problem stringlengths 10 5.15k | answer dict |
|---|---|
Find $\frac{a^{8}-256}{16 a^{4}} \cdot \frac{2 a}{a^{2}+4}$, if $\frac{a}{2}-\frac{2}{a}=3$. | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify completely: $$\sqrt[3]{40^3 + 70^3 + 100^3}.$$ | {
"answer": "10 \\sqrt[3]{1407}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of quadrilateral $EFGH$, given that $m\angle F = m \angle G = 135^{\circ}$, $EF=4$, $FG=6$, and $GH=8$. | {
"answer": "18\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \(0^{\circ} < \alpha < 30^{\circ}\), and \(\sin^6 \alpha + \cos^6 \alpha = \frac{7}{12}\), then \(1998 \cos \alpha = ?\) | {
"answer": "333 \\sqrt{30}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are some bullfinches in a pet store. One of the children exclaimed, "There are more than fifty bullfinches!" Another replied, "Don't worry, there are fewer than fifty bullfinches." The mother added, "At least there is one!" The father concluded, "Only one of your statements is true." Can you determine how many bullfinches are in the store, knowing that a bullfinch was purchased? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a table, there are 30 coins: 23 ten-ruble coins and 7 five-ruble coins. Out of these, 20 coins are heads up, and the remaining 10 are tails up. What is the smallest number \( k \) such that among any randomly chosen \( k \) coins, there will always be at least one ten-ruble coin that is heads up? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose a parabola has vertex $\left(\frac{3}{2},-\frac{25}{4}\right)$ and follows the equation $y = ax^2 + bx + c$, where $a < 0$ and the product $abc$ is an integer. Find the largest possible value of $a$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the graph of the power function $y=f(x)$ passes through the point $(9, \frac{1}{3})$, find the value of $f(25)$. | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2 \sqrt {3}\sin \frac {x}{3}\cos \frac {x}{3}-2\sin ^{2} \frac {x}{3}$.
(1) Find the range of the function $f(x)$;
(2) In $\triangle ABC$, angles $A$, $B$, $C$ correspond to sides $a$, $b$, $c$ respectively. If $f(C)=1$ and $b^{2}=ac$, find the value of $\sin A$. | {
"answer": "\\frac {\\sqrt {5}-1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $f(x)$ is an even function defined on $\mathbb{R}$ and satisfies $f(x+2)=- \frac{1}{f(x)}$. When $1 \leq x \leq 2$, $f(x)=x-2$. Find $f(6.5)$. | {
"answer": "-0.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point is randomly dropped onto the interval $[8, 13]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-2k-35\right)x^{2}+(3k-9)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$. | {
"answer": "0.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=a^{x}-(k+1)a^{-x}$ where $a > 0$ and $a\neq 1$, which is an odd function defined on $\mathbb{R}$.
1. Find the value of $k$.
2. If $f(1)= \frac {3}{2}$, and the minimum value of $g(x)=a^{2x}+a^{-2x}-2mf(x)$ on $[0,+\infty)$ is $-6$, find the value of $m$. | {
"answer": "2 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular hexagon with side length $1$ is given. Using a ruler construct points in such a way that among the given and constructed points there are two such points that the distance between them is $\sqrt7$ .
