question was solved by 45 teams, the second by 40 teams, the third by 35 teams, and the fourth by 30 teams. How many teams solved both the third and the fourth questions? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A coordinate system is established with the origin as the pole and the positive half of the x-axis as the polar axis. Given the curve $C_1: (x-2)^2 + y^2 = 4$, point A has polar coordinates $(3\sqrt{2}, \frac{\pi}{4})$, and the polar coordinate equation of line $l$ is $\rho \cos (\theta - \frac{\pi}{4}) = a$, with point A on line $l$.
(1) Find the polar coordinate equation of curve $C_1$ and the rectangular coordinate equation of line $l$.
(2) After line $l$ is moved 6 units to the left to obtain $l'$, the intersection points of $l'$ and $C_1$ are M and N. Find the polar coordinate equation of $l'$ and the length of $|MN|$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $F$ is a point on diagonal $BC$ of the unit square $ABCD$ such that $\triangle{ABF}$ is isosceles right triangle with $AB$ as the hypotenuse, consider a strip inside $ABCD$ parallel to $AD$ ranging from $y=\frac{1}{4}$ to $y=\frac{3}{4}$ of the unit square, calculate the area of the region $Q$ which lies inside the strip but outside of $\triangle{ABF}$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For some positive integer \( n \), the number \( 150n^3 \) has \( 150 \) positive integer divisors, including \( 1 \) and the number \( 150n^3 \). How many positive integer divisors does the number \( 108n^5 \) have? | {
"answer": "432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two distinct integers $x$ and $y$ are factors of 48. One of these integers must be even. If $x\cdot y$ is not a factor of 48, what is the smallest possible value of $x\cdot y$? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A function \( f \) is defined on the complex numbers by \( f(z) = (a + bi)z^2 \), where \( a \) and \( b \) are real numbers. The function has the property that for each complex number \( z \), \( f(z) \) is equidistant from both \( z \) and the origin. Given that \( |a+bi| = 5 \), find \( b^2 \). | {
"answer": "\\frac{99}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of real solutions to the equation:
\[
\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{150}{x - 150} = x^2.
\] | {
"answer": "151",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the integrals:
1) \(\int_{0}^{5} \frac{x \, dx}{\sqrt{1+3x}}\)
2) \(\int_{\ln 2}^{\ln 9} \frac{dx}{e^{x}-e^{-x}}\)
3) \(\int_{1}^{\sqrt{3}} \frac{(x^{3}+1) \, dx}{x^{2} \sqrt{4-x^{2}}}\)
4) \(\int_{0}^{\frac{\pi}{2}} \frac{dx}{2+\cos x}\) | {
"answer": "\\frac{\\pi}{3 \\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The local library has two service windows. In how many ways can eight people line up to be served if there are two lines, one for each window? | {
"answer": "40320",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A severe earthquake in Madrid caused a total of €50 million in damages. At the time of the earthquake, the exchange rate was such that 2 Euros were worth 3 American dollars. How much damage did the earthquake cause in American dollars? | {
"answer": "75,000,000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it satisfies $\sqrt{3}b\cos A - a\sin B = 0$.
$(1)$ Find the measure of angle $A$;
$(2)$ Given that $c=4$ and the area of $\triangle ABC$ is $6\sqrt{3}$, find the value of side length $a$. | {
"answer": "2\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$ and $f(x+ \frac{5}{2})=-\frac{1}{f(x)}$, and when $x \in [-\frac{5}{2}, 0]$, $f(x)=x(x+ \frac{5}{2})$, find $f(2016)=$ \_\_\_\_\_\_. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The left focus of the hyperbola $C$: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \, (a > 0, b > 0)$ is $F$. If the symmetric point $A$ of $F$ with respect to the line $\sqrt{3}x + y = 0$ is a point on the hyperbola $C$, then the eccentricity of the hyperbola $C$ is \_\_\_\_\_\_. | {
"answer": "\\sqrt{3} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a die is rolled, event \( A = \{1, 2, 3\} \) consists of rolling one of the faces 1, 2, or 3. Similarly, event \( B = \{1, 2, 4\} \) consists of rolling one of the faces 1, 2, or 4.
The die is rolled 10 times. It is known that event \( A \) occurred exactly 6 times.
a) Find the probability that under this condition, event \( B \) did not occur at all.
b) Find the expected value of the random variable \( X \), which represents the number of occurrences of event \( B \). | {
"answer": "\\frac{16}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A tangent line is drawn from a point on the line $y=x$ to the circle $(x-4)^2 + (y+2)^2 = 1$. Find the minimum length of the tangent line. | {
"answer": "\\sqrt{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, $c$ are the opposite sides of angles $A$, $B$, $C$ in triangle $\triangle ABC$, if $\triangle ABC$ simultaneously satisfies three of the following four conditions:①$a=\sqrt{3}$; ②$b=2$; ③$\frac{{sinB+sinC}}{{sinA}}=\frac{{a+c}}{{b-c}}$; ④${cos^2}({\frac{{B-C}}{2}})-sinBsinC=\frac{1}{4}$.<br/>$(1)$ What are the possible combinations of conditions that have a solution for the triangle?<br/>$(2)$ Among the combinations in $(1)$, choose one and find the area of the corresponding $\triangle ABC$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Circles of radius 3 and 4 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\pi$. | {
"answer": "24\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a > 0$, if $f(g(a)) = 18$, where $f(x) = x^2 + 10$ and $g(x) = x^2 - 6$, what is the value of $a$? | {
"answer": "\\sqrt{2\\sqrt{2} + 6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular chocolate bar is made of equal squares. Irena breaks off two complete strips of squares and eats the 12 squares she obtains. Later, Jack breaks off one complete strip of squares from the same bar and eats the 9 squares he obtains. How many squares of chocolate are left in the bar?
A) 72
B) 63
C) 54
D) 45
E) 36 | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The picture shows the same die in three different positions. When the die is rolled, what is the probability of rolling a 'YES'?
A) \(\frac{1}{3}\)
B) \(\frac{1}{2}\)
C) \(\frac{5}{9}\)
D) \(\frac{2}{3}\)
E) \(\frac{5}{6}\) | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In February 1983, $789$ millimeters of rain fell in Jorhat, India. What was the average rainfall in millimeters per hour during that particular month?
A) $\frac{789}{672}$
B) $\frac{789 \times 28}{24}$
C) $\frac{789 \times 24}{28}$
D) $\frac{28 \times 24}{789}$
E) $789 \times 28 \times 24$ | {
"answer": "\\frac{789}{672}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the parallelepiped $ABCD-{A'}{B'}{C'}{D'}$, the base $ABCD$ is a square with side length $2$, the length of the side edge $AA'$ is $3$, and $\angle {A'}AB=\angle {A'}AD=60^{\circ}$. Find the length of $AC'$. | {
"answer": "\\sqrt{29}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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