You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

 $DEB$  is a chord of a circle such that  $DE=3$  and  $EB=5$ . Let  $O$  be the centre of the circle. Join  $OE$  and extend  $OE$  to cut the circle at  $C$ . (See diagram). Given  $EC=1$ , find the radius of the circle.

[asy]
size(6cm);

pair O = (0,0), B = dir(110), D = dir(30), E = 0.4 * B + 0.6 * D, C = intersectionpoint(O--2*E, unitcircle);

draw(unitcircle);
draw(O--C);
draw(B--D);

dot(O);
dot(B);
dot(C);
dot(D);
dot(E);

label(" $B$ ", B, B);
label(" $C$ ", C, C);
label(" $D$ ", D, D);
label(" $E$ ", E, dir(280));
label(" $O$ ", O, dir(270));
[/asy]

Answer: $16$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let $p,$ $q,$ $r,$ and $s$ be the roots of \[x^4 + 6x^3 + 11x^2 + 6x + 3 = 0.\] Find the value of \[\frac{1}{pq} + \frac{1}{pr} + \frac{1}{ps} + \frac{1}{qr} + \frac{1}{qs} + \frac{1}{rs}.\]

Answer: $\frac{11}{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Una rolls 8 standard 6-sided dice simultaneously and calculates the product of the 8 numbers obtained. What is the probability that the product is divisible by 8?
A) $\frac{273}{288}$
B) $\frac{275}{288}$
C) $\frac{277}{288}$
D) $\frac{279}{288}$

Answer: $\frac{277}{288}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

The midpoints of the sides of a regular octagon \( ABCDEFGH \) are joined to form a smaller octagon. What fraction of the area of \( ABCDEFGH \) is enclosed by the smaller octagon?

Answer: $\frac{1}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

There is a magical tree with 123 fruits. On the first day, 1 fruit falls from the tree. From the second day onwards, the number of fruits falling each day increases by 1 compared to the previous day. However, if the number of fruits on the tree is less than the number of fruits that should fall on a given day, the falling process restarts from 1 fruit on that day and a new cycle begins. Following this pattern, on which day will all the fruits have fallen from the tree?

Answer: $17$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the function $f(x)=\cos 2x+2 \sqrt {3}\sin x\cos x$,
(1) Find the range of the function $f(x)$ and write out the interval where the function $f(x)$ is strictly increasing;
(2) If $0 < θ < \dfrac {π}{6}$ and $f(θ)= \dfrac {4}{3}$, compute the value of $\cos 2θ$.

Answer: $\dfrac { \sqrt {15}+2}{6}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Meteorological observations. At the weather station, it was noticed that during a certain period of time, if it rained in the morning, then the evening was clear, and if it rained in the evening, then the morning was clear. There were a total of 9 rainy days: 6 times there were clear evenings and 7 times there were clear mornings. How many days did this entire period of time cover?

Answer: $11$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the function $f(x) = \ln x - ax$, where $a \in \mathbb{R}$.
(1) If the line $y = 3x - 1$ is a tangent line to the graph of the function $f(x)$, find the value of the real number $a$.
(2) If the maximum value of the function $f(x)$ on the interval $[1, e^2]$ is $1 - ae$ (where $e$ is the base of the natural logarithm), find the value of the real number $a$.

Answer: $\frac{1}{e}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In a convex quadrilateral inscribed around a circle, the products of opposite sides are equal. The angle between a side and one of the diagonals is $20^{\circ}$. Find the angle between this side and the other diagonal.

Answer: $70$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given a tetrahedron \( P-ABC \) with its four vertices on the surface of sphere \( O \), where \( PA = PB = PC \) and \( \triangle ABC \) is an equilateral triangle with side length 2. \( E \) and \( F \) are the midpoints of \( AC \) and \( BC \) respectively, and \( \angle EPF = 60^\circ \). Determine the surface area of sphere \( O \).

Answer: $6\pi$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Point $A$ lies on the line $y = \frac{12}{5} x - 3$, and point $B$ lies on the parabola $y = x^2$. What is the minimum length of the segment $AB$?

Answer: $0.6$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Calculate the lengths of the arcs of the curves given by the parametric equations.

