problem stringlengths 10 5.15k | answer dict |
|---|---|
Determine the smallest positive real $K$ such that the inequality
\[ K + \frac{a + b + c}{3} \ge (K + 1) \sqrt{\frac{a^2 + b^2 + c^2}{3}} \]holds for any real numbers $0 \le a,b,c \le 1$ .
*Proposed by Fajar Yuliawan, Indonesia* | {
"answer": "\\frac{\\sqrt{6}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Amy and Bob choose numbers from $0,1,2,\cdots,81$ in turn and Amy choose the number first. Every time the one who choose number chooses one number from the remaining numbers. When all $82$ numbers are chosen, let $A$ be the sum of all the numbers Amy chooses, and let $B$ be the sum of all the numbers Bob chooses. During the process, Amy tries to make $\gcd(A,B)$ as great as possible, and Bob tries to make $\gcd(A,B)$ as little as possible. Suppose Amy and Bob take the best strategy of each one, respectively, determine $\gcd(A,B)$ when all $82$ numbers are chosen. | {
"answer": "41",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many unordered pairs of edges of a given square pyramid determine a plane? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube with an edge length of 6 units has the same volume as a triangular-based pyramid with a base having equilateral triangle sides of 10 units and a height of $h$ units. What is the value of $h$? | {
"answer": "\\frac{216\\sqrt{3}}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\tan \alpha$ and $\frac{1}{\tan \alpha}$ are the two real roots of the equation $x^2 - kx + k^2 - 3 = 0$, and $3\pi < \alpha < \frac{7}{2}\pi$, find $\cos \alpha + \sin \alpha$. | {
"answer": "-\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a rectangle $ABCD$ with all vertices on a sphere centered at $O$, where $AB = \sqrt{3}$, $BC = 3$, and the volume of the pyramid $O-ABCD$ is $4\sqrt{3}$, find the surface area of the sphere $O$. | {
"answer": "76\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We write one of the numbers $0$ and $1$ into each unit square of a chessboard with $40$ rows and $7$ columns. If any two rows have different sequences, at most how many $1$ s can be written into the unit squares? | {
"answer": "198",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right circular cone with three mutually perpendicular side edges, each with a length of $\sqrt{3}$, determine the surface area of the circumscribed sphere. | {
"answer": "9\\pi",
"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 $$\frac {sin2B}{ \sqrt {3}cos(B+C)-cosCsinB}= \frac {2b}{c}$$.
(I) Find the measure of angle A.
(II) If $$a= \sqrt {3}$$, find the maximum area of triangle ABC. | {
"answer": "\\frac { \\sqrt {3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the base edge length of a right prism is $1$ and the side edge length is $2$, and all the vertices of the prism lie on a sphere, find the radius of the sphere. | {
"answer": "\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four members of Barnett family, including two grandparents, one adult parent, and one child, visit a zoo. The grandparents, being senior citizens, get a 20% discount. The child receives a 60% discount due to being under the age of 12, while the adult pays the full ticket price. If the ticket for an adult costs $10.00, and one of the grandparents is paying for everyone, how much do they need to pay in total?
A) $38
B) $30
C) $42
D) $28
E) $34 | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\{a\_n\}$, where $a\_1+a\_2=3$, $a\_4+a\_5=5$.
(I) Find the general term formula of the sequence.
