problem stringlengths 10 5.15k | answer dict |
|---|---|
Consider a square ABCD with side length 8 units. On side AB, semicircles are constructed inside the square both with diameter AB. Inside the square and tangent to AB at its midpoint, another quarter circle with its center at the midpoint of AB is also constructed pointing inward. Calculate the ratio of the shaded area formed between the semicircles and the quarter-circle to the area of a circle with a radius equal to the radius of the quarter circle. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are three pairs of real numbers $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$ that satisfy both $x^3 - 3xy^2 = 2010$ and $y^3 - 3x^2y = 2000$. Compute $\left(1-\frac{x_1}{y_1}\right)\left(1-\frac{x_2}{y_2}\right)\left(1-\frac{x_3}{y_3}\right)$. | {
"answer": "\\frac{1}{100}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given events A, B, and C with respective probabilities $P(A) = 0.65$, $P(B) = 0.2$, and $P(C) = 0.1$, find the probability that the drawn product is not a first-class product. | {
"answer": "0.35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of $x$ for which $10^x \cdot 500^{x} = 1000000^{3}$.
A) $\frac{9}{1.699}$
B) $6$
C) $\frac{18}{3.699}$
D) $5$
E) $20$ | {
"answer": "\\frac{18}{3.699}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A segment \( AB \) of unit length, which is a chord of a sphere with radius 1, is positioned at an angle of \( \pi / 3 \) to the diameter \( CD \) of this sphere. The distance from the end \( C \) of the diameter to the nearest end \( A \) of the chord \( AB \) is \( \sqrt{2} \). Determine the length of segment \( BD \). | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)=\sin (2017x+\frac{\pi }{6})+\cos (2017x-\frac{\pi }{3})$, find the minimum value of $A|x_{1}-x_{2}|$ where $x_{1}$ and $x_{2}$ are real numbers such that $f(x_{1})\leq f(x)\leq f(x_{2})$ for all real $x$. | {
"answer": "\\frac{2\\pi}{2017}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin\alpha = \frac{1}{2} + \cos\alpha$, and $\alpha \in (0, \frac{\pi}{2})$, find the value of $\frac{\cos 2\alpha}{\sin(\alpha - \frac{\pi}{4})}$. | {
"answer": "-\\frac{\\sqrt{14}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC,$ $AB=AC=30$ and $BC=28.$ Points $G, H,$ and $I$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC},$ respectively, such that $\overline{GH}$ and $\overline{HI}$ are parallel to $\overline{AC}$ and $\overline{AB},$ respectively. What is the perimeter of parallelogram $AGHI$? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\cos (2x-φ)- \sqrt {3}\sin (2x-φ)(|φ| < \dfrac {π}{2})$, its graph is shifted to the right by $\dfrac {π}{12}$ units and is symmetric about the $y$-axis. Find the minimum value of $f(x)$ in the interval $\[- \dfrac {π}{2},0\]$. | {
"answer": "- \\sqrt {3}",
"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 is given that $A=\frac{\pi}{3}$, $c=4$, and $a=2\sqrt{6}$. Find the measure of angle $C$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During a secret meeting, 20 trainees elect their favorite supervisor. Each trainee votes for two supervisors. It is known that for any two trainees, there is always at least one supervisor for whom both have voted. What is the minimum number of votes received by the supervisor who wins the election? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a quadrilateral formed by the two foci and the two endpoints of the conjugate axis of a hyperbola $C$, one of its internal angles is $60^{\circ}$. Determine the eccentricity of the hyperbola $C$. | {
"answer": "\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Real numbers $X_1, X_2, \dots, X_{10}$ are chosen uniformly at random from the interval $[0,1]$ . If the expected value of $\min(X_1,X_2,\dots, X_{10})^4$ can be expressed as a rational number $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ , what is $m+n$ ?
