problem stringlengths 10 5.15k | answer dict |
|---|---|
Let the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$ be $a$, $b$, $c$ respectively, and $\cos B= \frac {3}{5}$, $b=2$
(Ⅰ) When $A=30^{\circ}$, find the value of $a$;
(Ⅱ) When the area of $\triangle ABC$ is $3$, find the value of $a+c$. | {
"answer": "2 \\sqrt {7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A store sells a batch of football souvenir books, with a cost price of $40$ yuan per book and a selling price of $44$ yuan per book. The store can sell 300 books per day. The store decides to increase the selling price, and after investigation, it is found that for every $1$ yuan increase in price, the daily sales decrease by 10 books. Let the new selling price after the increase be $x$ yuan $\left(44\leqslant x\leqslant 52\right)$, and let the daily sales be $y$ books.
$(1)$ Write down the functional relationship between $y$ and $x$;
$(2)$ At what price per book will the store maximize the profit from selling the souvenir books each day, and what is the maximum profit in yuan? | {
"answer": "2640",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cara is sitting at a circular table with her seven friends. Two of her friends, Alice and Bob, insist on sitting together but not next to Cara. How many different possible pairs of people could Cara be sitting between? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the distribution of the random variable $\xi$ is $P(\xi=x)= \dfrac{xk}{15}$, where $x$ takes values $(1,2,3,4,5)$, find the value of $P\left( \left. \dfrac{1}{2} < \xi < \dfrac{5}{2} \right. \right)$. | {
"answer": "\\dfrac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Throw 6 dice at a time, find the probability, in the lowest form, such that there will be exactly four kinds of the outcome. | {
"answer": "325/648",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $\sqrt {2}csinAcosB=asinC$.
(I) Find the measure of $\angle B$;
(II) If the area of $\triangle ABC$ is $a^2$, find the value of $cosA$. | {
"answer": "\\frac {3 \\sqrt {10}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a randomly selected number $x$ in the interval $[0,\pi]$, determine the probability of the event "$-1 \leqslant \tan x \leqslant \sqrt {3}$". | {
"answer": "\\dfrac{7}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given sets $A=\{-1,1,2\}$ and $B=\{-2,1,2\}$, a number $k$ is randomly selected from set $A$ and a number $b$ is randomly selected from set $B$. The probability that the line $y=kx+b$ does not pass through the third quadrant is $\_\_\_\_\_\_$. | {
"answer": "P = \\frac{2}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the sides opposite to the internal angles $A$, $B$, and $C$ of triangle $\triangle ABC$ be $a$, $b$, and $c$ respectively. It is known that $\left(\sin C+\sin B\right)\left(c-b\right)=a\left(\sin A-\sin B\right)$.
$(1)$ Find the measure of angle $C$.
$(2)$ If the angle bisector of $\angle ACB$ intersects $AB$ at point $D$ and $CD=2$, $AD=2DB$, find the area of triangle $\triangle ABC$. | {
"answer": "\\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the minute hand is moved back by 5 minutes, the number of radians it has turned is __________. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangular pyramid $SABC$, the height $SO$ passes through point $O$ - the center of the circle inscribed in the base $ABC$ of the pyramid. It is known that $\angle SAC = 60^\circ$, $\angle SCA = 45^\circ$, and the ratio of the area of triangle $AOB$ to the area of triangle $ABC$ is $\frac{1}{2 + \sqrt{3}}$. Find the angle $\angle BSC$. | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system, the equation of circle C is $x^2 + y^2 - 4x = 0$, and its center is point C. Consider the polar coordinate system with the origin as the pole and the non-negative half of the x-axis as the polar axis. Curve $C_1: \rho = -4\sqrt{3}\sin\theta$ intersects circle C at points A and B.
(1) Find the polar equation of line AB.