(II) Let $[x]$ denote the largest integer not greater than $x$ (e.g., $[0.6]=0$, $[1.2]=1$). Define $T\_n=[a\_1]+[a\_2]+…+[a\_n]$. Find the value of $T\_30$. | {
"answer": "175",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a cube $ABCDEFGH$ where each side has length $2$ units. Find $\sin \angle GAC$. (Consider this by extending the calculations needed for finding $\sin \angle HAC$) | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given circles ${C}_{1}:{x}^{2}+{y}^{2}=1$ and ${C}_{2}:(x-4)^{2}+(y-2)^{2}=1$, a moving point $M\left(a,b\right)$ is used to draw tangents $MA$ and $MB$ to circles $C_{1}$ and $C_{2}$ respectively, where $A$ and $B$ are the points of tangency. If $|MA|=|MB|$, calculate the minimum value of $\sqrt{(a-3)^{2}+(b+2)^{2}}$. | {
"answer": "\\frac{\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the numbers \\(1\\), \\(2\\), \\(3\\), \\(4\\), \\(5\\), \\(6\\), two numbers are selected to form an ordered pair of real numbers \\((x, y)\\). The probability that \\(\dfrac{x}{y+1}\\) is an integer is equal to \_\_\_\_\_\_ | {
"answer": "\\dfrac{4}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A burger at Ricky C's now weighs 180 grams, of which 45 grams are filler. What percent of the burger is not filler? Additionally, what percent of the burger is filler? | {
"answer": "25\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the set $\{1, a, \frac{b}{a}\}$ equals the set $\{0, a^2, a+b\}$, then find the value of $a^{2017} + b^{2017}$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of the eight letters in "GEOMETRY" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "ANGLE"? Express your answer as a common fraction. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$ , $b$ , $c$ , $d$ , $e$ , $f$ and $g$ be seven distinct positive integers not bigger than $7$ . Find all primes which can be expressed as $abcd+efg$ | {
"answer": "179",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=t \\ y=t+1 \end{cases}$ (where $t$ is the parameter), and the parametric equation of curve $C$ is $\begin{cases} x=2+2\cos \phi \\ y=2\sin \phi \end{cases}$ (where $\phi$ is the parameter). Establish a polar coordinate system with $O$ as the pole and the non-negative semi-axis of the $x$-axis as the polar axis.
(I) Find the polar coordinate equations of line $l$ and curve $C$;
(II) It is known that ray $OP$: $\theta_1=\alpha$ (where $0<\alpha<\frac{\pi}{2}$) intersects curve $C$ at points $O$ and $P$, and ray $OQ$: $\theta_2=\alpha+\frac{\pi}{2}$ intersects line $l$ at point $Q$. If the area of $\triangle OPQ$ is $1$, find the value of $\alpha$ and the length of chord $|OP|$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence ${a_n}$ satisfies $a_1=1$, $a_{n+1} \sqrt { \frac{1}{a_{n}^{2}}+4}=1$. Let $S_{n}=a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}$. If $S_{2n+1}-S_{n}\leqslant \frac{m}{30}$ holds for any $n\in\mathbb{N}^{*}$, find the minimum value of the positive integer $m$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An easel in a corner hosts three $30 \text{ cm} \times 40 \text{ cm}$ shelves, with equal distances between neighboring shelves. Three spiders resided where the two walls and the middle shelf meet. One spider climbed diagonally up to the corner of the top shelf on one wall, another climbed diagonally down to the corner of the lower shelf on the other wall. The third spider stayed in place and observed that from its position, the other two spiders appeared at an angle of $120^\circ$. What is the distance between the shelves? (The distance between neighboring shelves is the same.) | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given integers $a$ and $b$ satisfy: $a-b$ is a prime number, and $ab$ is a perfect square. When $a \geq 2012$, find the minimum value of $a$. | {
"answer": "2025",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a different factor tree, each value is also the product of the two values below it, unless the value is a prime number. Determine the value of $X$ for this factor tree:
[asy]
draw((-1,-.3)--(0,0)--(1,-.3),linewidth(1));
draw((-2,-1.3)--(-1.5,-.8)--(-1,-1.3),linewidth(1));
draw((1,-1.3)--(1.5,-.8)--(2,-1.3),linewidth(1));
label("X",(0,0),N);
label("Y",(-1.5,-.8),N);
label("2",(-2,-1.3),S);
label("Z",(1.5,-.8),N);
label("Q",(-1,-1.3),S);
label("7",(1,-1.3),S);
label("R",(2,-1.3),S);
draw((-1.5,-2.3)--(-1,-1.8)--(-.5,-2.3),linewidth(1));
draw((1.5,-2.3)--(2,-1.8)--(2.5,-2.3),linewidth(1));
label("5",(-1.5,-2.3),S);
label("3",(-.5,-2.3),S);
label("11",(1.5,-2.3),S);
label("2",(2.5,-2.3),S);
[/asy] | {
"answer": "4620",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When simplified, $\log_{16}{32} \cdot \log_{16}{\frac{1}{2}}$ becomes:
**A)** $-\frac{1}{4}$
**B)** $-\frac{5}{16}$
**C)** $\frac{5}{16}$
**D)** $-\frac{1}{16}$
**E)** $0$ | {
"answer": "-\\frac{5}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pentagon is formed by placing an equilateral triangle atop a square. Each side of the square is equal to the height of the equilateral triangle. What percent of the area of the pentagon is the area of the equilateral triangle? | {
"answer": "\\frac{3(\\sqrt{3} - 1)}{6} \\times 100\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a + 3 = (b-1)^2$ and $b + 3 = (a-1)^2$. Assuming $a \neq b$, determine the value of $a^2 + b^2$.