*2016 CCA Math Bonanza Lightning #4.4* | {
"answer": "1002",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For natural numbers \\(m\\) greater than or equal to \\(2\\) and their powers of \\(n\\), the following decomposition formula is given:
\\(2^{2}=1+3\\) \\(3^{2}=1+3+5\\) \\(4^{2}=1+3+5+7\\) \\(…\\)
\\(2^{3}=3+5\\) \\(3^{3}=7+9+11\\) \\(…\\)
\\(2^{4}=7+9\\) \\(…\\)
Following this pattern, the third number in the decomposition of \\(5^{4}\\) is \_\_\_\_\_\_． | {
"answer": "125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify and write the result as a common fraction: $$\sqrt[4]{\sqrt[3]{\sqrt{\frac{1}{65536}}}}$$ | {
"answer": "\\frac{1}{2^{\\frac{2}{3}}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice places six ounces of coffee into a twelve-ounce cup and two ounces of coffee plus four ounces of cream into a second twelve-ounce cup. She then pours half the contents from the first cup into the second and, after stirring thoroughly, pours half the liquid in the second cup back into the first. What fraction of the liquid in the first cup is now cream?
A) $\frac{1}{4}$
B) $\frac{4}{15}$
C) $\frac{1}{3}$
D) $\frac{1}{2}$
E) $\frac{1}{5}$ | {
"answer": "\\frac{4}{15}",
"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 $x_{1}, x_{2}$, 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"
} |
Rectangle $ABCD$ lies in a plane with $AB = CD = 3$ and $BC = DA = 8$. This rectangle is rotated $90^\circ$ clockwise around $D$, followed by another $90^\circ$ clockwise rotation around the new position of point $C$ after the first rotation. What is the length of the path traveled by point $A$?
A) $\frac{\pi(8 + \sqrt{73})}{2}$
B) $\frac{\pi(8 + \sqrt{65})}{2}$
C) $8\pi$
D) $\frac{\pi(7 + \sqrt{73})}{2}$
E) $\frac{\pi(9 + \sqrt{73})}{2}$ | {
"answer": "\\frac{\\pi(8 + \\sqrt{73})}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the numbers 0, 1, 2, 3, 4, 5, 6, determine the total number of 3-digit numbers that can be formed from these digits without repetition and divided by 5. | {
"answer": "55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define $f(x) = \frac{3}{27^x + 3}.$ Calculate the sum
\[ f\left(\frac{1}{2001}\right) + f\left(\frac{2}{2001}\right) + f\left(\frac{3}{2001}\right) + \dots + f\left(\frac{2000}{2001}\right). \] | {
"answer": "1000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiao Ming participated in the "Inheriting Classics, Building the Future" themed speech competition. His scores for speech image, speech content, and speech effect were 9, 8, and 9 respectively. If the scores for speech image, speech content, and speech effect are determined in a ratio of 2:5:3 to calculate the final score, then Xiao Ming's final competition score is ______ points. | {
"answer": "8.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A shooter hits the following scores in five consecutive shots: 9.7, 9.9, 10.1, 10.2, 10.1. The variance of this set of data is __________. | {
"answer": "0.032",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
All positive odd numbers are arranged in the following table (the number of numbers in the next row is twice the number of numbers in the previous row)
First row 1
Second row 3 5
Third row 7 9 11 13
…
Then, the third number in the sixth row is ． | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From a bottle containing 1 liter of alcohol, $\frac{1}{3}$ liter of alcohol is poured out, an equal amount of water is added and mixed thoroughly. Then, $\frac{1}{3}$ liter of the mixture is poured out, an equal amount of water is added and mixed thoroughly. Finally, 1 liter of the mixture is poured out and an equal amount of water is added. How much alcohol is left in the bottle? | {
"answer": "\\frac{8}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the triple integral \( I = \iiint_{G} \frac{d x d y}{1-x-y} \), where the region \( G \) is bounded by the planes:
1) \( x + y + z = 1 \), \( x = 0 \), \( y = 0 \), \( z = 0 \)
2) \( x = 0 \), \( x = 1 \), \( y = 2 \), \( y = 5 \), \( z = 2 \), \( z = 4 \). | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many alphabetic sequences (that is, sequences containing only letters from $a\cdots z$ ) of length $2013$ have letters in alphabetic order? | {
"answer": "\\binom{2038}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the line $y=kx+b$ is a common tangent to the curves $y=\ln \left(1+x\right)$ and $y=2+\ln x$, find the value of $k+b$. | {
"answer": "3-\\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a square ABCD with side length 4 units. Points P and R are the midpoints of sides AB and CD, respectively. Points Q is located at the midpoint of side BC, and point S is located at the midpoint of side AD. Calculate the fraction of the square's total area that is shaded when triangles APQ and CSR are shaded.
