problem stringlengths 10 5.15k | answer dict |
|---|---|
A trapezoid has side lengths 4, 6, 8, and 10. The trapezoid can be rearranged to form different configurations with sides 4 and 8 as the parallel bases. Calculate the total possible area of the trapezoid with its different configurations.
A) $24\sqrt{2}$
B) $36\sqrt{2}$
C) $42\sqrt{2}$
D) $48\sqrt{2}$
E) $54\sqrt{2}$ | {
"answer": "48\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The points \((2,3)\) and \((3, 7)\) lie on a circle whose center is on the \(x\)-axis. What is the radius of the circle? | {
"answer": "\\frac{\\sqrt{1717}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify the expression: $$\dfrac{\sqrt{450}}{\sqrt{288}} + \dfrac{\sqrt{245}}{\sqrt{96}}.$$ Express your answer as a common fraction. | {
"answer": "\\frac{30 + 7\\sqrt{30}}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Li is ready to complete the question over the weekend: Simplify and evaluate $(3-2x^{2}-5x)-(\square x^{2}+3x-4)$, where $x=-2$, but the coefficient $\square$ is unclearly printed.<br/>$(1)$ She guessed $\square$ as $8$. Please simplify $(3-2x^{2}-5x)-(8x^{2}+3x-4)$ and find the value of the expression when $x=-2$;<br/>$(2)$ Her father said she guessed wrong, the standard answer's simplification does not contain quadratic terms. Please calculate and determine the value of $\square$ in the original question. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Shift the graph of the function $y=\cos(\frac{π}{2}-2x)$ by an amount corresponding to the difference between the arguments of $y=\sin(2x-\frac{π}{4})$ and $y=\cos(\frac{π}{2}-2x)$. | {
"answer": "\\frac{\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of $A + B + C$, where $A$, $B$, and $C$ are the dimensions of a three-dimensional rectangular box with faces having areas $40$, $40$, $90$, $90$, $100$, and $100$ square units. | {
"answer": "\\frac{83}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \ln x + \ln (ax+1) - \frac {3a}{2}x + 1$ ($a \in \mathbb{R}$).
(1) Discuss the intervals of monotonicity for the function $f(x)$.
(2) When $a = \frac {2}{3}$, if the inequality $xe^{x-\frac {1}{2}} + m \geqslant f(x)$ holds true for all $x$, find the minimum value of $m$, where $e$ is the base of the natural logarithm. | {
"answer": "\\ln \\frac {2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For any positive integer \( n \), define
\( g(n) =\left\{\begin{matrix}\log_{4}{n}, &\text{if }\log_{4}{n}\text{ is rational,}\\ 0, &\text{otherwise.}\end{matrix}\right. \)
What is \( \sum_{n = 1}^{1023}{g(n)} \)?
**A** \( \frac{40}{2} \)
**B** \( \frac{42}{2} \)
**C** \( \frac{45}{2} \)
**D** \( \frac{48}{2} \)
**E** \( \frac{50}{2} \) | {
"answer": "\\frac{45}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given sets $A=\{2,3,4\}$ and $B=\{a+2,a\}$, if $A \cap B = B$, find $A^cB$ ___. | {
"answer": "\\{3\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A company is planning to increase the annual production of a product by implementing technical reforms in 2013. According to the survey, the product's annual production volume $x$ (in ten thousand units) and the technical reform investment $m$ (in million yuan, where $m \ge 0$) satisfy the equation $x = 3 - \frac{k}{m + 1}$ ($k$ is a constant). Without the technical reform, the annual production volume can only reach 1 ten thousand units. The fixed investment for producing the product in 2013 is 8 million yuan, and an additional investment of 16 million yuan is required for each ten thousand units produced. Due to favorable market conditions, all products produced can be sold. The company sets the selling price of each product at 1.5 times its production cost (including fixed and additional investments).
1. Determine the value of $k$ and express the profit $y$ (in million yuan) of the product in 2013 as a function of the technical reform investment $m$ (profit = sales revenue - production cost - technical reform investment).