(2) If line $C_2$ passing through point C(2, 0) is parameterized by $\begin{cases} x = 2 + \frac{\sqrt{3}}{2}t \\ y = \frac{1}{2}t \end{cases}$ (where t is a parameter) and meets line AB at point D and the y-axis at point E, find the value of $|CD|:|CE|$. | {
"answer": "1:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the expansion of \((x+y+z)^{8}\), what is the sum of the coefficients for all terms of the form \(x^{2} y^{a} z^{b}\) (where \(a, b \in \mathbf{N})\)? | {
"answer": "1792",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{m} = (a\cos x, \cos x)$ and $\overrightarrow{n} = (2\cos x, b\sin x)$, with a function $f(x) = \overrightarrow{m} \cdot \overrightarrow{n}$ that satisfies $f(0) = 2$ and $f\left(\frac{\pi}{3}\right) = \frac{1}{2} + \frac{\sqrt{3}}{2}$,
(1) If $x \in \left[0, \frac{\pi}{2}\right]$, find the maximum and minimum values of $f(x)$;
(2) If $f\left(\frac{\theta}{2}\right) = \frac{3}{2}$, and $\theta$ is an internal angle of a triangle, find $\tan \theta$. | {
"answer": "-\\frac{4 + \\sqrt{7}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a$ and $b$ are nonzero real numbers such that $\left| a \right| \ne \left| b \right|$ , compute the value of the expression
\[
\left( \frac{b^2}{a^2} + \frac{a^2}{b^2} - 2 \right) \times
\left( \frac{a + b}{b - a} + \frac{b - a}{a + b} \right) \times
\left(
\frac{\frac{1}{a^2} + \frac{1}{b^2}}{\frac{1}{b^2} - \frac{1}{a^2}}
- \frac{\frac{1}{b^2} - \frac{1}{a^2}}{\frac{1}{a^2} + \frac{1}{b^2}}
\right).
\] | {
"answer": "-8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a geometric sequence $\{a_n\}$ with positive terms, the sum of the first $n$ terms is $S_n$. If $-3$, $S_5$, and $S_{10}$ form an arithmetic sequence, calculate the minimum value of $S_{15} - S_{10}$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are four balls in a bag, each with the same shape and size, and their numbers are \\(1\\), \\(2\\), \\(3\\), and \\(4\\).
\\((1)\\) Draw two balls randomly from the bag. Calculate the probability that the sum of the numbers on the balls drawn is no greater than \\(4\\).
\\((2)\\) First, draw a ball randomly from the bag, and its number is \\(m\\). Put the ball back into the bag, then draw another ball randomly, and its number is \\(n\\). Calculate the probability that \\(n < m + 2\\). | {
"answer": "\\dfrac{13}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the expression \[2 - (-3) - 4 - (-5) - 6 - (-7) \times 2,\] calculate its value. | {
"answer": "-14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A basketball team has 15 available players. Initially, 5 players start the game, and the other 10 are available as substitutes. The coach can make up to 4 substitutions during the game, under the same rules as the soccer game—no reentry for substituted players and each substitution is distinct. Calculate the number of ways the coach can make these substitutions and find the remainder when divided by 100. | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are knights, liars, and followers living on an island; each knows who is who among them. All 2018 islanders were arranged in a row and asked to answer "Yes" or "No" to the question: "Are there more knights on the island than liars?". They answered in turn such that everyone else could hear. Knights told the truth, liars lied. Each follower gave the same answer as the majority of those who answered before them, and if "Yes" and "No" answers were equal, they gave either answer. It turned out that the number of "Yes" answers was exactly 1009. What is the maximum number of followers that could have been among the islanders? | {
"answer": "1009",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ $(a > b > 0)$, where $M$ and $N$ are the left and right vertices of the ellipse, and $P$ is any point on the ellipse. The slopes of the lines $PM$ and $PN$ are ${k_{1}}$ and ${k_{2}}$ respectively, and ${k_{1}}{k_{2}} \neq 0$. If the minimum value of $|{k_{1}}| + |{k_{2}}|$ is $1$, find the eccentricity of the ellipse. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $2B=A+C$ and $a+\sqrt{2}b=2c$, find the value of $\sin C$. | {
"answer": "\\frac{\\sqrt{6}+\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The base nine numbers $125_9$ and $33_9$ need to be multiplied and the result expressed in base nine. What is the base nine sum of the digits of their product? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Add 78.1563 to 24.3981 and round to the nearest hundredth. | {