A) 5
B) 10
C) 15
D) 20
E) 25 | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \frac {1}{2}x^{2}-2\ln x+a(a\in\mathbb{R})$, $g(x)=-x^{2}+3x-4$.
$(1)$ Find the intervals of monotonicity for $f(x)$;
$(2)$ Let $a=0$, the line $x=t$ intersects the graphs of $f(x)$ and $g(x)$ at points $M$ and $N$ respectively. When $|MN|$ reaches its minimum value, find the value of $t$;
$(3)$ If for any $x\in(m,n)$ (where $n-m\geqslant 1$), the graphs of the two functions are on opposite sides of the line $l$: $x-y+s=0$ (without intersecting line $l$), then these two functions are said to have an "EN channel". Investigate whether $f(x)$ and $g(x)$ have an "EN channel", and if so, find the range of $x$; if not, please explain why. | {
"answer": "\\frac {3+ \\sqrt {33}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $cosθ+cos(θ+\frac{π}{3})=\frac{\sqrt{3}}{3},θ∈(0,\frac{π}{2})$, find $\sin \theta$. | {
"answer": "\\frac{-1 + 2\\sqrt{6}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let point $O$ be inside $\triangle ABC$ and satisfy $4\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}=\overrightarrow{0}$. Determine the probability that a randomly thrown bean into $\triangle ABC$ lands in $\triangle OBC$. | {
"answer": "\\dfrac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $sin( \frac {\pi}{6}-\alpha)-cos\alpha= \frac {1}{3}$, find $cos(2\alpha+ \frac {\pi}{3})$. | {
"answer": "\\frac {7}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$ .
Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$ | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \frac {x}{1+x}$, then $f(1)+f(2)+f(3)+\ldots+f(2017)+f( \frac {1}{2})+f( \frac {1}{3})+\ldots+f( \frac {1}{2017})=$ \_\_\_\_\_\_ . | {
"answer": "\\frac {4033}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $A=30^{\circ}$, $AB=2$, $BC=1$, then the area of $\triangle ABC$ is equal to $\boxed{\text{answer}}$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(\frac{1}{2},\frac{1}{2}\sin x+\frac{\sqrt{3}}{2}\cos x)$, $\overrightarrow{b}=(1,y)$, if $\overrightarrow{a}\parallel\overrightarrow{b}$, let the function be $y=f(x)$.