2. When does the company's profit reach its maximum with the technical reform investment in 2013? Calculate the maximum profit. | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) If $\cos (\frac{\pi}{4}+x) = \frac{3}{5}$, and $\frac{17}{12}\pi < x < \frac{7}{4}\pi$, find the value of $\frac{\sin 2x + 2\sin^2 x}{1 - \tan x}$.
(2) Given the function $f(x) = 2\sqrt{3}\sin x\cos x + 2\cos^2 x - 1 (x \in \mathbb{R})$, if $f(x_0) = \frac{6}{5}$, and $x_0 \in [\frac{\pi}{4}, \frac{\pi}{2}]$, find the value of $\cos 2x_0$. | {
"answer": "\\frac{3 - 4\\sqrt{3}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\overrightarrow{m}=(2\sqrt{3},1)$, $\overrightarrow{n}=(\cos^2 \frac{A}{2},\sin A)$, where $A$, $B$, and $C$ are the interior angles of $\triangle ABC$;
$(1)$ When $A= \frac{\pi}{2}$, find the value of $|\overrightarrow{n}|$;
$(2)$ If $C= \frac{2\pi}{3}$ and $|AB|=3$, when $\overrightarrow{m} \cdot \overrightarrow{n}$ takes the maximum value, find the magnitude of $A$ and the length of side $BC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a match between two people is played with a best-of-five-games format, where the winner is the first to win three games, and that the probability of person A winning a game is $\dfrac{2}{3}$, calculate the probability that person A wins with a score of $3:1$. | {
"answer": "\\dfrac{8}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Select 4 out of 6 sprinters to participate in a 4×100 relay race. If neither A nor B runs the first leg, then there are $\boxed{\text{different}}$ possible team compositions. | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two points A (-2, 0), B (0, 2), and point C is any point on the circle $x^2+y^2-2x=0$, determine the minimum area of $\triangle ABC$. | {
"answer": "3- \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{\sqrt{5}}{5}$, and its right latus rectum equation is $x=5$.
(1) Find the equation of the ellipse;
(2) A line $l$ with a slope of $1$ passes through the right focus $F$ of the ellipse, intersecting the ellipse $C$ at points $A$ and $B$. $P$ is a moving point on the ellipse. Find the maximum area of $\triangle PAB$. | {
"answer": "\\frac{16\\sqrt{10}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin ^{2}x+ \frac{ \sqrt{3}}{2}\sin 2x$.
(1) Find the interval(s) where the function $f(x)$ is monotonically decreasing.
(2) In $\triangle ABC$, if $f(\frac{A}{2})=1$ and the area of the triangle is $3\sqrt{3}$, find the minimum value of side $a$. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In Geometrytown, all streets are $30$ feet wide and the blocks they enclose are rectangles with side lengths of $300$ feet and $500$ feet. Anne runs around the block on the $300$-foot side of the street, while Bob runs on the opposite side of the street. How many more feet than Anne does Bob run for every lap around the block? | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a triangle $\triangle ABC$, $\angle A = 45^\circ$ and $\angle B = 60^\circ$. A line segment $DE$, with $D$ on $AB$ and $\angle ADE = 30^\circ$, divides $\triangle ABC$ into two pieces of equal area. Determine the ratio $\frac{AD}{AB}$.