$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\left(t^{2}-2\right) \sin t+2 t \cos t \\
y=\left(2-t^{2}\right) \cos t+2 t \sin t
\end{array}\right. \\
& 0 \leq t \leq 2 \pi
\end{aligned}
$$

Answer: $\frac{8\pi^3}{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

There are 8 cards; one side of each card is blank, and the other side has the letters И, Я, Л, З, Г, О, О, О written on them. The cards are placed on the table with the blank side up, shuffled, and then turned over one by one in sequence. What is the probability that the letters will appear in the order to form the word "ЗООЛОГИЯ"?

Answer: $1/6720$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the function $f(x)=ax^{2}+bx+c(a,b,c∈R)$, if there exists a real number $a∈[1,2]$, for any $x∈[1,2]$, such that $f(x)≤slant 1$, then the maximum value of $7b+5c$ is _____.

Answer: $-6$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A store arranges a decorative tower of balls where the top level has 2 balls and each lower level has 3 more balls than the level above. The display uses 225 balls. What is the number of levels in the tower?

Answer: $12$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

On the sides of a unit square, points \( K, L, M, \) and \( N \) are marked such that line \( KM \) is parallel to two sides of the square, and line \( LN \) is parallel to the other two sides of the square. The segment \( KL \) cuts off a triangle from the square with a perimeter of 1. What is the area of the triangle cut off from the square by the segment \( MN \)?

Answer: $\frac{1}{4}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

How many solutions in natural numbers \( x, y \) does the system of equations have

\[
\left\{\begin{array}{l}
\gcd(x, y)=20! \\
\text{lcm}(x, y)=30!
\end{array} \quad (\text{where } n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n) ?\right.
\]

Answer: $256$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the geometric sequence ${a_{n}}$, $a_{n}=9$, $a_{5}=243$, find the sum of the first 4 terms.

Answer: $120$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

If two identical, uniform tetrahedral dice, each with the numbers $1$, $2$, $3$, $5$ on their faces, are rolled simultaneously and $x$ represents the sum of the numbers on the two faces facing down, determine the probability that $x$ is not less than $6$.

Answer: $\frac{1}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Jane starts to make a list, in increasing order, of the positive integers that have a first digit of 2. She writes $2, 20, 21, 22, \ldots$ and wants to find the three-digit number formed by the 998th, 999th, and 1000th digits. Help her determine what those digits are.

Answer: $216$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let $[x]$ denote the greatest integer not exceeding $x$, for example, $[3.14] = 3$. Then, find the value of $\left[\frac{2017 \times 3}{11}\right] + \left[\frac{2017 \times 4}{11}\right] + \left[\frac{2017 \times 5}{11}\right] + \left[\frac{2017 \times 6}{11}\right] + \left[\frac{2017 \times 7}{11}\right] + \left[\frac{2017 \times 8}{11}\right]$.

Answer: $6048$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

How many integers  $n$  are there such that  $0  \le n \le 720$  and  $n^2 \equiv 1$  (mod  $720$ )?

Answer: $16$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given a random variable $\xi$ that follows the normal distribution $N(0, \sigma^2)$, if $P(\xi > 2) = 0.023$, calculate $P(-2 \leq \xi \leq 2)$.

Answer: $0.954$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In a right triangle \( ABC \) with \( AC = 16 \) and \( BC = 12 \), a circle with center at \( B \) and radius \( BC \) is drawn. A tangent to this circle is constructed parallel to the hypotenuse \( AB \) (the tangent and the triangle lie on opposite sides of the hypotenuse). The leg \( BC \) is extended to intersect this tangent. Determine by how much the leg is extended.

Answer: $15$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

6 * cos(18°) + 2 * cos(36°) + 4 * cos(54°) + ... + 20 * cos(360°) = ?

Answer: $10$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A point moving in the positive direction of the $OX$ axis has its horizontal coordinate given by $x(t) = 5(t + 1)^2 + \frac{a}{(t + 1)^5}$, where $a$ is a positive constant. Find the minimum value of $a$ such that $x(t) \geqslant 24$ for all $t \geqslant 0$.

