problem stringlengths 10 5.15k | answer dict |
|---|---|
If $8^{2x} = 11$, evaluate $2^{x + 1.5}$. | {
"answer": "11^{1/6} \\cdot 2 \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation of the parabola $y^{2}=-4x$, and the equation of the line $l$ as $2x+y-4=0$. There is a moving point $A$ on the parabola. The distance from point $A$ to the $y$-axis is $m$, and the distance from point $A$ to the line $l$ is $n$. Find the minimum value of $m+n$. | {
"answer": "\\frac{6 \\sqrt{5}}{5}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The non-negative numbers \(a, b, c\) sum up to 1. Find the maximum possible value of the expression
$$
(a + 3b + 5c) \cdot \left(a + \frac{b}{3} + \frac{c}{5}\right)
$$ | {
"answer": "\\frac{9}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If A, B, and C stand in a row, calculate the probability that A and B are adjacent. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Bob chooses a $4$ -digit binary string uniformly at random, and examines an infinite sequence of uniformly and independently random binary bits. If $N$ is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of $N$ . For example, if Bob’s string is $0000$ and the stream of bits begins $101000001 \dots$ , then $N = 7$ . | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points $A, B$ and $C$ on a circle of radius $r$ are situated so that $AB=AC$, $AB>r$, and the length of minor arc $BC$ is $r$, calculate the ratio of the length of $AB$ to the length of $BC$. | {
"answer": "\\frac{1}{2}\\csc(\\frac{1}{4})",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle is inscribed in a square, and within this circle, a smaller square is inscribed such that one of its sides coincides with a side of the larger square and two vertices lie on the circle. Calculate the percentage of the area of the larger square that is covered by the smaller square. | {
"answer": "50\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Select a number from \\(1\\), \\(2\\), \\(3\\), \\(4\\), \\(5\\), \\(6\\), \\(7\\), and calculate the probability of the following events:
\\((1)\\) The selected number is greater than \\(3\\);
\\((2)\\) The selected number is divisible by \\(3\\);
\\((3)\\) The selected number is greater than \\(3\\) or divisible by \\(3\\). | {
"answer": "\\dfrac{5}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among all the five-digit numbers formed without repeating any of the digits 0, 1, 2, 3, 4, if they are arranged in ascending order, determine the position of the number 12340. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse E: $\\frac{x^{2}}{4} + \\frac{y^{2}}{2} = 1$, O is the coordinate origin, and a line with slope k intersects ellipse E at points A and B. The midpoint of segment AB is M, and the angle between line OM and AB is θ, with tanθ = 2 √2. Find the value of k. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular yard contains three flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, with the parallel sides measuring $10$ meters and $20$ meters. What fraction of the yard is occupied by the flower beds?
A) $\frac{1}{4}$
B) $\frac{1}{6}$
C) $\frac{1}{8}$
D) $\frac{1}{10}$
E) $\frac{1}{3}$ | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane rectangular coordinate system xOy, the origin is taken as the pole, the positive semi-axis of the x-axis is taken as the polar axis, and the polar coordinate system is established. The same unit length is adopted in both coordinate systems. The polar coordinate equation of the curve C is ρ = 4$\sqrt {2}$cos(θ - $\frac {π}{4}$), and the parameter equation of the line is $\begin{cases} x=2+ \frac {1}{2}t \\ y=1+ \frac { \sqrt {3}}{2}t\end{cases}$ (t is the parameter).