A) $\frac{\sqrt{3}}{4}$
B) $\frac{\sqrt{6} + \sqrt{2}}{4}$
C) $\frac{1}{4}$
D) $\frac{1}{2}$ | {
"answer": "\\frac{\\sqrt{6} + \\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the following equations:
a) $\log _{1 / 5} \frac{2+x}{10}=\log _{1 / 5} \frac{2}{x+1}$;
b) $\log _{3}\left(x^{2}-4 x+3\right)=\log _{3}(3 x+21)$;
c) $\log _{1 / 10} \frac{2 x^{2}-54}{x+3}=\log _{1 / 10}(x-4)$;
d) $\log _{(5+x) / 3} 3=\log _{-1 /(x+1)} 3$. | {
"answer": "-4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A piece of string fits exactly once around the perimeter of a rectangle with a length of 16 and a width of 10. Rounded to the nearest whole number, what is the area of the largest circle that can be formed from the piece of string? | {
"answer": "215",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $M$ be a point on the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, with $F_1$ and $F_2$ as its foci. If $\angle F_1MF_2 = \frac{\pi}{6}$, calculate the area of $\triangle MF_1F_2$. | {
"answer": "16(2 - \\sqrt{3})",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four buddies bought a ball. First one paid half of the ball price. Second one gave one third of money that other three gave. Third one paid a quarter of sum paid by other three. Fourth paid $5\$ $. How much did the ball cost? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let
\[ x^6 - 3x^3 - x^2 - x - 2 = q_1(x) q_2(x) \dotsm q_m(x), \]
where each non-constant polynomial $q_i(x)$ is monic with integer coefficients, and cannot be factored further over the integers. Compute $q_1(3) + q_2(3) + \dots + q_m(3)$. | {
"answer": "634",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Exactly three faces of a \(2 \times 2 \times 2\) cube are partially shaded. Calculate the fraction of the total surface area of the cube that is shaded. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute
$$\sum_{k=1}^{2000} k(\lceil \log_{2}{k}\rceil- \lfloor\log_{2}{k} \rfloor).$$ | {
"answer": "1998953",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Medians $\overline{AD}$ and $\overline{BE}$ of $\triangle ABC$ intersect at an angle of $45^\circ$. If $AD = 12$ and $BE = 16$, then calculate the area of $\triangle ABC$. | {
"answer": "64\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute and simplify the following expressions:
1. $(1)(1 \frac{1}{2})^{0}-(1-0.5^{-2})÷(\frac{27}{8})^{\frac{2}{3}}$
2. $\sqrt{2 \sqrt{2 \sqrt{2}}}$ | {
"answer": "2^{\\frac{7}{8}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Estimate
99×71≈
25×39≈
124÷3≈
398÷5≈ | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One hundred bricks, each measuring $3''\times 8''\times 15''$, are stacked to form a tower. Each brick can contribute $3''$, $8''$, or $15''$ to the height of the tower. How many different tower heights can be achieved using all one hundred bricks? | {
"answer": "1201",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\cos{\alpha}=-\frac{4}{5}$, where $\alpha$ is an angle in the third quadrant, find the value of $\sin{\left(\alpha-\frac{\pi}{4}\right)$. | {
"answer": "\\frac{\\sqrt{2}}{10}",
"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 $(2a-c)\cos B = b\cos C$;
(1) Find the magnitude of angle $B$;
(2) Let $\overrightarrow{m}=(\sin A, \cos 2A), \overrightarrow{n}=(4k,1)$ ($k>1$), and the maximum value of $\overrightarrow{m} \cdot \overrightarrow{n}$ is 5, find the value of $k$. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation of an ellipse is $\dfrac {x^{2}}{a^{2}} + \dfrac {y^{2}}{b^{2}} = 1 \; (a > b > 0)$, one of its vertices is $M(0,1)$, and its eccentricity $e = \dfrac {\sqrt {6}}{3}$.
$(1)$ Find the equation of the ellipse;
$(2)$ Suppose a line $l$ intersects the ellipse at points $A$ and $B$, and the distance from the origin $O$ to line $l$ is $\dfrac {\sqrt {3}}{2}$, find the maximum area of $\triangle AOB$. | {
"answer": "\\dfrac {\\sqrt {3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α \in \left( \frac{π}{2}, π \right)$ and $3\cos 2α = \sin \left( \frac{π}{4} - α \right)$, find the value of $\sin 2α$. | {
"answer": "-\\frac{17}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a\cos(B-C)+a\cos A=2\sqrt{3}b\sin C\cos A$.
$(1)$ Find angle $A$;
$(2)$ If the perimeter of $\triangle ABC$ is $8$ and the radius of the circumcircle is $\sqrt{3}$, find the area of $\triangle ABC$. | {
"answer": "\\frac{4\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=|2x+1|-|x|-2$.
(1) Solve the inequality $f(x)\geqslant 0$;
(2) If there exists a real number $x$ such that $f(x)-a\leqslant |x|$, find the minimum value of the real number $a$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a variant of SHORT BINGO, a $5\times5$ card is modified such that the middle square is marked as FREE, and the remaining 24 squares are populated with distinct numbers. The modification includes:
- First column contains 5 distinct numbers from the set $1-15$.