Notes: ''Using a ruler construct points $\ldots$ '' means: Newly constructed points arise only as the intersection of straight lines connecting two points that are given or already constructed. In particular, no length can be measured by the ruler. | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the numbers $2, 4, 12, 40$ each exactly once, you can perform operations to obtain 24. | {
"answer": "40 \\div 4 + 12 + 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $\left(\dfrac{a+2}{a^{2}-2a}+\dfrac{8}{4-a^{2}}\right)\div \dfrac{a^{2}-4}{a}$, where $a$ satisfies the equation $a^{2}+4a+1=0$. | {
"answer": "\\dfrac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $g(x)$ be a function defined for all positive real numbers such that $g(x) > 0$ for all $x > 0$ and
\[g(x + y) = \sqrt{g(xy) + 3}\] for all $x > y > 0.$ Determine $g(2023).$ | {
"answer": "\\frac{1 + \\sqrt{13}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a chessboard, $n$ white rooks and $n$ black rooks are arranged such that rooks of different colors do not attack each other. Find the maximum possible value of $n$. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the focus of the parabola $y^{2}=8x$ be $F$, and its directrix be $l$. Let $P$ be a point on the parabola, and $PA\perpendicular l$ with $A$ being the foot of the perpendicular. If the angle of inclination of the line $PF$ is $120^{\circ}$, then $|PF|=$ ______. | {
"answer": "\\dfrac{8}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the set \(T = \{0,1,2,3,4,5,6\}\) and \(M=\left\{\frac{a_{1}}{7}+\frac{a_{2}}{7^{2}}+\frac{a_{3}}{7^{3}}+\frac{a_{4}}{7^{4}}\right\}\), where \(a_{i} \in \mathbf{T}, i=\{1,2,3,4\}\). Arrange the numbers in \(M\) in descending order. Determine the 2005th number. | {
"answer": "\\frac{1}{7} + \\frac{1}{7^2} + \\frac{0}{7^3} + \\frac{4}{7^4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the function \( f(x) \) be defined on \( \mathbb{R} \), and for any \( x \), the condition \( f(x+2) + f(x) = x \) holds. It is also known that \( f(x) = x^3 \) on the interval \( (-2, 0] \). Find \( f(2012) \). | {
"answer": "1006",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set \( A=\left\{\left.\frac{a_{1}}{9}+\frac{a_{2}}{9^{2}}+\frac{a_{3}}{9^{3}}+\frac{a_{4}}{9^{4}} \right\rvert\, a_{i} \in\{0,1,2, \cdots, 8\}, i=1, 2, 3, 4\} \), arrange the numbers in \( A \) in descending order and find the 1997th number. | {
"answer": "\\frac{6}{9} + \\frac{2}{81} + \\frac{3}{729} + \\frac{1}{6561}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$, and the line $l: 4x - 5y + 40 = 0$. Is there a point on the ellipse for which the distance to line $l$ is minimal? If so, what is the minimal distance? | {
"answer": "\\frac{15}{\\sqrt{41}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sector $OAB$ is a quarter of a circle with radius 5 cm. Inside this sector, a circle is inscribed, tangent at three points. Find the radius of the inscribed circle in simplest radical form. | {
"answer": "5\\sqrt{2} - 5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\theta$ is an angle in the third quadrant, and $sin^{4}\theta+cos^{4}\theta= \frac {5}{9}$, then $sin2\theta= \_\_\_\_\_\_$. | {
"answer": "\\frac {2 \\sqrt {2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_1$ is $$\begin{cases} x=4t^{2} \\ y=4t \end{cases}$$ (where $t$ is the parameter). Taking the origin $O$ as the pole and the positive $x$-axis as the polar axis, and using the same unit length, the polar equation of curve $C_2$ is $$ρ\cos(θ+ \frac {π}{4})= \frac { \sqrt {2}}{2}$$.