Answer: $2 \sqrt{\left( \frac{24}{7} \right)^{7}}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Consider a $6\times 6\times h$ rectangular box containing a larger sphere of radius $3$ and four smaller spheres, each with radius $2$. The smaller spheres are placed at each corner of the bottom square face of the box and are tangent to two adjacent sides of the box. The larger sphere is tangent to all four smaller spheres. Determine the height $h$ of the box required to accommodate this configuration.
A) $5$
B) $5 + \sqrt{21}$
C) $5 + \sqrt{23}$
D) $5 + \sqrt{25}$
E) $5 + 2\sqrt{23}$

Answer: $5 + \sqrt{23}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Xiaoting's average score for five math tests is 85, the median is 86, and the mode is 88. What is the sum of the scores of the two lowest tests?

Answer: $163$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A ball thrown vertically upwards has a height above the ground that is a quadratic function of its travel time. Xiaohong throws two balls vertically upwards at intervals of 1 second. Assuming the height above the ground is the same at the moment of release for both balls, and both balls reach the same maximum height 1.1 seconds after being thrown, find the time $t$ seconds after the first ball is thrown such that the height above the ground of the first ball is equal to the height of the second ball. Determine $t = \qquad$ .

Answer: $1.6$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A segment connecting the centers of two intersecting circles is divided by their common chord into segments equal to 5 and 2. Find the length of the common chord, given that the radii of the circles are in the ratio \(4: 3\).

Answer: $2\sqrt{23}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that $ABCD$ is a rectangle with $AD = 10$ and the shaded area is $100, calculate the shortest distance between the semicircles.

Answer: $2.5 \pi$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Dima and Sergey were picking raspberries from a bush that had 900 berries. Dima alternated his actions while picking: he put one berry in the basket, and then he ate the next one. Sergey also alternated: he put two berries in the basket, and then he ate the next one. It is known that Dima picks berries twice as fast as Sergey. At some point, the boys collected all the raspberries from the bush.

Who ended up putting more berries in the basket? What will be the difference?

Answer: $100$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Pools $A$ and $B$ are both rectangular cuboids with a length of 3 meters, a width of 2 meters, and a depth of 1.2 meters. Valve 1 is used to fill pool $A$ with water and can fill an empty pool $A$ in 18 minutes. Valve 2 is used to transfer water from pool $A$ to pool $B$, taking 24 minutes to transfer a full pool $A$. If both Valve 1 and Valve 2 are opened simultaneously, how many cubic meters of water will be in pool $B$ when the water depth in pool $A$ is 0.4 meters?

Answer: $7.2$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

There are two docks, $A$ and $B$, on a river, where dock $A$ is upstream and dock $B$ is downstream. There are two boats, Boat 1 and Boat 2. The speed of Boat 1 in still water is twice the speed of Boat 2. Both boats start simultaneously from docks $A$ and $B$, respectively, and move towards each other. When Boat 1 departs, it leaves a floating cargo box on the water. After 20 minutes, the two boats meet and Boat 1 leaves another identical cargo box on the water. A while later, Boat 1 notices that it is missing cargo and turns around to search for it. When Boat 1 finds the second cargo box, Boat 2 encounters the first cargo box. Determine how many minutes have passed since Boat 1 departed when it realizes its cargo is missing. (Assume the time for turning around is negligible.)

Answer: $40$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let \(\omega\) denote the incircle of triangle \(ABC\). The segments \(BC, CA\), and \(AB\) are tangent to \(\omega\) at \(D, E\), and \(F\), respectively. Point \(P\) lies on \(EF\) such that segment \(PD\) is perpendicular to \(BC\). The line \(AP\) intersects \(BC\) at \(Q\). The circles \(\omega_1\) and \(\omega_2\) pass through \(B\) and \(C\), respectively, and are tangent to \(AQ\) at \(Q\); the former meets \(AB\) again at \(X\), and the latter meets \(AC\) again at \(Y\). The line \(XY\) intersects \(BC\) at \(Z\). Given that \(AB=15\), \(BC=14\), and \(CA=13\), find \(\lfloor XZ \cdot YZ \rfloor\).