- Second column contains 5 distinct numbers from the set $16-30$.
- Third column (excluding the FREE middle square) contains 4 distinct numbers from $31-45$.
- Fourth column contains 5 distinct numbers from the set $46-60$.
- Fifth column contains 5 distinct numbers from the set $61-75$.
How many distinct possibilities are there for the values in the first column of this modified SHORT BINGO card? | {
"answer": "360360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S_n$ and $T_n$ respectively be the sum of the first $n$ terms of the arithmetic sequences $\{a_n\}$ and $\{b_n\}$. Given that $\frac{S_n}{T_n} = \frac{n}{2n+1}$ for $n \in \mathbb{N}^*$, find the value of $\frac{a_5}{b_5}$. | {
"answer": "\\frac{9}{19}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given point A (-2, 1) and circle C: $(x-2)^2+(y-2)^2=1$, a ray of light is emitted from point A to the x-axis and then reflects in the direction of the tangent to the circle. The distance traveled by the ray of light from point A to the tangent point is ______. | {
"answer": "2\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the trapezoid $ABCD$, $CD$ is three times the length of $AB$. Given that the area of the trapezoid is $30$ square units, determine the area of $\triangle ABC$.
[asy]
draw((0,0)--(1,4)--(7,4)--(12,0)--cycle);
draw((1,4)--(0,0));
label("$A$",(1,4),NW);
label("$B$",(7,4),NE);
label("$C$",(12,0),E);
label("$D$",(0,0),W);
[/asy] | {
"answer": "7.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=28,$ find the smallest possible value of $|w^3+z^3|.$ | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given Laila had a 10-hour work day and attended three different meetings during this period, the first meeting took 50 minutes, the second meeting twice as long as the first, and the third meeting half as long as the second meeting, calculate the percentage of her work day that was spent attending meetings. | {
"answer": "33.33\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all values of $x$ with $0 \le x < \pi$ that satisfy $\sin x - \cos x = 1$. Enter all the solutions, separated by commas. | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the graph of the linear function $y=kx+b+2$ ($k \neq 0$) intersects the positive half of the x-axis at point A and the positive half of the y-axis at point B.
(1) Express the area $S_{\triangle AOB}$ of triangle $AOB$ in terms of $b$ and $k$.
(2) If the area $S_{\triangle AOB} = |OA| + |OB| + 3$,
① Express $k$ in terms of $b$ and determine the range of values for $b$.
② Find the minimum value of the area of triangle $AOB$. | {
"answer": "7 + 2\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of triangle $ABC$ is 1. Let $A_1, B_1, C_1$ be the midpoints of the sides $BC, CA, AB$ respectively. Points $K, L, M$ are taken on the segments $AB_1, CA_1, BC_1$ respectively. What is the minimum area of the common part of triangles $KLM$ and $A_1B_1C_1$? | {
"answer": "1/8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When written out in full, the number \((10^{2020}+2020)^{2}\) has 4041 digits. What is the sum of the digits of this 4041-digit number?
A) 9
B) 17
C) 25
D) 2048
E) 4041 | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the quadratic function $f(x)=3x^{2}-2x$, the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, and the point $(n,S_{n})$ (where $n \in \mathbb{N}^{*}$) is on the graph of the function $y=f(x)$.