$(1)$ Find the smallest positive period of the function $y=f(x)$;
$(2)$ Given an acute triangle $ABC$ with angles $A$, $B$, and $C$, if $f(A-\frac{\pi}{3})=\sqrt{3}$, side $BC= \sqrt{7}$, $\sin B=\frac{\sqrt{21}}{7}$, find the length of $AC$ and the area of $\triangle ABC$. | {
"answer": "\\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of circle $C$ is $$\rho=4 \sqrt {2}\sin\left( \frac {3\pi}{4}-\theta\right)$$
(1) Convert the polar equation of circle $C$ into a Cartesian coordinate equation;
(2) Draw a line $l$ with slope $\sqrt {3}$ through point $P(0,2)$, intersecting circle $C$ at points $A$ and $B$. Calculate the value of $$\left| \frac {1}{|PA|}- \frac {1}{|PB|}\right|.$$ | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The yearly changes in the population census of a city for five consecutive years are, respectively, 20% increase, 10% increase, 30% decrease, 20% decrease, and 10% increase. Calculate the net change over these five years, to the nearest percent. | {
"answer": "-19\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A class of 54 students in the fifth grade took a group photo. The fixed price is 24.5 yuan for 4 photos. Additional prints cost 2.3 yuan each. If every student in the class wants one photo, how much money in total needs to be paid? | {
"answer": "139.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the two foci of an ellipse and the endpoints of its minor axis precisely form the four vertices of a square, calculate the eccentricity of the ellipse. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that six coal freight trains are organized into two groups of three trains, with trains 'A' and 'B' in the same group, determine the total number of different possible departure sequences for the six trains. | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a woman was x years old in the year $x^2$, determine her birth year. | {
"answer": "1980",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a,$ $b,$ and $c$ be angles such that
\begin{align*}
\sin a &= \cot b, \\
\sin b &= \cot c, \\
\sin c &= \cot a.
\end{align*}
Find the largest possible value of $\cos a.$ | {
"answer": "\\sqrt{\\frac{3 - \\sqrt{5}}{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F$ is the right focus of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, and the line $l$ passing through the origin intersects the hyperbola at points $M$ and $N$, with $\overrightarrow{MF} \cdot \overrightarrow{NF} = 0$. If the area of $\triangle MNF$ is $ab$, find the eccentricity of the hyperbola. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two lines \\({{l}\_{1}}:(a-1)x+2y+3=0\\) and \\({{l}\_{2}}:x+ay+3=0\\) are parallel, then \\(a=\\)_______. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p,$ $q,$ $r$ be the roots of the cubic polynomial $x^3 - 3x - 2 = 0.$ Find
\[p(q - r)^2 + q(r - p)^2 + r(p - q)^2.\] | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a regular polygon with $2^n$ sides, for $n \ge 2$ , inscribed in a circle of radius $1$ . Denote the area of this polygon by $A_n$ . Compute $\prod_{i=2}^{\infty}\frac{A_i}{A_{i+1}}$ | {
"answer": "\\frac{2}{\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the cotangents of the three interior angles \(A, B, C\) of triangle \(\triangle ABC\), denoted as \(\cot A, \cot B, \cot C\), form an arithmetic sequence, then the maximum value of angle \(B\) is \(\frac{\pi}{3}\). | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At a regional science fair, 25 participants each have their own room in the same hotel, with room numbers from 1 to 25. All participants have arrived except those assigned to rooms 14 and 20. What is the median room number of the other 23 participants? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The following is Xiaoying's process of solving a linear equation. Please read carefully and answer the questions.
解方程:$\frac{{2x+1}}{3}-\frac{{5x-1}}{6}=1$
Solution:
To eliminate the denominators, we get $2\left(2x+1\right)-\left(5x-1\right)=1$ ... Step 1
Expanding the brackets, we get $4x+2-5x+1=1$ ... Step 2
Rearranging terms, we get $4x-5x=1-1-2$ ... Step 3
Combining like terms, we get $-x=-2$, ... Step 4
Dividing both sides of the equation by $-1$, we get $x=2$ ... Step 5
$(1)$ The basis of the third step in the above solution process is ______.