1. Find the rectangular coordinate equation of curve C and the general equation of line l.
2. Suppose point P(2, 1), if line l intersects curve C at points A and B, find the value of $| \frac {1}{|PA|} - \frac {1}{|PB|} |$. | {
"answer": "\\frac { \\sqrt {31}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the shortest distance from a point on the curve $y=x^{2}-\ln x$ to the line $x-y-2=0$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the system of equations $\begin{cases} x - 2y = z - 2u \\ 2yz = ux \end{cases}$, for every set of positive real number solutions $\{x, y, z, u\}$ where $z \geq y$, there exists a positive real number $M$ such that $M \leq \frac{z}{y}$. Find the maximum value of $M$. | {
"answer": "6 + 4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ n$ be a positive integer and $ [ \ n ] = a.$ Find the largest integer $ n$ such that the following two conditions are satisfied:
$ (1)$ $ n$ is not a perfect square;
$ (2)$ $ a^{3}$ divides $ n^{2}$ . | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function $f(x) = e^x + \frac{a}{e^x}$ has a derivative $y = f'(x)$ that is an odd function and the slope of a tangent line to the curve $y = f(x)$ is $\frac{3}{2}$, determine the abscissa of the tangent point. | {
"answer": "\\ln 2",
"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"
} |
Steve guesses randomly on a 20-question multiple-choice test where each question has two choices. What is the probability that he gets at least half of the questions correct? Express your answer as a common fraction. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a tetrahedron O-ABC, where $\angle BOC=90^\circ$, $OA \perpendicular$ plane BOC, and $AB= \sqrt{10}$, $BC= \sqrt{13}$, $AC= \sqrt{5}$. Points O, A, B, and C are all on the surface of sphere S. Find the surface area of sphere S. | {
"answer": "14\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four points are on a line segment.
If \( A B : B C = 1 : 2 \) and \( B C : C D = 8 : 5 \), then \( A B : B D \) equals | {
"answer": "4 : 13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points A and B are on a circle of radius 7 and AB = 8. Point C is the midpoint of the minor arc AB. What is the length of the line segment AC? | {
"answer": "\\sqrt{98 - 14\\sqrt{33}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square $EFGH$ has sides of length 4. A point $P$ on $EH$ is such that line segments $FP$ and $GP$ divide the square’s area into four equal parts. Find the length of segment $FP$.
A) $2\sqrt{3}$
B) $3$
C) $2\sqrt{5}$
D) $4$
E) $2\sqrt{7}$ | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C$ with its left and right foci at $F_{1}(-\sqrt{3},0)$ and $F_{2}(\sqrt{3},0)$, respectively, and the ellipse passes through the point $(-1, \frac{\sqrt{3}}{2})$.
(Ⅰ) Find the equation of the ellipse $C$;
(Ⅱ) Given a fixed point $A(1, \frac{1}{2})$, a line $l$ passing through the origin $O$ intersects the curve $C$ at points $M$ and $N$. Find the maximum area of $\triangle MAN$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The least positive integer with exactly \(2023\) distinct positive divisors can be written in the form \(m \cdot 10^k\), where \(m\) and \(k\) are integers and \(10\) is not a divisor of \(m\). What is \(m+k?\)
A) 999846
B) 999847
C) 999848
D) 999849
E) 999850 | {
"answer": "999846",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A $50$-gon \(Q_1\) is drawn in the Cartesian plane where the sum of the \(x\)-coordinates of the \(50\) vertices equals \(150\). A constant scaling factor \(k = 1.5\) applies only to the \(x\)-coordinates of \(Q_1\). The midpoints of the sides of \(Q_1\) form a second $50$-gon, \(Q_2\), and the midpoints of the sides of \(Q_2\) form a third $50$-gon, \(Q_3\). Find the sum of the \(x\)-coordinates of the vertices of \(Q_3\). | {
"answer": "225",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On an island, there are knights, liars, and followers; each person knows who is who. All 2018 island residents were lined up and each was asked to answer "Yes" or "No" to the question: "Are there more knights than liars on the island?" The residents responded one by one in such a way that the others could hear. Knights always told the truth, liars always lied. Each follower answered the same as the majority of the preceding respondents, and if the "Yes" and "No" answers were split equally, they could give either answer. It turned out that there were exactly 1009 "Yes" answers. What is the maximum number of followers that could be among the island residents? | {
"answer": "1009",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the absolute value of the difference of single-digit integers $C$ and $D$ such that in base 8:
$$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
& & & D & D & C_8 \\
-& & & \mathbf{6} & \mathbf{3} & D_8 \\
\cline{2-6}
& & C & \mathbf{3} & \mathbf{1} & \mathbf{5_8}
\end{array} $$
Express your answer in base $8$. | {
"answer": "5_8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the polar coordinate system, the polar equation of curve $C$ is $\rho =6\sin \theta$, and the polar coordinates of point $P$ are $(\sqrt{2},\frac{\pi }{4})$. Taking the pole as the origin of coordinates and the positive half-axis of the $x$-axis as the polar axis, a plane rectangular coordinate system is established.