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;
$(2)$ Let $b_{n}=\frac{3}{a_{n}a_{n+1}}$, and $T_{n}$ be the sum of the first $n$ terms of the sequence $\{b_{n}\}$. Find the smallest positive integer $m$ such that $T_{n} < \frac{m}{20}$ holds for all $n \in \mathbb{N}^{*}$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence \(\left\{a_{n}\right\}\) satisfies \(a_{1}=1, \sqrt{\frac{1}{a_{n}^{2}}+4}=\frac{1}{a_{n+1}}\). Let \(S_{n}=\sum_{i=1}^{n} a_{i}^{2}\). If \(S_{2 n+1}-S_{n} \leqslant \frac{t}{30}\) holds for any \(n \in \mathbf{N}^{*}\), what is the smallest positive integer \(t\)? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A starship enters an extraordinary meteor shower. Some of the meteors travel along a straight line at the same speed, equally spaced. Another group of meteors travels similarly along another straight line, parallel to the first, with the same speed but in the opposite direction, also equally spaced. The ship travels parallel to these lines. Astronaut Gavrila recorded that every 7 seconds the ship encounters meteors coming towards it, and every 13 seconds it overtakes meteors traveling in the same direction. He wondered how often the meteors would pass by if the ship were stationary. He thought that he should take the arithmetic mean of the two given times. Is Gavrila correct? If so, write down this arithmetic mean. If not, indicate the correct time in seconds, rounded to the nearest tenth. | {
"answer": "9.1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Around the outside of a $6$ by $6$ square, construct four semicircles with the four sides of the square as their diameters. Another square, $ABCD$, has its sides parallel to the corresponding sides of the original square, and each side of $ABCD$ is tangent to one of the semicircles. Calculate the area of square $ABCD$.
A) 100
B) 144
C) 196
D) 256
E) 324 | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\binom{21}{13}=20349$, $\binom{21}{14}=11628$, and $\binom{23}{15}=490314$, find $\binom{22}{15}$. | {
"answer": "458337",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x \) and \( y \) be positive integers such that
\[ x^2 + y^2 - 2017xy > 0 \]
and it is not a perfect square. Find the minimum value of \( x^2 + y^2 - 2017xy \). | {
"answer": "2019",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=A\sin^2(\omega x+\frac{\pi}{8})$ ($A>0, \omega>0$), the graph of which is symmetric with respect to the point $({\frac{\pi}{2},1})$, and its minimum positive period is $T$, where $\frac{\pi}{2}<T<\frac{3\pi}{2}$. Find the value of $\omega$. | {
"answer": "\\frac{5}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many times does the digit 9 appear in the list of all integers from 1 to 1000? | {
"answer": "301",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that positive real numbers $x$ and $y$ satisfy $e^{x}=y\ln x+y\ln y$, then the minimum value of $\frac{{e}^{x}}{x}-\ln y$ is ______. | {
"answer": "e-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of the function $y= \sqrt {3}\cos x+\sin x, (x\in \mathbb{R})$ is translated to the right by $\theta$ ($\theta>0$) units. Determine the minimum value of $\theta$ such that the resulting graph is symmetric about the y-axis. | {
"answer": "\\frac{5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose there exist constants $A$ , $B$ , $C$ , and $D$ such that \[n^4=A\binom n4+B\binom n3+C\binom n2 + D\binom n1\] holds true for all positive integers $n\geq 4$ . What is $A+B+C+D$ ?
*Proposed by David Altizio* | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $2\sin 2C\cdot\cos C-\sin 3C= \sqrt {3}(1-\cos C)$.
(1) Find the measure of angle $C$;
(2) If $AB=2$, and $\sin C+\sin (B-A)=2\sin 2A$, find the area of $\triangle ABC$. | {
"answer": "\\dfrac {2 \\sqrt {3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $|$$\overrightarrow {a}$$ $|=1$, $\overrightarrow {b}$ $=$ ($ $\frac { \sqrt {3}}{3} $, $ \frac { \sqrt {3}}{3}$), and $|$ $\overrightarrow {a}$ $+3 \overrightarrow {b}$ $|=2$, find the projection of $\overrightarrow {b}$ in the direction of $\overrightarrow {a}$. | {
"answer": "- \\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $R_n$ and $U_n$ be the respective sums of the first $n$ terms of two arithmetic sequences. If the ratio $R_n:U_n = (3n+5):(2n+13)$ for all $n$, find the ratio of the seventh term of the first sequence to the seventh term of the second sequence.
A) $7:6$
B) $4:3$
C) $3:2$
D) $5:4$ | {
"answer": "4:3",
"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"
} |
What is the value of $\frac{13!-12!+144}{11!}$? | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the equation of circle $M$ is given by $x^2+y^2-4x\cos \alpha-2y\sin \alpha+3\cos^2\alpha=0$ (where $\alpha$ is a parameter), and the parametric equation of line $l$ is $\begin{cases}x=\tan \theta \\ y=1+t\sin \theta\end{cases}$ (where $t$ is a parameter).