(Ⅰ) Convert the equation of curve $C_1$ into a standard equation, and the equation of curve $C_2$ into a Cartesian coordinate equation;
(Ⅱ) If curves $C_1$ and $C_2$ intersect at points $A$ and $B$, and the midpoint of $AB$ is $P$, and a perpendicular line to curve $C_2$ at point $P$ intersects curve $C_1$ at points $E$ and $F$, find $|PE|•|PF|$. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A company plans to advertise on TV stations A and B for a total time of no more than 300 minutes in 2011, with an advertising budget of no more than 90,000 yuan. The advertising rates for TV stations A and B are 500 yuan/minute and 200 yuan/minute, respectively. Assuming that for every minute of advertising done by TV stations A and B for the company, the revenue generated for the company is 0.3 million yuan and 0.2 million yuan, respectively. The question is: How should the company allocate its advertising time between TV stations A and B to maximize its revenue, and what is the maximum revenue? | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At 7:00, five sheep, designated as A, B, C, D, and E, have distances to Wolf Castle forming an arithmetic sequence with a common difference of 20 meters. At 8:00, these same five sheep have distances to Wolf Castle forming another arithmetic sequence, but with a common difference of 30 meters, and their order has changed to B, E, C, A, D. Find how many more meters the fastest sheep can run per hour compared to the slowest sheep. | {
"answer": "140",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the quadrilateral \(ABCD\), it is known that \(\angle BAC = \angle CAD = 60^\circ\) and \(AB + AD = AC\). It is also known that \(\angle ACD = 23^\circ\). What is the measure, in degrees, of \(\angle ABC\)? | {
"answer": "83",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The acronym XYZ is displayed on a rectangular grid similarly spaced 1 unit apart. The acronym starts X having length 2 units at the top and the bottom and slanted sides making triangles on both ends. Y is made with a vertical line of 3 units and two slanted lines extending from the midpoint down to 1 unit horizontally on each side. Z is formed by a horizontal top and bottom of 3 units and a diagonal connecting these. Determine the total length of line segments forming XYZ.
A) $13 + 5\sqrt{2} + \sqrt{10}$
B) $14 + 4\sqrt{2} + \sqrt{10}$
C) $13 + 4\sqrt{2} + \sqrt{10}$
D) $12 + 3\sqrt{2} + \sqrt{10}$
E) $14 + 3\sqrt{2} + \sqrt{11}$ | {
"answer": "13 + 4\\sqrt{2} + \\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, side $a$ is 2 units longer than side $b$, and side $b$ is 2 units longer than side $c$. If the sine of the largest angle is $\frac {\sqrt {3}}{2}$, then the area of triangle $\triangle ABC$ is \_\_\_\_\_\_. | {
"answer": "\\frac {15 \\sqrt {3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a math competition, 5 problems were assigned. There were no two contestants who solved exactly the same problems. However, for any problem that is disregarded, for each contestant there is another contestant who solved the same set of the remaining 4 problems. How many contestants participated in the competition? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The intersecting squares from left to right have sides of lengths 12, 9, 7, and 3, respectively. By how much is the sum of the black areas greater than the sum of the gray areas? | {
"answer": "103",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and point $A(1, \frac{3}{2})$ on ellipse $C$ has a sum of distances to these two points equal to $4$.
(I) Find the equation of the ellipse $C$ and the coordinates of its foci.
(II) Let point $P$ be a moving point on the ellipse obtained in (I), and point $Q(0, \frac{1}{2})$. Find the maximum value of $|PQ|$. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Fill in the blanks:
(1) In $\triangle ABC$, $A=60^{\circ}$, $b = 1$, and the area of $\triangle ABC$ is $\sqrt{3}$. Find the value of $\dfrac{a+b+c}{\sin A+\sin B+\sin C}$.
(2) In an arithmetic sequence with a common difference not equal to $0$, ${a_1}+{a_3}=8$, and $a_4$ is the geometric mean of $a_2$ and $a_9$. Find $a_5$.
(3) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $\sqrt{3}\sin A-a\cos B-2a=0$, find $\angle B$.