(1) Find the rectangular coordinate equation of curve $C$ and the rectangular coordinates of point $P$;
(2) A line $l$ passing through point $P$ intersects curve $C$ at points $A$ and $B$. If $|PA|=2|PB|$, find the value of $|AB|$. | {
"answer": "3 \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the wavelength of each light quantum is approximately $688$ nanometers and $1$ nanometer is equal to $0.000000001$ meters, express the wavelength of each light quantum in scientific notation. | {
"answer": "6.88\\times 10^{-7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a hyperbola with eccentricity $e$ and an ellipse with eccentricity $\frac{\sqrt{2}}{2}$ share the same foci $F_{1}$ and $F_{2}$. If $P$ is a common point of the two curves and $\angle F_{1}PF_{2}=60^{\circ}$, then $e=$ ______. | {
"answer": "\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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 she will use. | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=\cos(2x- \frac{\pi}{6})$, find the horizontal shift required to transform the graph of $y=\sin 2x$ into the graph of $y=\cos(2x- \frac{\pi}{6})$. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a rectangular sheet of paper, a picture is drawn in the shape of a "cross" formed by two rectangles $ABCD$ and $EFGH$, where the sides are parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, and $FG=10$. Find the area of the quadrilateral $AFCH$. | {
"answer": "52.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Extend the definition of the binomial coefficient to $C_x^m = \frac{x(x-1)\dots(x-m+1)}{m!}$ where $x\in\mathbb{R}$ and $m$ is a positive integer, with $C_x^0=1$. This is a generalization of the binomial coefficient $C_n^m$ (where $n$ and $m$ are positive integers and $m\leq n$).
1. Calculate the value of $C_{-15}^3$.
2. Let $x > 0$. For which value of $x$ does $\frac{C_x^3}{(C_x^1)^2}$ attain its minimum value?
3. Can the two properties of binomial coefficients $C_n^m = C_n^{n-m}$ (Property 1) and $C_n^m + C_n^{m-1} = C_{n+1}^m$ (Property 2) be extended to $C_x^m$ where $x\in\mathbb{R}$ and $m$ is a positive integer? If so, write the extended form and provide a proof. If not, explain why. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\lfloor x \rfloor$ represent the integer part of the real number $x$, and $\{x\}$ represent the fractional part of the real number $x$, e.g., $\lfloor 3.1 \rfloor = 3, \{3.1\} = 0.1$. It is known that all terms of the sequence $\{a\_n\}$ are positive, $a\_1 = \sqrt{2}$, and $a\_{n+1} = \lfloor a\_n \rfloor + \frac{1}{\{a\_n\}}$. Find $a\_{2017}$. | {
"answer": "4032 + \\sqrt{2}",
"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. It is given that $a \neq b$, $c= \sqrt{3}$, and $\cos^2A - \cos^2B = \sqrt{3}\sin A\cos A - \sqrt{3}\sin B\cos B$.
$(I)$ Find the magnitude of angle $C$.
$(II)$ If $\sin A= \frac{4}{5}$, find the area of $\triangle ABC$. | {
"answer": "\\frac{8\\sqrt{3}+18}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \frac {1}{2}x^{2} + x - 2\ln{x}$ ($x > 0$):
(1) Find the intervals of monotonicity for $f(x)$.