$A$. the basic property of equations
$B$. the basic property of inequalities
$C$. the basic property of fractions
$D$. the distributive property of multiplication
$(2)$ Errors start to appear from the ______ step;
$(3)$ The correct solution to the equation is ______. | {
"answer": "x = -3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangle, one of the acute angles $\beta$ satisfies \[\tan \frac{\beta}{2} = \frac{1}{\sqrt[3]{3}}.\] Let $\phi$ be the angle between the median and the angle bisector drawn from this acute angle $\beta$. Calculate $\tan \phi.$ | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The increasing sequence of positive integers $b_1,$ $b_2,$ $b_3,$ $\dots$ follows the rule
\[b_{n + 2} = b_{n + 1} + b_n\]for all $n \ge 1.$ If $b_9 = 544,$ then find $b_{10}.$ | {
"answer": "883",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$? | {
"answer": "16.\\overline{6}\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the polar coordinate system, let the point on the circle $ \begin{cases}x= \frac{ \sqrt{6}}{2}\cos \theta \\ y= \frac{ \sqrt{6}}{2}\sin \theta \end{cases} (\theta \text{ is a parameter}) $ have a distance $d$ from the line $ρ( \sqrt{7}\cos θ-\sin θ)= \sqrt{2}$. Find the maximum value of $d$. | {
"answer": "\\frac{ \\sqrt{6}}{2} + \\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a restaurant offers 15 different dishes, and Yann and Camille each decide to order either one or two different dishes, how many different combinations of meals can they order? Assume that the dishes can be repeated but the order in which each person orders the dishes matters. | {
"answer": "57600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the area of rectangle $ABCD$ is $8$, when the perimeter of the rectangle is minimized, fold $\triangle ACD$ along the diagonal $AC$, then the surface area of the circumscribed sphere of the pyramid $D-ABC$ is ______. | {
"answer": "16\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that 70% of the light bulbs are produced by Factory A with a pass rate of 95%, and 30% are produced by Factory B with a pass rate of 80%, calculate the probability of buying a qualified light bulb produced by Factory A from the market. | {
"answer": "0.665",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the curve $C$ is given by the parametric equations $\begin{cases} x= \sqrt {3}+2\cos \alpha \\ y=1+2\sin \alpha\end{cases}$ (where $\alpha$ is the parameter). A polar coordinate system is established with the origin of the Cartesian coordinate system as the pole and the positive $x$-axis as the polar axis.
$(1)$ Find the polar equation of curve $C$;
$(2)$ Lines $l_{1}$ and $l_{2}$ pass through the origin $O$ and intersect curve $C$ at points $A$ and $B$ other than the origin. If $\angle AOB= \dfrac {\pi}{3}$, find the maximum value of the area of $\triangle AOB$. | {
"answer": "3 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A tourist attraction estimates that the number of tourists $p(x)$ (in ten thousand people) from January 2013 onwards in the $x$-th month is approximately related to $x$ as follows: $p(x)=-3x^{2}+40x (x \in \mathbb{N}^{*}, 1 \leqslant x \leqslant 12)$. The per capita consumption $q(x)$ (in yuan) in the $x$-th month is approximately related to $x$ as follows: $q(x)= \begin{cases}35-2x & (x \in \mathbb{N}^{*}, 1 \leqslant x \leqslant 6) \\ \frac{160}{x} & (x \in \mathbb{N}^{*}, 7 \leqslant x \leqslant 12)\end{cases}$. Find the month in 2013 with the maximum total tourism consumption and the maximum total consumption for that month. | {
"answer": "3125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For any two non-zero plane vectors $\overrightarrow{\alpha}$ and $\overrightarrow{\beta}$, define $\overrightarrow{\alpha}○\overrightarrow{\beta}=\dfrac{\overrightarrow{\alpha}⋅\overrightarrow{\beta}}{\overrightarrow{\beta}⋅\overrightarrow{\beta}}$. If plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|\geqslant |\overrightarrow{b}| > 0$, the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\theta\in(0,\dfrac{\pi}{4})$, and both $\overrightarrow{a}○\overrightarrow{b}$ and $\overrightarrow{b}○\overrightarrow{a}$ are in the set $\{\dfrac{n}{2}|n\in\mathbb{Z}\}$, find the value of $\overrightarrow{a}○\overrightarrow{b}$. | {
"answer": "\\dfrac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle has radius $52$ and center $O$ . Points $A$ is on the circle, and point $P$ on $\overline{OA}$ satisfies $OP = 28$ . Point $Q$ is constructed such that $QA = QP = 15$ , and point $B$ is constructed on the circle so that $Q$ is on $\overline{OB}$ . Find $QB$ .