[asy]
filldraw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)--cycle,gray,linewidth(1));
filldraw((0,2)--(2,4)--(0,4)--(0,2)--cycle,white,linewidth(1));
filldraw((4,2)--(2,0)--(4,0)--(4,2)--cycle,white,linewidth(1));
label("P",(0,2),W);
label("Q",(2,4),N);
label("R",(4,2),E);
label("S",(2,0),S);
[/asy] | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function
\[ x(t)=5(t+1)^{2}+\frac{a}{(t+1)^{5}}, \]
where \( a \) is a constant. Find the minimum value of \( a \) such that \( x(t) \geqslant 24 \) for all \( t \geqslant 0 \). | {
"answer": "2 \\sqrt{\\left(\\frac{24}{7}\\right)^7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $b_1, b_2, \ldots$ be a sequence determined by the rule $b_n= \frac{b_{n-1}}{3}$ if $b_{n-1}$ is divisible by 3, and $b_n = 2b_{n-1} + 2$ if $b_{n-1}$ is not divisible by 3. Determine how many positive integers $b_1 \le 3000$ are such that $b_1$ is less than each of $b_2$, $b_3$, and $b_4$. | {
"answer": "2000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x$ is a root of the equation $x^{2}+x-6=0$, simplify $\frac{x-1}{\frac{2}{{x-1}}-1}$ and find its value. | {
"answer": "\\frac{8}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the set of three-digit numbers composed of \\(0\\), \\(1\\), \\(2\\), and \\(3\\) without repeating digits be \\(A\\). If a number is randomly selected from \\(A\\), the probability that the number is exactly even is. | {
"answer": "\\dfrac{5}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\cos \alpha =\dfrac{\sqrt{5}}{5}$ and $\sin (\alpha -\beta )=\dfrac{\sqrt{10}}{10}$, calculate the value of $\cos \beta$. | {
"answer": "\\dfrac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The equation of the directrix of the parabola $y^{2}=6x$ is $x=\frac{3}{2}$. | {
"answer": "-\\dfrac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A$ be a set of positive integers satisfying the following : $a.)$ If $n \in A$ , then $n \le 2018$ . $b.)$ If $S \subset A$ such that $|S|=3$ , then there exists $m,n \in S$ such that $|n-m| \ge \sqrt{n}+\sqrt{m}$ What is the maximum cardinality of $A$ ? | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A parallelogram has side lengths of 10, 12, $10y-2$, and $4x+6$. Determine the value of $x+y$. | {
"answer": "2.7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, and point $P$ is a point on the right branch of the hyperbola. $M$ is the incenter of $\triangle PF\_1F\_2$, satisfying $S\_{\triangle MPF\_1} = S\_{\triangle MPF\_2} + \lambda S\_{\triangle MF\_1F\_2}$. If the eccentricity of this hyperbola is $3$, then $\lambda = \_\_\_\_\_\_$.