(2) Find the extreme values of the function $f(x)$. | {
"answer": "\\frac {3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five friends — Sarah, Lily, Emma, Nora, and Kate — performed in a theater as quartets, with one friend sitting out each time. Nora performed in 10 performances, which was the most among all, and Sarah performed in 6 performances, which was the fewest among all. Calculate the total number of performances. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A beam of light is emitted from point $P(1,2,3)$, reflected by the $Oxy$ plane, and then absorbed at point $Q(4,4,4)$. The distance traveled by the light beam is ______. | {
"answer": "\\sqrt{62}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify the product \[\frac{9}{3}\cdot\frac{15}{9}\cdot\frac{21}{15} \dotsm \frac{3n+6}{3n} \dotsm \frac{3003}{2997}.\] | {
"answer": "1001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four equal circles with diameter $6$ are arranged such that three circles are tangent to one side of a rectangle and the fourth circle is tangent to the opposite side. All circles are tangent to at least one other circle with their centers forming a straight line that is parallel to the sides of the rectangle they touch. The length of the rectangle is twice its width. Calculate the area of the rectangle. | {
"answer": "648",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number $1025$ can be written as $23q + r$ where $q$ and $r$ are positive integers. What is the greatest possible value of $q - r$? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Simplify: $\dfrac{\sin(\pi -\alpha)\cos(\pi +\alpha)\sin(\dfrac{\pi}{2}+\alpha)}{\sin(-\alpha)\sin(\dfrac{3\pi}{2}+\alpha)}$.
(2) Given $\alpha \in (\dfrac{\pi}{2}, \pi)$, and $\sin(\pi -\alpha) + \cos \alpha = \dfrac{7}{13}$, find $\tan \alpha$. | {
"answer": "-\\dfrac{12}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given: The curve $C$ has the polar coordinate equation: $ρ=a\cos θ (a>0)$, and the line $l$ has the parametric equations: $\begin{cases}x=1+\frac{\sqrt{2}}{2}t\\y=\frac{\sqrt{2}}{2}t\end{cases}$ ($t$ is the parameter)
1. Find the Cartesian equation of the curve $C$ and the line $l$;
2. If the line $l$ is tangent to the curve $C$, find the value of $a$. | {
"answer": "a=2(\\sqrt{2}-1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin(\alpha - \beta) = \frac{1}{3}$ and $\cos \alpha \sin \beta = \frac{1}{6}$, calculate the value of $\cos(2\alpha + 2\beta)$. | {
"answer": "\\frac{1}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the vectors $\overrightarrow{m}=(\cos x,\sin x)$ and $\overrightarrow{n}=(2 \sqrt {2}+\sin x,2 \sqrt {2}-\cos x)$, and the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$, where $x\in R$.
(I) Find the maximum value of the function $f(x)$;
(II) If $x\in(-\frac {3π}{2},-π)$ and $f(x)=1$, find the value of $\cos (x+\frac {5π}{12})$. | {
"answer": "-\\frac {3 \\sqrt {5}+1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$ and $b$ are positive real numbers satisfying $9a^{2}+b^{2}=1$, find the maximum value of $\frac{ab}{3a+b}$. | {
"answer": "\\frac{\\sqrt{2}}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Wang Lei and her older sister walk from home to the gym to play badminton. It is known that the older sister walks 20 meters more per minute than Wang Lei. After 25 minutes, the older sister reaches the gym, and then realizes she forgot the racket. She immediately returns along the same route to get the racket and meets Wang Lei at a point 300 meters away from the gym. Determine the distance between Wang Lei's home and the gym in meters. | {
"answer": "1500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\alpha$ be a root of $x^6-x-1$ , and call two polynomials $p$ and $q$ with integer coefficients $\textit{equivalent}$ if $p(\alpha)\equiv q(\alpha)\pmod3$ . It is known that every such polynomial is equivalent to exactly one of $0,1,x,x^2,\ldots,x^{727}$ . Find the largest integer $n<728$ for which there exists a polynomial $p$ such that $p^3-p-x^n$ is equivalent to $0$ . | {
"answer": "727",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the area of the circle described by the equation \(3x^2 + 3y^2 - 15x + 9y + 27 = 0\) in terms of \(\pi\). | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, $c\cos A= \frac{4}{b}$, and the area of $\triangle ABC$, $S \geq 2$.