*Proposed by Justin Hsieh* | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right circular cone is inverted and filled with water to 2/3 of its height. What percent of the cone's volume and surface area (not including the base) are filled with water and exposed to air, respectively? Express your answer as a decimal to the nearest ten-thousandth. | {
"answer": "55.5556\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a basketball player has a probability of $a$ for scoring 3 points in a shot, $b$ for scoring 2 points, and $c$ for not scoring any points, where $a, b, c \in (0, 1)$, and the mathematical expectation for scoring points in one shot is 2, determine the minimum value of $\frac{2}{a} + \frac{1}{3b}$. | {
"answer": "\\frac{16}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Players A and B participate in a two-project competition, with each project adopting a best-of-five format (the first player to win 3 games wins the match, and the competition ends), and there are no ties in each game. Based on the statistics of their previous matches, player A has a probability of $\frac{2}{3}$ of winning each game in project $A$, and a probability of $\frac{1}{2}$ of winning each game in project $B$, with no influence between games.
$(1)$ Find the probability of player A winning in project $A$ and project $B$ respectively.
$(2)$ Let $X$ be the number of projects player A wins. Find the distribution and mathematical expectation of $X$. | {
"answer": "\\frac{209}{162}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What percent of the square $EFGH$ is shaded? All angles in the diagram are right angles, and the side length of the square is 8 units. In this square:
- A smaller square in one corner measuring 2 units per side is shaded.
- A larger square region, excluding a central square of side 3 units, occupying from corners (2,2) to (6,6) is shaded.
- The remaining regions are not shaded. | {
"answer": "17.1875\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the two lines $ax+2y+6=0$ and $x+(a-1)y+(a^{2}-1)=0$ are parallel, determine the set of possible values for $a$. | {
"answer": "\\{-1\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In quadrilateral $EFGH$, $EF = 6$, $FG = 18$, $GH = 6$, and $HE = x$ where $x$ is an integer. Calculate the value of $x$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The houses on the south side of Crazy Street are numbered in increasing order starting at 1 and using consecutive odd numbers, except that odd numbers that contain the digit 3 are missed out. What is the number of the 20th house on the south side of Crazy Street?
A) 41
B) 49
C) 51
D) 59
E) 61 | {
"answer": "59",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle ABC$ have sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$ respectively, given that $a^{2}+2b^{2}=c^{2}$, then $\dfrac {\tan C}{\tan A}=$ ______ ; the maximum value of $\tan B$ is ______. | {
"answer": "\\dfrac { \\sqrt {3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $$\frac{\cos\alpha + \sin\alpha}{\cos\alpha - \sin\alpha} = 2$$, find the value of $$\frac{1 + \sin4\alpha - \cos4\alpha}{1 + \sin4\alpha + \cos4\alpha}$$. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a box containing red, blue, green, and yellow balls. It is known that the number of red balls is twice the number of blue balls, the number of blue balls is twice the number of green balls, and the number of yellow balls is more than seven. How many yellow balls are in the box if there are 27 balls in total? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \begin{cases} kx^{2}+2x-1, & x\in (0,1] \\ kx+1, & x\in (1,+\infty) \end{cases}$ has two distinct zeros $x_{1}$ and $x_{2}$, then the maximum value of $\dfrac {1}{x_{1}}+ \dfrac {1}{x_{2}}$ is ______. | {
"answer": "\\dfrac {9}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b=2$.