"answer": "102.55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of all the even integers between $200$ and $400$? | {
"answer": "30300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three vertices of parallelogram $ABCD$ are $A(-1, 3), B(2, -1), D(7, 6)$ with $A$ and $D$ diagonally opposite. Calculate the product of the coordinates of vertex $C$. | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A wooden block is 5 inches long, 5 inches wide, and 1 inch high. The block is painted red on all six sides and then cut into twenty-five 1-inch cubes. How many of the resulting cubes each have a total number of red faces that is an odd number? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, where $\angle A = 90^\circ$, $BC = 20$, and $\tan C = 4\cos B$. Find the length of $AB$. | {
"answer": "5\\sqrt{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a function $f(x) = m\ln{x} + nx$ whose tangent at point $(1, f(1))$ is parallel to the line $x + y - 2 = 0$, and $f(1) = -2$, where $m, n \in \mathbb{R}$,
(Ⅰ) Find the values of $m$ and $n$, and determine the intervals of monotonicity for the function $f(x)$;
(Ⅱ) Let $g(x)= \frac{1}{t}(-x^{2} + 2x)$, for a positive real number $t$. If there exists $x_0 \in [1, e]$ such that $f(x_0) + x_0 \geq g(x_0)$ holds, find the maximum value of $t$. | {
"answer": "\\frac{e(e - 2)}{e - 1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sally's salary in 2006 was $\$ 37,500 $. For 2007 she got a salary increase of $ x $ percent. For 2008 she got another salary increase of $ x $ percent. For 2009 she got a salary decrease of $ 2x $ percent. Her 2009 salary is $ \ $34,825$ . Suppose instead, Sally had gotten a $2x$ percent salary decrease for 2007, an $x$ percent salary increase for 2008, and an $x$ percent salary increase for 2009. What would her 2009 salary be then? | {
"answer": "34825",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A container holds $47\frac{2}{3}$ cups of sugar. If one recipe requires $1\frac{1}{2}$ cups of sugar, how many full recipes can be made with the sugar in the container? Express your answer as a mixed number. | {
"answer": "31\\frac{7}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of these two circles? | {
"answer": "\\frac{9\\pi - 18}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of the expression \(\cos(x+y)\) given that \(\cos x - \cos y = \frac{1}{4}\). | {
"answer": "31/32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the curve $ C: y = \frac {1}{1 + x^2}$ , Let $ A(\alpha ,\ f(\alpha)),\ B\left( - \frac {1}{\alpha},\ f\left( - \frac {1}{\alpha} \right)\right)\ (\alpha > 0).$ Find the minimum area bounded by the line segments $ OA,\ OB$ and $ C,$ where $ O$ is the origin.
Note that you are not allowed to use the integral formula of $ \frac {1}{1 + x^2}$ for the problem. | {
"answer": "\\frac{\\pi}{2} - \\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the set \( A = \{1, 2, \cdots, 2016\} \). For any 1008-element subset \( X \) of \( A \), if there exist \( x \) and \( y \in X \) such that \( x < y \) and \( x \mid y \), then \( X \) is called a "good set". Find the largest positive integer \( a \) (where \( a \in A \)) such that any 1008-element subset containing \( a \) is a good set. | {
"answer": "1008",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sphere is inscribed in a right cone with base radius \(15\) cm and height \(30\) cm. Find the radius \(r\) of the sphere, which can be expressed as \(b\sqrt{d} - b\) cm. What is the value of \(b + d\)? | {
"answer": "12.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $x^{10} + x + 1 = 0$ and $x^100 = a_0 + a_1x +... + a_9x^9$ . Find $a_5$ . | {
"answer": "-252",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a basketball match, Natasha attempted only three-point shots, two-point shots, and free-throw shots. She was successful on $25\%$ of her three-point shots and $40\%$ of her two-point shots. Additionally, she had a free-throw shooting percentage of $50\%$. Natasha attempted $40$ shots in total, given that she made $10$ free-throw shot attempts. How many points did Natasha score? | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the square roots of a number are $2a+3$ and $a-18$, then this number is ____. | {
"answer": "169",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b\sin A+a\cos B=0$.