$(I)$ Find the parametric equation of the trajectory $C$ of the center of circle $M$, and explain what curve it represents;
$(II)$ Find the maximum chord length cut by the trajectory $C$ on line $l$. | {
"answer": "\\dfrac {4 \\sqrt {3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the probability that a triangle formed by randomly selecting 3 vertices from a regular octagon is a right triangle? | {
"answer": "\\frac{2}{7}",
"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 $a = 2c \cos A$ and $\sqrt{5} \sin A = 1$, find:
1. $\sin C$
2. $\frac{b}{c}$ | {
"answer": "\\frac{2\\sqrt{5} + 5\\sqrt{3}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Bill buys a stock for $100. On the first day, the stock decreases by $25\%$, on the second day it increases by $35\%$ from its value at the end of the first day, and on the third day, it decreases again by $15\%$. What is the overall percentage change in the stock's value over the three days? | {
"answer": "-13.9375\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\theta$ is an angle in the third quadrant, if $\sin^{4}{\theta} + \cos^{4}{\theta} = \frac{5}{9}$, find the value of $\sin{2\theta}$. | {
"answer": "\\frac{2\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the sum of the fourth powers of the roots of the equation \[ x\sqrt{x} - 8x + 9\sqrt{x} - 2 = 0, \] assuming all roots are real and nonnegative. | {
"answer": "2018",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a > 0$ and $b > 0$, they satisfy the equation $3a + b = a^2 + ab$. Find the minimum value of $2a + b$. | {
"answer": "3 + 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $y=a (0 < a < 1)$ and the function $f(x)=\sin \omega x$ intersect at 12 points on the right side of the $y$-axis. These points are denoted as $(x\_1)$, $(x\_2)$, $(x\_3)$, ..., $(x\_{12})$ in order. It is known that $x\_1= \dfrac {\pi}{4}$, $x\_2= \dfrac {3\pi}{4}$, and $x\_3= \dfrac {9\pi}{4}$. Calculate the sum $x\_1+x\_2+x\_3+...+x\_{12}$. | {
"answer": "66\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $m=(\sqrt{3}\cos x,-1)$, $n=(\sin x,\cos ^{2}x)$.
$(1)$ When $x=\frac{\pi}{3}$, find the value of $m\cdot n$;
$(2)$ If $x\in\left[ 0,\frac{\pi}{4} \right]$, and $m\cdot n=\frac{\sqrt{3}}{3}-\frac{1}{2}$, find the value of $\cos 2x$. | {
"answer": "\\frac{3 \\sqrt{2}- \\sqrt{3}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a certain basketball player has a 50% chance of making each shot, we use a random simulation method to estimate the probability that the player makes exactly two out of four shots: First, we generate a random integer between 0 and 9 using a calculator, where 0, 1, 2, 3, and 4 represent a successful shot, and 5, 6, 7, 8, and 9 represent a missed shot; then, we group every four random numbers to represent the results of four shots. After conducting the random simulation, 20 groups of random numbers are generated:
9075 9660 1918 9257 2716 9325 8121 4589 5690 6832
4315 2573 3937 9279 5563 4882 7358 1135 1587 4989
Based on this, estimate the probability that the athlete makes exactly two out of four shots. | {
"answer": "0.35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the graph of the exponential function $y=f(x)$ passes through the point $(\frac{1}{2}, \frac{\sqrt{2}}{2})$, determine the value of $\log_{2}f(2)$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( x \) is a real number and \( \lceil x \rceil = 14 \), how many possible values are there for \( \lceil x^2 \rceil \)? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the following propositions, the true one is marked by \_\_\_\_\_\_.
\\((1)\\) The negation of the proposition "For all \\(x > 0\\), \\(x^{2}-x \leqslant 0\\)" is "There exists an \\(x > 0\\) such that \\(x^{2}-x > 0\\)."
\\((2)\\) If \\(A > B\\), then \\(\sin A > \sin B\\).
\\((3)\\) Given a sequence \\(\{a_{n}\}\\), "The sequence \\(a_{n}\\), \\(a_{n+1}\\), \\(a_{n+2}\\) forms a geometric sequence" is a necessary and sufficient condition for "\\(a_{n+1}^{2} = a_{n}a_{n+2}\\)."