(1) If angles $A$, $B$, $C$ form an arithmetic progression, find the radius of the circumcircle of $\triangle ABC$.
(2) If sides $a$, $b$, $c$ form an arithmetic progression, find the maximum area of $\triangle ABC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the height of a cylinder is $1$, and the circumferences of its two bases are on the surface of the same sphere with a diameter of $2$, calculate the volume of the cylinder. | {
"answer": "\\dfrac{3\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many distinct four-digit positive integers are there such that the product of their digits equals 18? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function \[f(x) = \left\{ \begin{aligned} x-3 & \quad \text{ if } x < 5 \\ \sqrt{x} & \quad \text{ if } x \ge 5 \end{aligned} \right.\]has an inverse $f^{-1}.$ Find the value of $f^{-1}(-6) + f^{-1}(-5) + \dots + f^{-1}(5) + f^{-1}(6).$ | {
"answer": "94",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An ellipse satisfies the property that a light ray emitted from one focus of the ellipse, after reflecting off the ellipse, will pass through the other focus. Consider a horizontally placed elliptical billiards table that satisfies the equation $\frac{x^2}{16} + \frac{y^2}{9} = 1$. Let points A and B correspond to its two foci. If a stationary ball is placed at point A and then sent along a straight line, it bounces off the elliptical wall and returns to point A. Calculate the maximum possible distance the ball has traveled. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For an upcoming holiday, the weather forecast indicates a probability of $30\%$ chance of rain on Monday and a $60\%$ chance of rain on Tuesday. Moreover, once it starts raining, there is an additional $80\%$ chance that the rain will continue into the next day without interruption. Calculate the probability that it rains on at least one day during the holiday period. Express your answer as a percentage. | {
"answer": "72\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a large 15 by 20 rectangular region, one quarter area of the rectangle is shaded. If the shaded quarter region itself represents one fourth of its quarter area, calculate the fraction of the total area that is shaded.
A) $\frac{1}{16}$
B) $\frac{1}{12}$
C) $\frac{1}{4}$
D) $\frac{3}{20}$
E) $\frac{1}{5}$ | {
"answer": "\\frac{1}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x)=2\sin x\cos x-2\cos ^{2}(x+\frac{π}{4})$.
$(1)$ Find the intervals where $f(x)$ is monotonically increasing and its center of symmetry.
$(2)$ Given $x\in (0,\frac{π}{2})$, if $f(x+\frac{π}{6})=\frac{3}{5}$, find the value of $\cos 2x$. | {
"answer": "\\frac{4\\sqrt{3}-3}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the graph of $$f(x)=-\cos^{2} \frac {ω}{2}x+ \frac { \sqrt {3}}{2}\sinωx$$ has a distance of $$\frac {π}{2}(ω＞0)$$ between two adjacent axes of symmetry.
(Ⅰ) Find the intervals where $f(x)$ is strictly decreasing;
(Ⅱ) In triangle ABC, a, b, and c are the sides opposite to angles A, B, and C, respectively. If $$f(A)= \frac {1}{2}$$, $c=3$, and the area of triangle ABC is $$3 \sqrt {3}$$, find the value of a. | {
"answer": "\\sqrt {13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $10$ points in the space such that each $4$ points are not lie on a plane. Connect some points with some segments such that there are no triangles or quadrangles. Find the maximum number of the segments. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a tower with a height of $8$ cubes, where a blue cube must always be at the top, determine the number of different towers the child can build using $2$ red cubes, $4$ blue cubes, and $3$ green cubes. | {
"answer": "210",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, where $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = \sqrt{2}$, and $\overrightarrow{a} \cdot \overrightarrow{b} = 1$, calculate the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $xOy$, a pole coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $\rho\cos^2\theta=2a\sin\theta$ ($a>0$). The parameter equation of the line $l$ passing through the point $P(-1,-2)$ is $$\begin{cases} x=-1+ \frac { \sqrt {2}}{2}t \\ y=-2+ \frac { \sqrt {2}}{2}t\end{cases}$$ ($t$ is the parameter). The line $l$ intersects the curve $C$ at points $A$ and $B$.