(Note: $S\_{\triangle MPF\_1}$, $S\_{\triangle MPF\_2}$, $S\_{\triangle MF\_1F\_2}$ represent the area of $\triangle MPF\_1$, $\triangle MPF\_2$, $\triangle MF\_1F\_2$ respectively.) | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest possible value of \(| |a_1 - a_2| - a_3| - \ldots - a_{1990}|\), where \(a_1, a_2, \ldots, a_{1990}\) is a permutation of \(1, 2, 3, \ldots, 1990\)? | {
"answer": "1989",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle $ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, if $\cos B= \frac{1}{4}, b=3$, and $\sin C=2\sin A$, find the area of triangle $ABC$. | {
"answer": "\\frac{9\\sqrt{15}}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function $f(x) = |\log_3 x|$ has a range of $[0,1]$ on the interval $[a, b]$. Find the minimum value of $b - a$. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular room measures 15-feet by 8-feet and has a triangular extension with a base of 8-feet and a height of 5-feet. How many square yards of carpet are needed to cover the entire floor of the room, including the triangular extension? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the solution set of the system of linear inequalities in one variable $x$, $\left\{{\begin{array}{l}{x-1≥2x+1}\\{2x-1＜a}\end{array}}\right.$, is $x\leqslant -2$, and the solution of the fractional equation in variable $y$, $\frac{{y-1}}{{y+1}}=\frac{a}{{y+1}}-2$, is negative, then the sum of all integers $a$ that satisfy the conditions is ______. | {
"answer": "-8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin \left(\alpha+ \frac {\pi}{3}\right)=- \frac {4}{5}$, and $- \frac {\pi}{2} < \alpha < 0$, find $\cos \alpha=$ ______. | {
"answer": "\\frac {3-4 \\sqrt {3}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $-1 < a < 0$, find the maximum value of the inequality $\frac{2}{a} - \frac{1}{1+a}$. | {
"answer": "-3 - 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points P and Q are on a circle of radius 7 and the chord PQ=8. Point R is the midpoint of the minor arc PQ. Calculate the length of the line segment PR. | {
"answer": "\\sqrt{98 - 14\\sqrt{33}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equations
​
(1) $x^{2}-4x+3=0$
​
(2) $(x+1)(x-2)=4$
​
(3) $3x(x-1)=2-2x$
​
(4) $2x^{2}-4x-1=0$ | {
"answer": "\\frac{2- \\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A solid has a triangular base with sides of lengths $s$, $s$, $s \sqrt{2}$. Two opposite vertices of the triangle extend vertically upward by a height $h$ where $h = 3s$. Given $s = 2\sqrt{2}$, what is the volume of this solid? | {
"answer": "24\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle $ \triangle ABC $ with sides $ a $, $ b $, and $ c $ opposite angles $ A $, $ B $, and $ C $, respectively, consider vectors $ \overrightarrow{m} = (a, \sqrt{3}b) $ and $ \overrightarrow{n} = (\cos A, \sin B) $ being parallel.
(I) Find the angle $ A $.
(II) If $ a = \sqrt{7} $ and the area of $ \triangle ABC $ is $ \frac{3\sqrt{3}}{2} $, determine the perimeter of the triangle. | {
"answer": "5 + \\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular cake with dimensions $4$ inches, $3$ inches, and $2$ inches (length, width, height respectively) is iced on the sides, the top, and the bottom. The cake is cut from the top center vertex across to the midpoint of the bottom edge on the opposite side face, creating one triangular piece. If the top center vertex is $T$, and the midpoint on the opposite bottom edge is denoted as $M$, find the volume of the cake on one side of the cut ($c$ cubic inches) and the area of icing on this piece ($s$ square inches). Calculate the sum $c+s$. | {
"answer": "38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the natural numbers 1 to 100, each time we take out two different numbers so that their sum is greater than 100, how many different ways are there to do this? | {
"answer": "2500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\alpha$, $\beta$, $\gamma$ be the roots of the cubic polynomial $x^3 - 3x - 2 = 0.$ Find
\[
\alpha(\beta - \gamma)^2 + \beta(\gamma - \alpha)^2 + \gamma(\alpha - \beta)^2.