(1) Find the measure of angle $B$;
(2) If $b=2$, find the maximum area of $\triangle ABC$. | {
"answer": "\\sqrt{2}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane rectangular coordinate system $xOy$, the parametric equations of curve $C$ are $\left\{{\begin{array}{l}{x=2\cos\alpha,}\\{y=\sin\alpha}\end{array}}\right.$ ($\alpha$ is the parameter). Taking the coordinate origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the line $l$ is $2\rho \cos \theta -\rho \sin \theta +2=0$.
$(1)$ Find the Cartesian equation of curve $C$ and the rectangular coordinate equation of line $l$;
$(2)$ If line $l$ intersects curve $C$ at points $A$ and $B$, and point $P(0,2)$, find the value of $\frac{1}{{|PA|}}+\frac{1}{{|PB|}}$. | {
"answer": "\\frac{8\\sqrt{5}}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the random variable $ξ$ follows a normal distribution $N(1,σ^{2})$, and $P(ξ\leqslant 4)=0.79$, determine the value of $P(-2\leqslant ξ\leqslant 1)$. | {
"answer": "0.29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that all faces of the tetrahedron P-ABC are right triangles, and the longest edge PC equals $2\sqrt{3}$, the surface area of the circumscribed sphere of this tetrahedron is \_\_\_\_\_\_. | {
"answer": "12\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle $C$ is defined by the equation $x^2 + y^2 = 1$. After the transformation $\begin{cases} x' = 2x \\ y' = \sqrt{2}y \end{cases}$, we obtain the curve $C_1$. Establish a polar coordinate system with the coordinate origin as the pole and the positive half of the $x$-axis as the polar axis. The polar coordinate of the line $l$ is $\rho \cos \left( \theta + \frac{\pi}{3} \right) = \frac{1}{2}$.
(1) Write the parametric equation of $C_1$ and the normal equation of $l$.
(2) Let point $M(1,0)$. The line $l$ intersects with the curve $C_1$ at two points $A$ and $B$. Compute $|MA| \cdot |MB|$ and $|AB|$. | {
"answer": "\\frac{12\\sqrt{2}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number of sets $A$ that satisfy $\{1, 2\} \subset A \subseteq \{1, 2, 3, 4, 5, 6\}$ must be determined. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a fixed point P (-2, 0) and a line $l: (1+3\lambda)x + (1+2\lambda)y - (2+5\lambda) = 0$, where $\lambda \in \mathbb{R}$, find the maximum distance $d$ from point P to line $l$. | {
"answer": "\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: $\sqrt{25}-\left(-1\right)^{2}+|2-\sqrt{5}|$. | {
"answer": "2+\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that line $l$ intersects circle $C$: $x^2+y^2+2x-4y+a=0$ at points $A$ and $B$, and the midpoint of chord $AB$ is $P(0,1)$.
(I) If the radius of circle $C$ is $\sqrt{3}$, find the value of the real number $a$;
(II) If the length of chord $AB$ is $6$, find the value of the real number $a$;
(III) When $a=1$, circles $O$: $x^2+y^2=4$ and $C$ intersect at points $M$ and $N$, find the length of the common chord $MN$. | {
"answer": "\\sqrt{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a tetrahedron P-ABC, if PA, PB, and PC are mutually perpendicular, and PA=2, PB=PC=1, then the radius of the inscribed sphere of the tetrahedron P-ABC is \_\_\_\_\_\_. | {
"answer": "\\frac {1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square and four circles, each with a radius of 8 inches, are arranged as in the previous problem. What is the area, in square inches, of the square? | {
"answer": "1024",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a > 2b$ ($a, b \in \mathbb{R}$), the range of the function $f(x) = ax^2 + x + 2b$ is $[0, +\infty)$. Determine the minimum value of $$\frac{a^2 + 4b^2}{a - 2b}$$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $(xOy)$, the parametric equations of the curve $C_{1}$ are given by $\begin{cases} x=2\cos \alpha \\ y=2+2\sin \alpha \end{cases}$ ($\alpha$ is the parameter). Point $M$ moves on curve $C_{1}$, and point $P$ satisfies $\overrightarrow{OP}=2\overrightarrow{OM}$. The trajectory of point $P$ forms the curve $C_{2}$.