(1) Find the rectangular coordinate equation of $C$ and the general equation of $l$;
(2) If $|PA|$, $|AB|$, and $|PB|$ form a geometric sequence, find the value of $a$. | {
"answer": "\\frac {3+ \\sqrt {10}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | {
"answer": "\\sqrt{55}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $$α∈(0， \frac {π}{3})$$ and vectors $$a=( \sqrt {6}sinα， \sqrt {2})$$, $$b=(1，cosα- \frac { \sqrt {6}}{2})$$ are orthogonal,
(1) Find the value of $$tan(α+ \frac {π}{6})$$;
(2) Find the value of $$cos(2α+ \frac {7π}{12})$$. | {
"answer": "\\frac { \\sqrt {2}- \\sqrt {30}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among all right triangles \(ABC\) with \( \angle C = 90^\circ\), find the maximum value of \( \sin A + \sin B + \sin^2 A \). | {
"answer": "\\sqrt{2} + \\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a parabola $C: y^2 = 2px (p > 0)$ that passes through the point $(1, -2)$, a line $l$ through focus $F$ intersects the parabola $C$ at points $A$ and $B$. If $Q$ is the point $(-\frac{7}{2}, 0)$ and $BQ \perp BF$, find the value of $|BF| - |AF|$. | {
"answer": "-\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A mathematician is working on a geospatial software and comes across a representation of a plot's boundary described by the equation $x^2 + y^2 + 8x - 14y + 15 = 0$. To correctly render it on the map, he needs to determine the diameter of this plot. | {
"answer": "10\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $|x|=3$, $y^{2}=4$, and $x < y$, find the value of $x+y$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $A=\{x|x^{3}+3x^{2}+2x > 0\}$, $B=\{x|x^{2}+ax+b\leqslant 0\}$ and $A\cap B=\{x|0 < x\leqslant 2\}$, $A\cup B=\{x|x > -2\}$, then $a+b=$ ______. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a round-robin chess tournament with $x$ players, two players dropped out after playing three matches each. The tournament ended with a total of 84 matches played. How many players were there initially? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the sum of the distances from a point in space to the vertices of a regular tetrahedron with edge length 1. | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the expression \(\frac{a}{b}+\frac{c}{d}+\frac{e}{f}\), where each letter is replaced by a different digit from \(1, 2, 3, 4, 5,\) and \(6\), determine the largest possible value of this expression. | {
"answer": "9\\frac{5}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a blue ball and an orange ball are randomly and independently tossed into bins numbered with the positive integers, where for each ball the probability that it is tossed into bin k is 3^(-k) for k = 1, 2, 3, ..., determine the probability that the blue ball is tossed into a higher-numbered bin than the orange ball. | {
"answer": "\\frac{7}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A room is 25 feet long and 15 feet wide. Find the ratio of the length of the room to its perimeter and the ratio of the width of the room to its perimeter. Express both your answers in the form $a:b$. | {
"answer": "3:16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a long straight section of a two-lane highway where cars travel in both directions, cars all travel at the same speed and obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for every 10 kilometers per hour of speed or fraction thereof. Assuming cars are 5 meters long and can travel at any speed, let $N$ be the maximum whole number of cars that can pass a photoelectric eye placed beside the road in one hour in one direction. Find $N$ divided by $10$. | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the length of a rectangle is increased by $15\%$ and the width is increased by $25\%$, by what percent is the area increased? | {
"answer": "43.75\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A trirectangular tetrahedron $M-ABC$ has three pairs of adjacent edges that are perpendicular, and a point $N$ inside the base triangle $ABC$ is at distances of $2\sqrt{2}$, $4$, and $5$ from the three faces respectively. Find the surface area of the smallest sphere that passes through both points $M$ and $N$. | {
"answer": "49\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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