\\((4)\\) Given the function \\(f(x) = \lg x + \frac{1}{\lg x}\\), the minimum value of the function \\(f(x)\\) is \\(2\\). | {
"answer": "(1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least possible positive four-digit palindrome that is divisible by 4? | {
"answer": "1881",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}, \overrightarrow{b}$ satisfy $\overrightarrow{a}=(1, \sqrt {3}), |\overrightarrow{b}|=1, |\overrightarrow{a}+ \overrightarrow{b}|= \sqrt {3}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______. | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $a$, $b$, $c > 0$ and $$a(a+b+c)+bc=4-2 \sqrt {3}$$, calculate the minimum value of $2a+b+c$. | {
"answer": "2\\sqrt{3}-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles triangle \(ABC\), the base \(AC\) is equal to 1, and the angle \(\angle ABC\) is \(2 \arctan \frac{1}{2}\). Point \(D\) lies on the side \(BC\) such that the area of triangle \(ABC\) is four times the area of triangle \(ADC\). Find the distance from point \(D\) to the line \(AB\) and the radius of the circle circumscribed around triangle \(ADC\). | {
"answer": "\\frac{\\sqrt{265}}{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a district of Shanghai, the government convened the heads of 5 companies for an annual experience exchange meeting. Among them, Company A had 2 representatives attending, while the other 4 companies each had 1 representative attending. If 3 representatives are to be selected to speak at the meeting, the number of possible situations where these 3 representatives come from 3 different companies is ____. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An exchange point conducts two types of operations:
1) Give 2 euros - receive 3 dollars and a candy as a gift;
2) Give 5 dollars - receive 3 euros and a candy as a gift.
When the wealthy Buratino came to the exchange point, he had only dollars. When he left, he had fewer dollars, he did not acquire any euros, but he received 50 candies. How many dollars did this "gift" cost Buratino? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = (2-a)(x-1) - 2\ln x$
(1) When $a=1$, find the intervals of monotonicity for $f(x)$.
(2) If the function $f(x)$ has no zeros in the interval $\left(0, \frac{1}{2}\right)$, find the minimum value of $a$. | {
"answer": "2 - 4\\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A survey of participants was conducted at the Olympiad. $50\%$ of the participants liked the first round, $60\%$ of the participants liked the second round, $70\%$ of the participants liked the opening of the Olympiad. It is known that each participant liked either one option or all three. Determine the percentage of participants who rated all three events positively. | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x_{1}, x_{2}, \cdots, x_{n} \) and \( a_{1}, a_{2}, \cdots, a_{n} (n>2) \) be sequences of arbitrary real numbers satisfying the following conditions:
1. \( x_{1}+x_{2}+\cdots+x_{n}=0 \);
2. \( \left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{n}\right|=1 \);
3. \( a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n} \).
To ensure that the inequality \( \mid a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n} \mid \leqslant A\left(a_{1}-a_{n}\right) \) holds, find the minimum value of \( A \). | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a trapezoid with $AB\parallel DC$ . Let $M$ be the midpoint of $CD$ . If $AD\perp CD, AC\perp BM,$ and $BC\perp BD$ , find $\frac{AB}{CD}$ .
[i]Proposed by Nathan Ramesh | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{m}=(\cos \frac {x}{2},-1)$ and $\overrightarrow{n}=( \sqrt {3}\sin \frac {x}{2},\cos ^{2} \frac {x}{2})$, and the function $f(x)= \overrightarrow{m} \cdot \overrightarrow{n}+1$.
(I) If $x \in [\frac{\pi}{2}, \pi]$, find the minimum value of $f(x)$ and the corresponding value of $x$.