(1) Determine the range of possible values for angle $A$.
(2) Find the maximum value of the function $f(x) = \cos^2 A + \sqrt{3}\sin^2\left(\frac{\pi}{2}+ \frac{A}{2}\right) - \frac{\sqrt{3}}{2}$. | {
"answer": "\\frac{1}{2} + \\frac{\\sqrt{6}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a new diagram below, we have $\cos \angle XPY = \frac{3}{5}$. A point Z is placed such that $\angle XPZ$ is a right angle. What is $\sin \angle YPZ$?
[asy]
pair X, P, Y, Z;
P = (0,0);
X = Rotate(-aCos(3/5))*(-2,0);
Y = (2,0);
Z = Rotate(-90)*(2,0);
dot("$Z$", Z, S);
dot("$Y$", Y, S);
dot("$X$", X, W);
dot("$P$", P, S);
draw(X--P--Y--Z--cycle);
[/asy] | {
"answer": "\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Schools A and B are having a sports competition with three events. In each event, the winner gets 10 points and the loser gets 0 points, with no draws. The school with the highest total score after the three events wins the championship. It is known that the probabilities of school A winning in the three events are 0.5, 0.4, and 0.8, respectively, and the results of each event are independent.<br/>$(1)$ Find the probability of school A winning the championship;<br/>$(2)$ Let $X$ represent the total score of school B, find the distribution table and expectation of $X$. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Quadrilateral $ALEX,$ pictured below (but not necessarily to scale!)
can be inscribed in a circle; with $\angle LAX = 20^{\circ}$ and $\angle AXE = 100^{\circ}:$ | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set $A=\{x|0<x+a\leq5\}$, and the set $B=\{x|-\frac{1}{2}\leq x<6\}$
(Ⅰ) If $A\subseteq B$, find the range of the real number $a$;
(Ⅱ) If $A\cap B$ is a singleton set, find the value of the real number $a$. | {
"answer": "\\frac {11}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To obtain the graph of the function $$y=2\sin(x+ \frac {\pi}{6})\cos(x+ \frac {\pi}{6})$$, determine the horizontal shift required to transform the graph of the function $y=\sin 2x$. | {
"answer": "\\frac {\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the vertices of the regular triangular prism $ABC-A_{1}B_{1}C_{1}$ lie on the surface of a sphere $O$, the lateral area of the regular triangular prism $ABC-A_{1}B_{1}C_{1}$ is $6$, and the base area is $\sqrt{3}$, calculate the surface area of the sphere $O$. | {
"answer": "\\frac{19\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\{b_k\}$ be a sequence of integers such that $b_1=2$ and $b_{m+n}=b_m+b_n+m^2+n^2,$ for all positive integers $m$ and $n.$ Find $b_{12}.$ | {
"answer": "160",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose a sequence $\{a\_n\}$ satisfies $\frac{1}{a\_{n+1}} - \frac{1}{a\_n} = d (n \in \mathbb{N}^*, d$ is a constant), then the sequence $\{a\_n\}$ is called a "harmonic sequence". It is known that the sequence $\{\frac{1}{x\_n}\}$ is a "harmonic sequence", and $x\_1 + x\_2 + ... + x\_{20} = 200$, find the maximum value of $x\_3 x\_{18}$. | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangles $ABC$ and $AFG$ have areas $3009$ and $9003,$ respectively, with $B=(0,0), C=(331,0), F=(800,450),$ and $G=(813,463).$ What is the sum of all possible $x$-coordinates of $A$? | {
"answer": "1400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sums of the first n terms of two arithmetic sequences $\{a_n\}$ and $\{b_n\}$ denoted as $S_n$ and $T_n$, respectively, if $\frac {S_{n}}{T_{n}} = \frac {2n}{3n+1}$, calculate the value of $\frac {a_{6}}{b_{6}}$. | {