(4) Given a sequence $\{a_n\}$ where ${a_1}=-60$ and $a_{n+1}=a_n+3$, find the sum $|a_1|+|a_2|+|a_3|+\ldots+|a_{30}|$. | {
"answer": "765",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\{a_n\}$ be a geometric sequence, $\{b_n\}$ be an arithmetic sequence, and $b_1=0$, $c_n=a_n+b_n$. If $\{c_n\}$ is $1$, $1$, $2$, $\ldots$, find the sum of the first $10$ terms of the sequence $\{c_n\}$. | {
"answer": "978",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find \( b \) if \( b \) is the remainder when \( 1998^{10} \) is divided by \( 10^{4} \). | {
"answer": "1024",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( x \in \mathbb{R} \), find the maximum value of \(\frac{\sin x(2-\cos x)}{5-4 \cos x}\). | {
"answer": "\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the trapezoid \( ABCD \), angles \( A \) and \( D \) are right angles, \( AB = 1 \), \( CD = 4 \), \( AD = 5 \). Point \( M \) is taken on side \( AD \) such that \(\angle CMD = 2 \angle BMA \).
In what ratio does point \( M \) divide side \( AD \)? | {
"answer": "2:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An inscribed dodecagon. A convex dodecagon is inscribed in a circle. The lengths of some six sides of the dodecagon are $\sqrt{2}$, and the lengths of the remaining six sides are $\sqrt{24}$. What is the radius of the circle? | {
"answer": "\\sqrt{38}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\dfrac {x^{2}}{a^{2}}-\dfrac {y^{2}}{16}=1$ with a point $P$ on its right branch. The difference in distances from $P$ to the left and right foci is $6$, and the distance from $P$ to the left directrix is $\dfrac {34}{5}$. Calculate the distance from $P$ to the right focus. | {
"answer": "\\dfrac {16}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a_{1}, a_{2}, \cdots, a_{6}$ be any permutation of $\{1,2, \cdots, 6\}$. If the sum of any three consecutive numbers is not divisible by 3, how many such permutations exist? | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In acute triangle $ \Delta ABC $, the sides opposite angles $A, B, C$ are denoted as $a, b, c$, respectively. Let vector $ \overrightarrow{m}=(2,c) $ and vector $ \overrightarrow{n} =\left(\frac{b}{2}\cos C-\sin A, \cos B\right) $, given that $b=\sqrt{3}$ and $ \overrightarrow{m} \bot \overrightarrow{n} $.
(Ⅰ) Find angle $B$;
(Ⅱ) Find the maximum area of $ \Delta ABC $ and the lengths of the other two sides $a, c$ under this condition. | {
"answer": "\\frac{3\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in $\triangle ABC$, $B= \frac{\pi}{4}$ and the height to side $BC$ is equal to $\frac{1}{3}BC$, calculate the value of $\sin A$. | {
"answer": "\\frac{3\\sqrt{10}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During intervals, students played table tennis. Any two students played against each other no more than one game. At the end of the week, it turned out that Petya played half, Kolya played a third, and Vasya played a fifth of all the games played during the week. How many games could have been played during the week if it is known that Vasya did not play with either Petya or Kolya? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When dividing the numbers 312837 and 310650 by some three-digit natural number, the remainders are the same. Find this remainder. | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle with radius $R$ and a fixed point $A$ on the circumference, a point is randomly chosen on the circumference and connected to point $A$. The probability that the length of the chord formed is between $R$ and $\sqrt{3}R$ is ______. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( x, y, z \in \mathbf{R}_{+} \) and \( x^{2} + y^{2} + z^{2} = 1 \), find the value of \( z \) when \(\frac{(z+1)^{2}}{x y z} \) reaches its minimum. | {
"answer": "\\sqrt{2} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the real numbers \( x \) and \( y \) that satisfy \( xy + 6 = x + 9y \) and \( y \in (-\infty, 1) \), find the maximum value of \((x+3)(y+1)\). | {
"answer": "27 - 12\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Eight balls are randomly and independently painted either black or white with equal probability. What is the probability that each ball is painted a color such that it is different from the color of at least half of the other 7 balls?