(I) Find the equation of $C_{2}$;
(II) In the polar coordinate system with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the ray $\theta = \dfrac{\pi}{3}$ intersects $C_{1}$ at point $A$ and $C_{2}$ at point $B$. Find the length of the segment $|AB|$. | {
"answer": "2 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Six friends earn $25, $30, $35, $45, $50, and $60. Calculate the amount the friend who earned $60 needs to distribute to the others when the total earnings are equally shared among them. | {
"answer": "19.17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to the interior angles $A$, $B$, and $C$, respectively. If $a\cos \left(B-C\right)+a\cos A=2\sqrt{3}c\sin B\cos A$ and $b^{2}+c^{2}-a^{2}=2$, then the area of $\triangle ABC$ is ____. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the number of geometric sequences of length $3$ where each number is a positive integer no larger than $10$ . | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a new arcade game, the "monster" is the shaded region of a semicircle with radius 2 cm as shown in the diagram. The mouth, which is an unshaded piece within the semicircle, subtends a central angle of 90°. Compute the perimeter of the shaded region.
A) $\pi + 3$ cm
B) $2\pi + 2$ cm
C) $\pi + 4$ cm
D) $2\pi + 4$ cm
E) $\frac{5}{2}\pi + 2$ cm | {
"answer": "\\pi + 4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $a \gt 0$, $b \gt 0$, and $a+2b=1$, find the minimum value of $\frac{{b}^{2}+a+1}{ab}$. | {
"answer": "2\\sqrt{10} + 6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The integer sequence \(a_1, a_2, a_3, \dots\) is defined as follows: \(a_1 = 1\). For \(n \geq 1\), \(a_{n+1}\) is the smallest integer greater than \(a_n\) such that for all \(i, j, k \in \{1, 2, \dots, n+1\}\), the condition \(a_i + a_j \neq 3a_k\) is satisfied. Find the value of \(a_{22006}\). | {
"answer": "66016",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Complex numbers $d$, $e$, and $f$ are zeros of a polynomial $Q(z) = z^3 + sz^2 + tz + u$, and $|d|^2 + |e|^2 + |f|^2 = 300$. The points corresponding to $d$, $e$, and $f$ in the complex plane are the vertices of an equilateral triangle. Find the square of the length of each side of the triangle. | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 2 boys and 3 girls, a total of 5 students standing in a row. If boy A does not stand at either end, and among the 3 girls, exactly 2 girls stand next to each other, calculate the number of different arrangements. | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x, y, z$ be positive real numbers such that $x + 2y + 3z = 1$. Find the maximum value of $x^2 y^2 z$. | {
"answer": "\\frac{4}{16807}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given circles $P, Q,$ and $R$ where $P$ has a radius of 1 unit, $Q$ a radius of 2 units, and $R$ a radius of 1 unit. Circles $Q$ and $R$ are tangent to each other externally, and circle $R$ is tangent to circle $P$ externally. Compute the area inside circle $Q$ but outside circle $P$ and circle $R$. | {
"answer": "2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$, where $S_n=n^2+n$, and the general term of the sequence $\{b_n\}$ is given by $b_n=x^{n-1}$.
(1) Find the general term formula for the sequence $\{a_n\}$;
(2) Let $c_n=a_nb_n$, and the sum of the first $n$ terms of the sequence $\{c_n\}$ be $T_n$.