(II) If $x \in [0, \frac{\pi}{2}]$ and $f(x)= \frac{11}{10}$, find the value of $\sin x$. | {
"answer": "\\frac{3\\sqrt{3}+4}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be an inscribed trapezoid such that the sides $[AB]$ and $[CD]$ are parallel. If $m(\widehat{AOD})=60^\circ$ and the altitude of the trapezoid is $10$ , what is the area of the trapezoid? | {
"answer": "100\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a pile of eggs. Joan counted the eggs, but her count was way off by $1$ in the $1$ 's place. Tom counted in the eggs, but his count was off by $1$ in the $10$ 's place. Raoul counted the eggs, but his count was off by $1$ in the $100$ 's place. Sasha, Jose, Peter, and Morris all counted the eggs and got the correct count. When these seven people added their counts together, the sum was $3162$ . How many eggs were in the pile? | {
"answer": "439",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The line $y=kx$ intersects the graph of the function $y=\tan x$ ($-\frac{π}{2}＜x＜\frac{π}{2}$) at points $M$ and $N$ (not coinciding with the origin $O$). The coordinates of point $A$ are $(-\frac{π}{2},0)$. Find $(\overrightarrow{AM}+\overrightarrow{AN})\cdot\overrightarrow{AO}$. | {
"answer": "\\frac{\\pi^2}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin \omega x (\omega > 0)$, translate the graph of this function to the left by $\dfrac{\pi}{4\omega}$ units to obtain the graph of the function $g(x)$. If the graph of $g(x)$ is symmetric about the line $x=\omega$ and is monotonically increasing in the interval $(-\omega,\omega)$, determine the value of $\omega$. | {
"answer": "\\dfrac{\\sqrt{\\pi}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify the expression:
\[
\frac{1}{\dfrac{3}{\sqrt{5}+2} - \dfrac{4}{\sqrt{7}+2}}.
\] | {
"answer": "\\frac{3(9\\sqrt{5} + 4\\sqrt{7} + 10)}{(9\\sqrt{5} - 4\\sqrt{7} - 10)(9\\sqrt{5} + 4\\sqrt{7} + 10)}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a positive integer \( n \geq 3 \), for an \( n \)-element real array \(\left(x_{1}, x_{2}, \cdots, x_{n}\right)\), if every permutation \( y_{1}, y_{2}, \cdots, y_{n} \) of it satisfies \(\sum_{i=1}^{n-1} y_{i} y_{i+1} \geq -1\), then the real array \(\left(x_{1}, x_{2}, \cdots, x_{n}\right)\) is called "glowing". Find the largest constant \( C = C(n) \) such that for every glowing \( n \)-element real array, \(\sum_{1 \leq i < j \leq n} x_{i} x_{j} \geq C \). | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of every third odd integer between $200$ and $500$? | {
"answer": "17400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive number that is both prime and a palindrome, and is exactly $8$ less than a perfect square? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five volunteers and two elderly people need to line up in a row, with the two elderly people next to each other but not at the ends. How many different ways can they arrange themselves? | {
"answer": "960",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose \( S = \{1, 2, \cdots, 2005\} \). If any subset of \( S \) containing \( n \) pairwise coprime numbers always includes at least one prime number, find the minimum value of \( n \). | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two unit vectors \\( \overrightarrow{a} \\) and \\( \overrightarrow{b} \\). If \\( |3 \overrightarrow{a} - 2 \overrightarrow{b}| = 3 \\), find the value of \\( |3 \overrightarrow{a} + \overrightarrow{b}| \\). | {
"answer": "2 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers \(a_{1}, a_{2}, \cdots, a_{n}\) which are all greater than 0 (where \(n\) is a natural number no less than 4) and satisfy the equation \(a_{1} + a_{2} + \cdots + a_{n} = 1\), find the maximum value of the sum \(S = \sum_{k=1}^{n} \frac{a_{k}^{2}}{a_{k} + a_{k+1} + a_{k+2}} \quad\text{with}\ (a_{n+1} = a_{1}, a_{n+2} = a_{2})\). | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Numbers from 1 to 100 are written in a vertical row in ascending order. Fraction bars of different sizes are inserted between them. The calculation starts with the smallest fraction bar and ends with the largest one, for example, $\frac{1}{\frac{5}{3}}=\frac{15}{4}$. What is the greatest possible value that the resulting fraction can have? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a,$ $b,$ $c,$ $d$ be real numbers such that $a + b + c + d = 10$ and
\[ab + ac + ad + bc + bd + cd = 20.\]
Find the largest possible value of $d.$ | {
"answer": "\\frac{5 + \\sqrt{105}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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