A) $\frac{21}{128}$
B) $\frac{28}{128}$
C) $\frac{35}{128}$
D) $\frac{42}{128}$ | {
"answer": "\\frac{35}{128}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a \in \mathbf{R}_{+} \). If the function
\[
f(x)=\frac{a}{x-1}+\frac{1}{x-2}+\frac{1}{x-6} \quad (3 < x < 5)
\]
achieves its maximum value at \( x=4 \), find the value of \( a \). | {
"answer": "-\\frac{9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Dragoons take up \(1 \times 1\) squares in the plane with sides parallel to the coordinate axes such that the interiors of the squares do not intersect. A dragoon can fire at another dragoon if the difference in the \(x\)-coordinates of their centers and the difference in the \(y\)-coordinates of their centers are both at most 6, regardless of any dragoons in between. For example, a dragoon centered at \((4,5)\) can fire at a dragoon centered at the origin, but a dragoon centered at \((7,0)\) cannot. A dragoon cannot fire at itself. What is the maximum number of dragoons that can fire at a single dragoon simultaneously? | {
"answer": "168",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest constant $N$, such that for any triangle with sides $a, b,$ and $c$, and perimeter $p = a + b + c$, the inequality holds:
\[
\frac{a^2 + b^2 + k}{c^2} > N
\]
where $k$ is a constant. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the infinite series: $1 - \frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \cdots$. Let $T$ be the limiting sum of this series. Find $T$.
**A)** $\frac{3}{26}$
**B)** $\frac{15}{26}$
**C)** $\frac{27}{26}$
**D)** $\frac{1}{26}$
**E)** $\frac{40}{26}$ | {
"answer": "\\frac{15}{26}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the integral $\int_{0}^{\frac{\pi}{2}} \sin^{2} \frac{x}{2} dx =$ \_\_\_\_\_\_. | {
"answer": "\\frac{\\pi}{4} - \\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $y=-x+1$ and the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$, which intersect at points $A$ and $B$.
(1) If the eccentricity of the ellipse is $\frac{\sqrt{3}}{3}$ and the focal length is $2$, find the length of the line segment $AB$.
(2) If vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are perpendicular to each other (where $O$ is the origin), find the maximum length of the major axis of the ellipse when its eccentricity $e \in [\frac{1}{2}, \frac{\sqrt{2}}{2}]$. | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $P$ lies on the line $3x+4y+8=0$, and $PA$ and $PB$ are the two tangents drawn from $P$ to the circle $x^{2}+y^{2}-2x-2y+1=0$. Let $A$ and $B$ be the points of tangency, and $C$ be the center of the circle. Find the minimum possible area of the quadrilateral $PACB$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given vectors $m=(\sin \frac{A}{2},\cos \frac{A}{2})$ and $n=(\cos \frac{A}{2},-\cos \frac{A}{2})$, and $2m\cdot n+|m|=\frac{ \sqrt{2}}{2}$, find $\angle A=$____. | {
"answer": "\\frac{5\\pi }{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( \theta_{1}, \theta_{2}, \theta_{3}, \theta_{4} \in \mathbf{R}^{+} \) and \( \theta_{1} + \theta_{2} + \theta_{3} + \theta_{4} = \pi \), find the minimum value of \( \left(2 \sin^{2} \theta_{1} + \frac{1}{\sin^{2} \theta_{1}}\right)\left(2 \sin^{2} \theta_{2} + \frac{1}{\sin^{2} \theta_{2}}\right)\left(2 \sin^{2} \theta_{3} + \frac{1}{\sin^{2} \theta_{3}}\right)\left(2 \sin^{2} \theta_{4} + \frac{1}{\sin^{2} \theta_{1}}\right) \). | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
a) What is the maximum number of squares on an $8 \times 8$ board that can be painted black such that in each corner of three squares, there is at least one unpainted square?