"answer": "\\frac {11}{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2010,0),(2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$? Express your answer as a common fraction. | {
"answer": "\\frac{335}{2011}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the domain of the function $f(x)=x^{2}$ is $D$, and its range is ${0,1,2,3,4,5}$, then there are \_\_\_\_\_\_ such functions $f(x)$ (answer with a number). | {
"answer": "243",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A $40$ feet high screen is put on a vertical wall $10$ feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye? | {
"answer": "10\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( a, b > 0 \). Find the maximum values of \( \sqrt{\frac{a}{2a+b}} + \sqrt{\frac{b}{2b+a}} \) and \( \sqrt{\frac{a}{a+2b}} + \sqrt{\frac{b}{b+2a}} \). | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The coefficient of $x^2$ in the expansion of $(x+1)^5(x-2)$ is $\boxed{5}$. | {
"answer": "-15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If two points are randomly selected from the eight vertices of a cube, the probability that the line determined by these two points intersects each face of the cube is ______. | {
"answer": "\\frac{1}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $|\overrightarrow {a}|= \sqrt {3}$, $|\overrightarrow {b}|=2$, and $(\overrightarrow {a}- \overrightarrow {b}) \perp \overrightarrow {a}$, find the projection of $\overrightarrow {a}$ on $\overrightarrow {b}$. | {
"answer": "\\frac {3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a} = (4\cos \alpha, \sin \alpha)$, $\overrightarrow{b} = (\sin \beta, 4\cos \beta)$, and $\overrightarrow{c} = (\cos \beta, -4\sin \beta)$, where $\alpha, \beta \in \mathbb{R}$ and neither $\alpha$, $\beta$, nor $\alpha + \beta$ equals $\frac{\pi}{2} + k\pi, k \in \mathbb{Z}$:
1. Find the maximum value of $|\overrightarrow{b} + \overrightarrow{c}|$.
2. When $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$ and perpendicular to $(\overrightarrow{b} - 2\overrightarrow{c})$, find the value of $\tan \alpha + \tan \beta$. | {
"answer": "-30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 7 people standing in a row. How many different arrangements are there according to the following requirements?
(1) Among them, A, B, and C cannot stand next to each other;
(2) Among them, A and B have exactly one person between them;
(3) A does not stand at the head of the row, and B does not stand at the end of the row. | {
"answer": "3720",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\left\{ a_n \right\}$ such that $a_{n+1}+a_n={(-1)}^n\cdot n$ ($n\in \mathbb{N}^*$), find the sum of the first 20 terms of $\left\{ a_n \right\}$. | {
"answer": "-100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles A, B, and C are a, b, and c, respectively. If $C= \frac {\pi}{3}, b= \sqrt {2}, c= \sqrt {3}$, find the measure of angle A. | {
"answer": "\\frac{5\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $DEF,$ $\cot D \cot F = \frac{1}{3}$ and $\cot E \cot F = \frac{1}{8}.$ Find $\tan F.$ | {
"answer": "12 + \\sqrt{136}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that f(x) = |log₃x|, if a and b satisfy f(a - 1) = f(2b - 1), and a ≠ 2b, then the minimum value of a + b is ___. | {
"answer": "\\frac{3}{2} + \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Choose one of the three conditions in $①$ $ac=\sqrt{3}$, $②$ $c\sin A=3$, $③$ $c=\sqrt{3}b$, and supplement it in the following question. If the triangle in the question exists, find the value of $c$; if the triangle in the question does not exist, explain the reason.<br/>Question: Does there exist a $\triangle ABC$ where the internal angles $A$, $B$, $C$ have opposite sides $a$, $b$, $c$, and $\sin A=\sqrt{3}\sin B$, $C=\frac{π}{6}$, _______ $?$<br/>Note: If multiple conditions are selected to answer separately, the first answer will be scored. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all solutions to