\] | {
"answer": "-18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C$: $\frac{{x}^{2}}{3}+{y}^{2}=1$ with left focus and right focus as $F_{1}$ and $F_{2}$ respectively. The line $y=x+m$ intersects $C$ at points $A$ and $B$. If the area of $\triangle F_{1}AB$ is twice the area of $\triangle F_{2}AB$, find the value of $m$. | {
"answer": "-\\frac{\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a random variable $\xi$ follows the normal distribution $N(0, \sigma^2)$. If $P(\xi > 2) = 0.023$, calculate $P(-2 \leq \xi \leq 2)$. | {
"answer": "0.954",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point $Q$ is chosen inside $\triangle DEF$ such that lines drawn through $Q$ parallel to $\triangle DEF$'s sides decompose it into three smaller triangles $u_1$, $u_2$, and $u_3$, which have areas $3$, $12$, and $15$ respectively. Determine the area of $\triangle DEF$. | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A student, Leo, needs to earn 30 study points for a special credit. For the first 6 points, he needs to complete 1 project each. For the next 6 points, he needs 2 projects each; for the next 6 points, 3 projects each, and so on. Determine the minimum number of projects Leo needs to complete to earn 30 study points. | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, if $AB=2$, $AC=\sqrt{2}BC$, find the maximum value of $S_{\triangle ABC}$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Thirty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $\log_2 q.$ | {
"answer": "409",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f\left(x\right)=4\cos x\sin \left(x-\frac{π}{3}\right)+a$ with a maximum value of $2$.
(1) Find the value of $a$ and the minimum positive period of the function $f\left(x\right)$;
(2) In $\triangle ABC$, if $A < B$, and $f\left(A\right)=f\left(B\right)=1$, find the value of $\frac{BC}{AB}$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(\alpha) = \frac{\sin(\pi - \alpha)\cos(\pi + \alpha)\sin(-\alpha + \frac{3\pi}{2})}{\cos(-\alpha)\cos(\alpha + \frac{\pi}{2})}$.
$(1)$ Simplify $f(\alpha)$;
$(2)$ If $\alpha$ is an angle in the third quadrant, and $\cos(\alpha - \frac{3\pi}{2}) = \frac{1}{5}$, find the value of $f(\alpha)$; | {
"answer": "\\frac{2\\sqrt{6}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\cos\alpha = \frac{1}{7}$, $\cos(\alpha-\beta) = \frac{13}{14}$, and $0<\beta<\alpha<\frac{\pi}{2}$,
(1) find the value of $\tan 2\alpha$;
(2) determine $\beta$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $x^{2}+y^{2}=4$, find the minimum value of $\sqrt{2-y}+\sqrt{5-2x}$. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the area of the circle described by the graph of the equation
\[r = 4 \cos \theta - 3 \sin \theta.\] | {
"answer": "\\frac{25\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Engan alphabet of a fictional region contains 15 letters: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O. Suppose license plates are to be formed with six letters using only the letters in the Engan alphabet. How many license plates of six letters are possible that begin with either A or B, end with O, cannot contain the letter I, and have no letters that repeat? | {
"answer": "34320",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A die has faces numbered $1$, $2$, $3$, $3$, $4$, and $4$. Another die has faces numbered $2$, $3$, $5$, $6$, $7$, and $8$. Determine the probability that the sum of the top two numbers when both dice are rolled will be either $6$, $8$, or $10$.
A) $\frac{1}{36}$
B) $\frac{1}{18}$
C) $\frac{11}{36}$
D) $\frac{5}{36}$
E) $\frac{1}{6}$ | {
"answer": "\\frac{11}{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the graph of $y=\sqrt{2}\sin 3x$, calculate the horizontal shift required to obtain the graph of $y=\sin 3x+\cos 3x$. | {
"answer": "\\dfrac{\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x$ and $y$ are real numbers, and $x^{2}+2xy-y^{2}=7$, find the minimum value of $x^{2}+y^{2}$. | {
"answer": "\\frac{7\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a=3$, and $\left(a+b\right)\sin B=\left(\sin A+\sin C\right)\left(a+b-c\right)$.<br/>$(1)$ Find angle $A$;<br/>$(2)$ If $acosB+bcosA=\sqrt{3}$, find the area of $\triangle ABC$. | {
"answer": "\\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equations:
(1) $3x^2 -32x -48=0$
(2) $4x^2 +x -3=0$
(3) $(3x+1)^2 -4=0$
(4) $9(x-2)^2 =4(x+1)^2.$ | {
"answer": "\\frac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five friends all brought some cakes with them when they met. Each of them gave a cake to each of the others. They then ate all the cakes they had just been given. As a result, the total number of cakes they had between them decreased by half. How many cakes did the five friends have at the start? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given positive real numbers $a$ and $b$ satisfying $a+b=1$, find the maximum value of $\dfrac {2a}{a^{2}+b}+ \dfrac {b}{a+b^{2}}$. | {
"answer": "\\dfrac {2 \\sqrt {3}+3}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Maximum difference in weights of two bags is achieved by taking the largest and smallest possible values of the two different brands. Given the weights of three brands of flour are $\left(25\pm 0.1\right)kg$, $\left(25\pm 0.2\right)kg$, and $\left(25\pm 0.3\right)kg$, calculate the maximum possible difference in weights. | {
"answer": "0.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a 6×6 grid, park 3 identical red cars and 3 identical black cars such that there is only one car in each row and each column, with each car occupying one cell. There are ______ possible parking arrangements. (Answer with a number) | {
"answer": "14400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a game, two wheels are present. The first wheel has six segments with numbers 1 through 6. The second wheel has four segments, numbered 1, 1, 2, and 2. The game is to spin both wheels and add the resulting numbers. The player wins if the sum is a number less than 5. What is the probability of winning the game?