b) What is the minimum number of squares on an $8 \times 8$ board that need to be painted black such that in each corner of three squares, there is at least one black square? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Minimize \(\boldsymbol{F}=\boldsymbol{x}_{2}-\boldsymbol{x}_{1}\) for non-negative \(x_{1}\) and \(x_{2}\), subject to the system of constraints:
$$
\left\{\begin{aligned}
-2 x_{1}+x_{2}+x_{3} &=2 \\
x_{1}-2 x_{2}+x_{4} &=2 \\
x_{1}+x_{2}+x_{5} &=5
\end{aligned}\right.
$$ | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bus arrives randomly sometime between 2:00 and 3:00 and waits for 15 minutes before leaving. If Carla arrives randomly between 2:00 and 3:00, what is the probability that the bus will still be there when Carla arrives? | {
"answer": "\\frac{7}{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that four people A, B, C, D are randomly selected for a volunteer activity, find the probability that A is selected and B is not. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Construct the cross-section of a triangular pyramid \( A B C D \) with a plane passing through the midpoints \( M \) and \( N \) of edges \( A C \) and \( B D \) and the point \( K \) on edge \( C D \), for which \( C K: K D = 1: 2 \). In what ratio does this plane divide edge \( A B \)? | {
"answer": "1:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Grain warehouses A and B each originally stored a certain number of full bags of grain. If 90 bags are transferred from warehouse A to warehouse B, the number of bags in warehouse B will be twice the number in warehouse A. If an unspecified number of bags are transferred from warehouse B to warehouse A, the number of bags in warehouse A will be six times the number in warehouse B. What is the minimum number of bags originally stored in warehouse A? | {
"answer": "153",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $$\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1(a＞0，b＞0)$$, the sum of the two line segments that are perpendicular to the two asymptotes and pass through one of its foci is $a$. Find the eccentricity of the hyperbola. | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin\alpha = \frac{3}{5}$, and $\alpha \in \left(\frac{\pi}{2}, \pi \right)$.
(1) Find the value of $\tan\left(\alpha+\frac{\pi}{4}\right)$;
(2) If $\beta \in (0, \frac{\pi}{2})$, and $\cos(\alpha-\beta) = \frac{1}{3}$, find the value of $\cos\beta$. | {
"answer": "\\frac{6\\sqrt{2} - 4}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Assume that savings banks offer the same interest rate as the inflation rate for a year to deposit holders. The government takes away $20 \%$ of the interest as tax. By what percentage does the real value of government interest tax revenue decrease if the inflation rate drops from $25 \%$ to $16 \%$, with the real value of the deposit remaining unchanged? | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three boys \( B_{1}, B_{2}, B_{3} \) and three girls \( G_{1}, G_{2}, G_{3} \) are to be seated in a row according to the following rules:
1) A boy will not sit next to another boy and a girl will not sit next to another girl,
2) Boy \( B_{1} \) must sit next to girl \( G_{1} \).
If \( s \) is the number of different such seating arrangements, find the value of \( s \). | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( ABC \) be a triangle in which \( \angle ABC = 60^\circ \). Let \( I \) and \( O \) be the incentre and circumcentre of \( ABC \), respectively. Let \( M \) be the midpoint of the arc \( BC \) of the circumcircle of \( ABC \), which does not contain the point \( A \). Determine \( \angle BAC \) given that \( MB = OI \). | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}(1-\ln (1+\sqrt[3]{x}))^{\frac{x}{\sin ^{4} \sqrt[3]{x}}}