\[\sqrt{x + 2 - 2 \sqrt{x - 4}} + \sqrt{x + 12 - 8 \sqrt{x - 4}} = 4.\] | {
"answer": "[13]",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum real number \( M \) such that for all real numbers \( x \) and \( y \) satisfying \( x + y \geqslant 0 \), the following inequality holds:
$$
\left(x^{2}+y^{2}\right)^{3} \geqslant M\left(x^{3}+y^{3}\right)(xy - x - y). | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $b^2=ac$ and $a^2-c^2=ac-bc$, find the value of $$\frac{c}{b\sin B}$$. | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f\left( x \right)=\sin \left( 2x+\phi \right)\left(\left| \phi \right| < \dfrac{\pi }{2} \right)$ whose graph is symmetric about the point $\left( \dfrac{\pi }{3},0 \right)$, and $f\left( {{x}_{1}} \right)+f\left( {{x}_{2}} \right)=0$ when ${{x}_{1}},{{x}_{2}}\in \left( \dfrac{\pi }{12},\dfrac{7\pi }{12} \right)$ $\left( {{x}_{1}}\ne {{x}_{2}} \right)$, find $f\left( {{x}_{1}}+{{x}_{2}} \right)$. | {
"answer": "-\\dfrac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system, $O$ is the origin, and points $A(-1,0)$, $B(0, \sqrt{3})$, $C(3,0)$. A moving point $D$ satisfies $|\overrightarrow{CD}|=1$, then the maximum value of $|\overrightarrow{OA}+ \overrightarrow{OB}+ \overrightarrow{OD}|$ is ______. | {
"answer": "\\sqrt{7}+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive four-digit integers of the form $\_\_90$ are divisible by 90? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $c=2$, $C=\dfrac{\pi }{3}$, and $\sin B=2\sin A$, find the area of $\triangle ABC$. | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $(x+1)^4(x+4)^8 = a + a_1(x+3) + a_2(x+3)^2 + \ldots + a_{12}(x+3)^{12}$, find the value of $a_2 + a_4 + \ldots + a_{12}$. | {
"answer": "112",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a} = (\sin x, \cos x)$, $\overrightarrow{b} = (\sin x, \sin x)$, and $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$
(1) If $x \in \left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$, find the range of the function $f(x)$.
(2) Let the sides opposite the acute angles $A$, $B$, and $C$ of triangle $\triangle ABC$ be $a$, $b$, and $c$, respectively. If $f(B) = 1$, $b = \sqrt{2}$, and $c = \sqrt{3}$, find the value of $a$. | {
"answer": "\\frac{\\sqrt{6} + \\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\alpha \in (0, \frac{\pi}{2})$, and $\tan 2\alpha = \frac{\cos \alpha}{2-\sin \alpha}$, calculate the value of $\tan \alpha$. | {
"answer": "\\frac{\\sqrt{15}}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given in the tetrahedron P-ABC, PA is perpendicular to the plane ABC, AB=AC=PA=2, and in triangle ABC, ∠BAC=120°, then the volume of the circumscribed sphere of the tetrahedron P-ABC is \_\_\_\_\_\_. | {
"answer": "\\frac{20\\sqrt{5}\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $$f(x)= \begin{cases} ( \frac {1}{2})^{x} & ,x≥4 \\ f(x+1) & ,x<4\end{cases}$$, find the value of $f(\log_{2}3)$. | {
"answer": "\\frac {1}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$ | {
"answer": "-\\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Quadrilateral $EFGH$ is a parallelogram. A line through point $G$ makes a $30^\circ$ angle with side $GH$. Determine the degree measure of angle $E$.