(i) Find $T_n$;
(ii) If $x=2$, find the minimum value of the sequence $\left\{\dfrac{nT_{n+1}-2n}{T_{n+2}-2}\right\}$. | {
"answer": "\\dfrac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F\_1$, $F\_2$. Let $P$ be a point on the ellipse, and $\triangle F\_1 P F\_2$ have centroid $G$ and incenter $I$. If $\overrightarrow{IG} = λ(1,0) (λ ≠ 0)$, find the eccentricity $e$ of the ellipse.
A) $\frac{1}{2}$
B) $\frac{\sqrt{2}}{2}$
C) $\frac{1}{4}$
D) $\frac{\sqrt{5}-1}{2}$ | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a particular sequence, the first term is $a_1 = 1009$ and the second term is $a_2 = 1010$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = 2n$ for all $n \ge 1$. Determine $a_{1000}$. | {
"answer": "1675",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a weekend volleyball tournament, Team E plays against Team F, and Team G plays against Team H on Saturday. On Sunday, the winners of Saturday's matches face off in a final, while the losers compete for the consolation prize. Furthermore, there is a mini tiebreaker challenge between the losing teams on Saturday to decide the starting server for Sunday's consolation match. One possible ranking of the team from first to fourth at the end of the tournament is EGHF. Determine the total number of possible four-team ranking sequences at the end of the tournament. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin \alpha = 3 \sin \left(\alpha + \frac{\pi}{6}\right)$, find the value of $\tan \left(\alpha + \frac{\pi}{12}\right)$. | {
"answer": "2 \\sqrt{3} - 4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a household, when someone is at home, the probability of a phone call being answered at the first ring is 0.1, at the second ring is 0.3, at the third ring is 0.4, and at the fourth ring is 0.1. Calculate the probability of the phone call being answered within the first four rings. | {
"answer": "0.9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the slant height of a certain cone is $4$ and the height is $2\sqrt{3}$, calculate the total surface area of the cone. | {
"answer": "12\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=\sin (2x+\frac{π}{3})$, determine the horizontal shift required to obtain this graph from the graph of the function $y=\sin 2x$. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of the function $y=\sin (2x+\varphi)$ is translated to the left by $\dfrac {\pi}{8}$ units along the $x$-axis and results in a graph of an even function, then determine one possible value of $\varphi$. | {
"answer": "\\dfrac {\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 5, 7, and 8. What is the area of the triangle? | {
"answer": "\\frac{119.84}{\\pi^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Right $\triangle PQR$ has sides $PQ = 5$, $QR = 12$, and $PR = 13$. Rectangle $ABCD$ is inscribed in $\triangle PQR$ such that $A$ and $B$ are on $\overline{PR}$, $D$ on $\overline{PQ}$, and $C$ on $\overline{QR}$. If the height of the rectangle (parallel to side $\overline{PQ}$) is half its length (parallel to side $\overline{PR}$), find the length of the rectangle. | {
"answer": "7.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A box contains $12$ ping-pong balls, of which $9$ are new and $3$ are old. Three balls are randomly drawn from the box for use, and then returned to the box. Let $X$ denote the number of old balls in the box after this process. What is the value of $P(X = 4)$? | {
"answer": "\\frac{27}{220}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: ${2}^{0}-|-3|+(-\frac{1}{2})=\_\_\_\_\_\_$. | {
"answer": "-2\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers 1, 2, ..., 2002 are written in order on a blackboard. Then the 1st, 4th, 7th, ..., 3k+1th, ... numbers in the list are erased. The process is repeated on the remaining list (i.e., erase the 1st, 4th, 7th, ... 3k+1th numbers in the new list). This continues until no numbers are left. What is the last number to be erased? | {
"answer": "1598",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a point $Q$ on a rectangular piece of paper $DEF$, where $D, E, F$ are folded onto $Q$. Let $Q$ be a fold point of $\triangle DEF$ if the creases, which number three unless $Q$ is one of the vertices, do not intersect within the triangle. Suppose $DE=24, DF=48,$ and $\angle E=90^\circ$. Determine the area of the set of all possible fold points $Q$ of $\triangle DEF$. | {