$$ | {
"answer": "e^{-1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$, respectively. Given that $\frac{{c\sin C}}{{\sin A}} - c = \frac{{b\sin B}}{{\sin A}} - a$ and $b = 2$, find:
$(1)$ The measure of angle $B$;
$(2)$ If $a = \frac{{2\sqrt{6}}}{3}$, find the area of triangle $\triangle ABC$. | {
"answer": "1 + \\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the volume of solid $T$ defined by the inequalities $|x| + |y| \leq 2$, $|x| + |z| \leq 2$, and $|y| + |z| \leq 2$. | {
"answer": "\\frac{32}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point \( P \) moves on the circle \( C: x^{2}+(y+2)^{2}=\frac{1}{4} \), and point \( Q \) moves on the curve \( y=a x^{2} \) (where \( a > 0 \) and \( -1 \leq x \leq 2 \)), if the maximum value of \(|PQ|\) is \(\frac{9}{2}\), then find \( a \). | {
"answer": "\\frac{\\sqrt{3} - 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum sample size for which the precision of the estimate of the population mean $a$ based on the sample mean with a confidence level of 0.975 is $\delta=0.3$, given that the standard deviation $\sigma=1.2$ of the normally distributed population is known. | {
"answer": "62",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all three-digit numbers \( \overline{\mathrm{MGU}} \) consisting of distinct digits \( M, \Gamma, \) and \( U \) for which the equality \( \overline{\mathrm{MGU}} = (M + \Gamma + U) \times (M + \Gamma + U - 2) \) holds. | {
"answer": "195",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $503, 1509, 3015, 6021, \dots$, determine how many of the first $1500$ numbers in this sequence are divisible by $503$. | {
"answer": "1500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cara is sitting at a circular table with her seven friends. Determine the different possible pairs of people Cara could be sitting between on this table. | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the following diagram, \(\angle ACB = 90^\circ\), \(DE \perp BC\), \(BE = AC\), \(BD = \frac{1}{2} \mathrm{~cm}\), and \(DE + BC = 1 \mathrm{~cm}\). Suppose \(\angle ABC = x^\circ\). Find the value of \(x\). | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \( f(x) = \frac{1}{\sqrt[3]{1 - x^3}} \). Find \( f(f(f( \ldots f(19)) \ldots )) \), calculated 95 times. | {
"answer": "\\sqrt[3]{1 - \\frac{1}{19^3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sample data set $3$, $3$, $4$, $4$, $5$, $6$, $6$, $7$, $7$, calculate the standard deviation of the data set. | {
"answer": "\\frac{2\\sqrt{5}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $d$ and $e$ are digits. For how many pairs of $(d, e)$ is $2.0d06e > 2.006$? | {
"answer": "99",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the operation $x*y$ be defined as $x*y = (x+1)(y+1)$. The operation $x^{*2}$ is defined as $x^{*2} = x*x$. Calculate the value of the polynomial $3*(x^{*2}) - 2*x + 1$ when $x=2$. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that the angles of triangle $PQR$ satisfy
\[\cos 3P + \cos 3Q + \cos 3R = 1.\]Two sides of the triangle have lengths 12 and 15. Find the maximum length of the third side. | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let's call a number "remarkable" if it has exactly 4 different natural divisors, among which there are two such that neither is a multiple of the other. How many "remarkable" two-digit numbers exist? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $(xOy)$, the parametric equations of curve $C_{1}$ are given by $\begin{cases}x=2t-1 \\ y=-4t-2\end{cases}$ $(t$ is the parameter$)$, and in the polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis, the polar equation of curve $C_{2}$ is $\rho= \frac{2}{1-\cos \theta}$.