A) $\frac{1}{6}$
B) $\frac{1}{4}$
C) $\frac{1}{3}$
D) $\frac{1}{2}$
E) $\frac{2}{3}$ | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the polynomials $P\left(x\right)=16x^4+40x^3+41x^2+20x+16$ and $Q\left(x\right)=4x^2+5x+2$ . If $a$ is a real number, what is the smallest possible value of $\frac{P\left(a\right)}{Q\left(a\right)}$ ?
*2016 CCA Math Bonanza Team #6* | {
"answer": "4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive three-digit integers with a $7$ in the units place are divisible by $21$? | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $S_{n}$ is the sum of the first $n$ terms of the sequence $\{a_{n}\}$, if ${a_1}=\frac{5}{2}$, and ${a_{n+1}}({2-{a_n}})=2$ for $n\in\mathbb{N}^*$, then $S_{22}=$____. | {
"answer": "-\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $P, Q, R, S, T, U, V,$ and $W$ lie, in that order, on line $\overline{PW}$, dividing it into seven equal segments, each of length 1. Point $X$ is not on line $PW$. Points $Y$ and $Z$ lie on line segments $\overline{XR}$ and $\overline{XW}$ respectively. The line segments $\overline{YQ}, \overline{ZT},$ and $\overline{PX}$ are parallel. Determine the ratio $\frac{YQ}{ZT}$. | {
"answer": "\\frac{7}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The integer $n$ has exactly six positive divisors, and they are: $1<a<b<c<d<n$ . Let $k=a-1$ . If the $k$ -th divisor (according to above ordering) of $n$ is equal to $(1+a+b)b$ , find the highest possible value of $n$ . | {
"answer": "2009",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the average of a sample $m$, $4$, $6$, $7$ is $5$, then the variance of this sample is ______. | {
"answer": "\\frac{5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify the product \[\frac{9}{3}\cdot\frac{15}{9}\cdot\frac{21}{15}\dotsm\frac{3n+6}{3n}\dotsm\frac{3003}{2997}.\] | {
"answer": "1001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the first $15$ positive even integers is also the sum of six consecutive even integers. What is the smallest of these six integers? | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $(xOy)$, with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system. Consider the curve $C\_1$: $ρ^{2}-4ρ\cos θ+3=0$, $θ∈[0,2π]$, and the curve $C\_2$: $ρ= \frac {3}{4\sin ( \frac {π}{6}-θ)}$, $θ∈[0,2π]$.