"answer": "147",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of $a + b$ if the points $(2,a,b),$ $(a,3,b),$ and $(a,b,4)$ are collinear. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twelve chairs are evenly spaced around a round table and numbered clockwise from $1$ through $12$. Six married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse or next to someone of the same profession. Determine the number of seating arrangements possible. | {
"answer": "2880",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points are drawn on the sides of a square, dividing each side into \( n \) equal parts. The points are joined to form several small squares and some triangles. How many small squares are formed when \( n=7 \)? | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$ a point $P$ on the ellipse makes lines connecting it to the two foci $F1$ and $F2$ perpendicular to each other. Then, the area of $\triangle PF1F2$ is ________. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In quadrilateral $EFGH$, $\angle F$ is a right angle, diagonal $\overline{EG}$ is perpendicular to $\overline{GH}$, $EF=20$, $FG=24$, and $GH=16$. Find the perimeter of $EFGH$. | {
"answer": "60 + 8\\sqrt{19}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a cube with side length 4 units. Determine the volume of the set of points that are inside or within 2 units outside of the cube. | {
"answer": "1059",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of the function $f(x)=\sin(2x+\varphi)$ is translated to the right by $\frac{\pi}{12}$ units and then becomes symmetric about the $y$-axis. Determine the maximum value of the function $f(x)$ in the interval $\left[0, \frac{\pi}{4}\right]$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways are there to arrange numbers from 1 to 8 in circle in such way the adjacent numbers are coprime?
Note that we consider the case of rotation and turn over as distinct way. | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
I'm going to dinner at a large restaurant which my friend recommended, unaware that I am vegan and have both gluten and dairy allergies. Initially, there are 6 dishes that are vegan, which constitutes one-sixth of the entire menu. Unfortunately, 4 of those vegan dishes contain either gluten or dairy. How many dishes on the menu can I actually eat? | {
"answer": "\\frac{1}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A huge number $y$ is given by $2^33^24^65^57^88^39^{10}11^{11}$. What is the smallest positive integer that, when multiplied with $y$, results in a product that is a perfect square? | {
"answer": "110",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the country Betia, there are 125 cities, some of which are connected by express trains that do not stop at intermediate stations. It is known that any four cities can be visited in a circular order. What is the minimum number of city pairs connected by express trains? | {
"answer": "7688",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed, leaving the remaining six cards in either ascending or descending order. | {
"answer": "74",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangle \( \triangle ABC \), it is given that the angles are in the ratio \(\angle A : \angle B : \angle C = 3 : 5 : 10\). Also, it is known that \(\triangle A'B'C \cong \triangle ABC\). What is the ratio \(\angle BCA' : \angle BCB'\)? | {
"answer": "1:4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two machine tools, A and B, produce the same product. The products are divided into first-class and second-class according to quality. In order to compare the quality of the products produced by the two machine tools, each machine tool produced 200 products. The quality of the products is as follows:<br/>
| | First-class | Second-class | Total |
|----------|-------------|--------------|-------|
| Machine A | 150 | 50 | 200 |
| Machine B | 120 | 80 | 200 |
| Total | 270 | 130 | 400 |
$(1)$ What are the frequencies of first-class products produced by Machine A and Machine B, respectively?<br/>
$(2)$ Can we be $99\%$ confident that there is a difference in the quality of the products produced by Machine A and Machine B?<br/>
Given: $K^{2}=\frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$.<br/>
| $P(K^{2}\geqslant k)$ | 0.050 | 0.010 | 0.001 |
|-----------------------|-------|-------|-------|
| $k$ | 3.841 | 6.635 | 10.828 | | {