[asy]
size(100);
draw((0,0)--(5,2)--(6,7)--(1,5)--cycle);
draw((5,2)--(7.5,3)); // transversal line
draw(Arc((5,2),1,-60,-20)); // transversal angle
label("$H$",(0,0),SW); label("$G$",(5,2),SE); label("$F$",(6,7),NE); label("$E$",(1,5),NW);
label("$30^\circ$",(6.3,2.8), E);
[/asy] | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a $123 \times 123$ board, each cell is painted either purple or blue according to the following conditions:
- Each purple cell that is not on the edge of the board has exactly 5 blue cells among its 8 neighbors.
- Each blue cell that is not on the edge of the board has exactly 4 purple cells among its 8 neighbors.
Note: Two cells are neighbors if they share a side or a vertex.
(a) Consider a $3 \times 3$ sub-board within the $123 \times 123$ board. How many cells of each color can there be in this $3 \times 3$ sub-board?
(b) Calculate the number of purple cells on the $123 \times 123$ board. | {
"answer": "6724",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If three angles \(x, y, z\) form an arithmetic sequence with a common difference of \(\frac{\pi}{2}\), then \(\tan x \tan y + \tan y \tan z + \tan z \tan x = \) ______. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two arithmetic sequences \\(\{a_n\}\) and \\(\{b_n\}\) with the sum of the first \\(n\) terms denoted as \\(S_n\) and \\(T_n\) respectively. If \\( \dfrac {S_n}{T_n}= \dfrac {2n}{3n+1}\), then \\( \dfrac {a_2}{b_3+b_7}+ \dfrac {a_8}{b_4+b_6}=\) ______. | {
"answer": "\\dfrac {9}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 99 children standing in a circle, each initially holding a ball. Every minute, each child with a ball throws their ball to one of their two neighbors. If a child receives two balls, one of the balls is irrevocably lost. What is the minimum amount of time after which only one ball can remain with the children? | {
"answer": "98",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For certain real values of $p, q, r,$ and $s,$ the equation $x^4+px^3+qx^2+rx+s=0$ has four non-real roots. The product of two of these roots is $17 + 2i$ and the sum of the other two roots is $2 + 5i,$ where $i^2 = -1.$ Find $q.$ | {
"answer": "63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\tan (\alpha+\beta)= \frac {3}{4}$ and $\tan (\alpha- \frac {\pi}{4})= \frac {1}{2}$, find the value of $\tan (\beta+ \frac {\pi}{4})$. | {
"answer": "\\frac {2}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, find the maximum value of $x-y$. | {
"answer": "1+3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c respectively. Given that $(a+c)^2 = b^2 + 2\sqrt{3}ac\sin C$.
1. Find the measure of angle B.
2. If $b=8$, $a>c$, and the area of triangle ABC is $3\sqrt{3}$, find the value of $a$. | {
"answer": "5 + \\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate $7 \cdot 9\frac{2}{5}$. | {
"answer": "65\\frac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a quadratic function $f(x) = ax^2 - 4bx + 1$.
(1) Let set $P = \{-1,1,2,3,4,5\}$ and set $Q = \{-2,-1,1,2,3,4\}$. Randomly select a number from set $P$ as $a$ and from set $Q$ as $b$. Calculate the probability that the function $y = f(x)$ is increasing on the interval $[1,+\infty)$.
(2) Suppose the point $(a, b)$ is a random point within the region defined by $\begin{cases} x+y-8\leqslant 0, \\ x > 0, \\ y > 0 \end{cases}$. Calculate the probability that the function $y = f(x)$ is increasing on the interval $[1,+\infty)$. | {
"answer": "\\dfrac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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