(I) Find a parametric equation of the curve $C\_1$;
(II) If the curves $C\_1$ and $C\_2$ intersect at points $A$ and $B$, find the value of $|AB|$. | {
"answer": "\\frac { \\sqrt {15}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $AB = BC = 2$, $\angle ABC = 120^\circ$. A point $P$ is outside the plane of $\triangle ABC$, and a point $D$ is on the line segment $AC$, such that $PD = DA$ and $PB = BA$. Find the maximum volume of the tetrahedron $PBCD$. | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Susie Q has $2000 to invest. She invests part of the money in Alpha Bank, which compounds annually at 4 percent, and the remainder in Beta Bank, which compounds annually at 6 percent. After three years, Susie's total amount is $\$2436.29$. Determine how much Susie originally invested in Alpha Bank. | {
"answer": "820",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a triangle with sides of lengths 13, 14, and 15, the orthocenter is denoted by \( H \). The altitude from vertex \( A \) to the side of length 14 is \( A D \). What is the ratio \( \frac{H D}{H A} \)? | {
"answer": "5:11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the number 105,000, express it in scientific notation. | {
"answer": "1.05\\times 10^{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of all two-digit positive integers whose squares end with the digits 25? | {
"answer": "644",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given circle $C_1$: $(x-1)^2+(y-2)^2=1$
(1) Find the equation of the tangent line to circle $C_1$ passing through point $P(2,4)$.
(2) If circle $C_1$ intersects with circle $C_2$: $(x+1)^2+(y-1)^2=4$ at points $A$ and $B$, find the length of segment $AB$. | {
"answer": "\\frac {4 \\sqrt {5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, the sums of the first $n$ terms are $S_n$ and $T_n$, respectively. For any positive integer $n$, it holds that $$\frac {S_{n}}{T_{n}} = \frac {3n+5}{2n+3}$$, then $$\frac {a_{7}}{b_{7}} = \_\_\_\_\_\_ ．$$ | {
"answer": "\\frac {44}{29}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 8.$ Find the maximum value of
\[\sqrt{3x + 2} + \sqrt{3y + 2} + \sqrt{3z + 2}.\] | {
"answer": "3\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two lines $ l_1: x + my + 6 = 0 $ and $ l_2: (m-2)x + 3y + 2m = 0 $, if $ l_1 \parallel l_2 $, then the distance between $ l_1 $ and $ l_2 $ is __________. | {
"answer": "\\frac{8\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$ and $b$ be nonnegative real numbers such that
\[\sin (ax + b) = \sin 15x\]for all integers $x.$ Find the smallest possible value of $a.$ | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A secret facility is in the shape of a rectangle measuring $200 \times 300$ meters. There is a guard at each of the four corners outside the facility. An intruder approached the perimeter of the secret facility from the outside, and all the guards ran towards the intruder by the shortest paths along the external perimeter (while the intruder remained in place). Three guards ran a total of 850 meters to reach the intruder. How many meters did the fourth guard run to reach the intruder? | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify $\sqrt{9800}$. | {
"answer": "70\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A net for hexagonal pyramid is constructed by placing a triangle with side lengths $x$ , $x$ , and $y$ on each side of a regular hexagon with side length $y$ . What is the maximum volume of the pyramid formed by the net if $x+y=20$ ? | {
"answer": "128\\sqrt{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $a, b \in \mathbb{R}$ and $a^{2}+2b^{2}=6$, find the minimum value of $a+ \sqrt{2}b$. | {
"answer": "-2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $(xOy)$, an ellipse $(C)$ is defined by the equation $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $e = \frac{\sqrt{2}}{2}$. The point $P(2, 1)$ lies on the ellipse $(C)$.
(1) Find the equation of the ellipse $(C)$;
(2) If points $A$ and $B$ both lie on the ellipse $(C)$ and the midpoint $M$ of $AB$ lies on the line segment $OP$ (excluding endpoints).
$\quad\quad$ (a) Find the slope of the line $AB$;
$\quad\quad$ (b) Find the maximum area of $\triangle AOB$. | {
"answer": "\\frac{3 \\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite to the angles $A$, $B$, and $C$ are labeled as $a$, $b$, and $c$ respectively. Given $a=5, B= \frac {\pi}{3},$ and $\cos A= \frac {11}{14}$, find the area $S$ of the triangle $ABC$. | {
"answer": "10 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let P be a moving point on the line $3x+4y+3=0$, and through point P, two tangents are drawn to the circle $C: x^2+y^2-2x-2y+1=0$, with the points of tangency being A and B, respectively. Find the minimum value of the area of quadrilateral PACB. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the length of the common chord of the circle $x^{2}+y^{2}=50$ and $x^{2}+y^{2}-12x-6y+40=0$. | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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