"answer": "99\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Technology changes the world. Express sorting robots have become popular from Weibo to Moments. According to the introduction, these robots can not only automatically plan the optimal route, accurately place packages in the corresponding compartments, sense and avoid obstacles, automatically return to the team to pick up packages, but also find charging piles to charge themselves when they run out of battery. A certain sorting warehouse plans to sort an average of 200,000 packages per day. However, the actual daily sorting volume may deviate from the plan. The table below shows the situation of sorting packages in the third week of October in this warehouse (the part exceeding the planned amount is recorded as positive, and the part that does not reach the planned amount is recorded as negative):
| Day of the Week | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday |
|-----------------|--------|---------|-----------|----------|--------|----------|--------|
| Sorting Situation (in 10,000s) | +6 | +4 | -6 | +8 | -1 | +7 | -4 |
$(1)$ The day with the most sorted packages in the warehouse this week is ______; the day with the least sorted packages is ______; the day with the most sorted packages has ______ more packages than the day with the least sorted packages;<br/>$(2)$ How many packages, on average, did the warehouse actually sort per day this week? | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the Riemann function defined on the interval $\left[0,1\right]$ is: $R\left(x\right)=\left\{\begin{array}{l}{\frac{1}{q}, \text{when } x=\frac{p}{q} \text{(p, q are positive integers, } \frac{p}{q} \text{ is a reduced proper fraction)}}\\{0, \text{when } x=0,1, \text{or irrational numbers in the interval } (0,1)}\end{array}\right.$, and the function $f\left(x\right)$ is an odd function defined on $R$ with the property that for any $x$ we have $f\left(2-x\right)+f\left(x\right)=0$, and $f\left(x\right)=R\left(x\right)$ when $x\in \left[0,1\right]$, find the value of $f\left(-\frac{7}{5}\right)-f\left(\frac{\sqrt{2}}{3}\right)$. | {
"answer": "\\frac{5}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the data in the table, where the number of No. 5 batteries is $x$ and the number of No. 7 batteries is $y$, the masses of one No. 5 battery and one No. 7 battery are $x$ grams and $y$ grams, respectively. By setting up and solving a system of equations, express the value of $x$ obtained by elimination. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When any set of $k$ consecutive positive integers necessarily includes at least one positive integer whose digit sum is a multiple of 11, we call each of these sets of $k$ consecutive positive integers a "dragon" of length $k." Find the shortest dragon length. | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the volumes of the bodies bounded by the surfaces.
$$
z = 2x^2 + 18y^2, \quad z = 6
$$ | {
"answer": "6\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points $A(-2,0)$ and $B(2,0)$, the slope of line $PA$ is $k_1$, and the slope of line $PB$ is $k_2$, with the product $k_1k_2=-\frac{3}{4}$.
$(1)$ Find the equation of the locus $C$ for point $P$.
$(2)$ Let $F_1(-1,0)$ and $F_2(1,0)$. Extend line segment $PF_1$ to meet the locus $C$ at another point $Q$. Let point $R$ be the midpoint of segment $PF_2$, and let $O$ be the origin. Let $S$ represent the sum of the areas of triangles $QF_1O$ and $PF_1R$. Find the maximum value of $S$. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $0 < \alpha < \frac{\pi}{2}$ and $0 < \beta < \frac{\pi}{2}$, if $\sin\left(\frac{\pi}{3}-\alpha\right) = \frac{3}{5}$ and $\cos\left(\frac{\beta}{2} - \frac{\pi}{3}\right) = \frac{2\sqrt{5}}{5}$,
(I) find the value of $\sin \alpha$;
(II) find the value of $\cos\left(\frac{\beta}{2} - \alpha\right)$. | {
"answer": "\\frac{11\\sqrt{5}}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In right triangle $\triangle ABC$ with hypotenuse $\overline{AB}$, $AC = 15$, $BC = 20$, and $\overline{CD}$ is the altitude to $\overline{AB}$. Let $\omega$ be the circle having $\overline{CD}$ as a diameter. Let $I$ be a point outside $\triangle ABC$ such that $\overline{AI}$ and $\overline{BI}$ are both tangent to circle $\omega$. Find the ratio of the perimeter of $\triangle ABI$ to the length $AB$ and express it in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. | {
"answer": "97",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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