(1) Write the Cartesian equation of curve $C_{2}$;
(2) Let $M_{1}$ be a point on curve $C_{1}$, and $M_{2}$ be a point on curve $C_{2}$. Find the minimum value of $|M_{1}M_{2}|$. | {
"answer": "\\frac{3 \\sqrt{5}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A four-digit integer $m$ and the four-digit integer obtained by reversing the order of the digits of $m$ are both divisible by 63. If $m$ is also divisible by 11, what is the greatest possible value of $m$? | {
"answer": "9702",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$\Phi$ is the union of all triangles that are symmetric of the triangle $ABC$ wrt a point $O$ , as point $O$ moves along the triangle's sides. If the area of the triangle is $E$ , find the area of $\Phi$ . | {
"answer": "2E",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the sum $\sum_{n=1}^{9} \frac{1}{n(n+1)(n+2)}$ written in its lowest terms be $\frac{p}{q}$ . Find the value of $q - p$ . | {
"answer": "83",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Select 5 different letters from the word "equation" to arrange in a row, including the condition that the letters "qu" are together and in the same order. | {
"answer": "480",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Another trapezoid \(ABCD\) has \(AD\) parallel to \(BC\). \(AC\) and \(BD\) intersect at \(P\). If \(\frac{[ADP]}{[BCP]} = \frac{1}{2}\), find \(\frac{[ADP]}{[ABCD]}\). (Here, the notation \(\left[P_1 \cdots P_n\right]\) denotes the area of the polygon \(P_1 \cdots P_n\)). | {
"answer": "3 - 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What positive integer can $n$ represent if it is known that by erasing the last three digits of the number $n^{3}$, we obtain the number $n$? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \( O \) is located inside an isosceles right triangle \( ABC \). The distance from \( O \) to the vertex \( A \) of the right angle is 6, to the vertex \( B \) is 4, and to the vertex \( C \) is 8. Find the area of triangle \( ABC \). | {
"answer": "20 + 6\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic-geometric sequence $\{a\_n\}$, where $a\_1 + a\_3 = 10$ and $a\_4 + a\_6 = \frac{5}{4}$, find its fourth term and the sum of the first five terms. | {
"answer": "\\frac{31}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mr. and Mrs. Smith have three children. They own a family van with a driver's seat, a passenger seat in the front, and two seats in the back. Either Mr. Smith or Mrs. Smith must sit in the driver's seat. How many seating arrangements are possible if one of the children insists on sitting in the front passenger seat? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jill has 8 red marbles and 4 blue marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 8 marbles. What is the probability that exactly four of the marbles that she removes are red? Express your answer as a decimal rounded to the nearest thousandth. | {
"answer": "0.171",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right triangle \( ABC \) with \(\angle A = 60^\circ\) and hypotenuse \( AB = 2 + 2\sqrt{3} \), a line \( p \) is drawn through vertex \( B \) parallel to \( AC \). Points \( D \) and \( E \) are placed on line \( p \) such that \( AB = BD \) and \( BC = BE \). Let \( F \) be the intersection point of lines \( AD \) and \( CE \). Find the possible value of the perimeter of triangle \( DEF \).
Options:
\[
\begin{gathered}
2 \sqrt{2} + \sqrt{6} + 9 + 5 \sqrt{3} \\
\sqrt{3} + 1 + \sqrt{2} \\
9 + 5 \sqrt{3} + 2 \sqrt{6} + 3 \sqrt{2}
\end{gathered}
\]
\[
1 + \sqrt{3} + \sqrt{6}
\] | {
"answer": "1 + \\sqrt{3} + \\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $AC=8$, $BC=7$, $\cos B=-\frac{1}{7}$.
(1) Find the measure of angle $A$;
(2) Find the area of $\triangle ABC$. | {
"answer": "6\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Jessie moves from 0 to 24 in six steps, and travels four steps to reach point x, then one more step to reach point z, and finally one last step to point y, calculate the value of y. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $(xOy)$, the parametric equations of the curve $C$ are given by: $\begin{cases} x=2\cos\theta \\ y=\sin\theta \end{cases}$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis.
(1) If the horizontal coordinate of each point on the curve $C$ remains unchanged and the vertical coordinate is stretched to twice its original length, obtain the curve $C_{1}$. Find the polar equation of $C_{1}$;
(2) The polar equation of the straight line $l$ is $\rho\sin\left(\theta+\frac{\pi}{3}\right)=\sqrt{3}$, which intersects the curve $C_{1}$ at points $A$ and $B$. Calculate the area of triangle $AOB$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(a \leq b < c\) are the side lengths of a right triangle, find the maximum constant \(M\) such that
$$
\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq \frac{M}{a+b+c} .
$$ | {
"answer": "5 + 3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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