Answer: $196$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

There are 16 different cards, including 4 red, 4 yellow, 4 blue, and 4 green cards. If 3 cards are drawn at random, the requirement is that these 3 cards cannot all be of the same color, and at most 1 red card is allowed. The number of different ways to draw the cards is \_\_\_\_\_\_ . (Answer with a number)

Answer: $472$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$, the symmetric point $R$ of the left focus $F(-c,0)$ with respect to the line $bx + cy = 0$ is on the ellipse, find the eccentricity of the ellipse.

Answer: $\dfrac{\sqrt{2}}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Four mice: White, Gray, Fat, and Thin were dividing a cheese wheel. They cut it into 4 apparently identical slices. Some slices had more holes, so Thin's slice weighed 20 grams less than Fat's slice, and White's slice weighed 8 grams less than Gray's slice. However, White wasn't upset because his slice weighed exactly one-quarter of the total cheese weight.

Gray cut 8 grams from his piece, and Fat cut 20 grams from his piece. How should the 28 grams of removed cheese be divided so that all mice end up with equal amounts of cheese? Don't forget to explain your answer.

Answer: $14$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Observe that $7 = 5 \times 1 + 2$, $12 = 5 \times 2 + 2$, and $17 = 5 \times 3 + 2$. Here, 7, 12, and 17 are called "3 consecutive numbers that leave a remainder of 2 when divided by 5." If the sum of 3 consecutive numbers that leave a remainder of 2 when divided by 5 is equal to 336, find the smallest of these numbers.

Answer: $107$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

The side \( AB \) of triangle \( ABC \) is longer than side \( AC \), and \(\angle A = 40^\circ\). Point \( D \) lies on side \( AB \) such that \( BD = AC \). Points \( M \) and \( N \) are the midpoints of segments \( BC \) and \( AD \) respectively. Find the angle \( \angle BNM \).

Answer: $20$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Divide the natural numbers from 1 to 8 into two groups such that the product of all numbers in the first group is divisible by the product of all numbers in the second group. What is the minimum value of the quotient of the product of the first group divided by the product of the second group?

Answer: $70$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

We want to set up an electric bell. The location of the bell is at a distance of $30 \mathrm{~m}$ from the ringing spot. The internal resistance of the bell is 2 ohms. We plan to use 2 Leclanché cells connected in series, each with an electromotive force of 1.5 Volts and an internal resistance of 1 ohm. What diameter of copper wire conductor can we use, if the bell will only operate reliably if at least 0.4 amperes flow through it? The resistance of 1 meter of copper wire with a cross-sectional area of $1 \mathrm{~mm}^{2}$ is $\frac{1}{55} \mathrm{ohm}$.

Answer: $0.63$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

On the side $BC$ of the triangle $ABC$, a point $D$ is chosen such that $\angle BAD = 50^\circ$, $\angle CAD = 20^\circ$, and $AD = BD$. Find $\cos \angle C$.

Answer: $\frac{\sqrt{3}}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

How many times will a clock strike over the course of 12 hours if it chimes on the half-hours as well?

Answer: $90$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let $\overrightarrow{{e_1}}$ and $\overrightarrow{{e_2}}$ be non-collinear vectors. If $k\overrightarrow{{e_1}}+4\overrightarrow{{e_2}}$ and $\overrightarrow{{e_1}}+k\overrightarrow{{e_2}}$ are collinear and have opposite directions, then the value of $k$ is ____.

Answer: $-2$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Four circles, each with a radius of 7 inches, are arranged such that they touch each other and the sides of a square as shown. Determine the area, in square inches, of the square.

Answer: $784$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

There are 20 cards, each with a number from 1 to 20. These cards are placed in a box, and 4 people each draw one card without replacement. The two people who draw the smaller numbers form one group, and the two people who draw the larger numbers form another group. If two people draw the numbers 5 and 14, what is the probability that these two people are in the same group? Answer in the simplest fraction form.

Answer: $7/51$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the Cartesian coordinate system $xOy$, given the parabola $(E): y^2 = 2px (p > 0)$ with focus $F$, $P$ is an arbitrary point on the parabola $(E)$ in the first quadrant, and $Q$ is a point on the line segment $PF$ such that $\overrightarrow{OQ} = \frac{2}{3} \overrightarrow{OP} + \frac{1}{3} \overrightarrow{OF}$. Determine the maximum slope of the line $OQ$.

Answer: $\sqrt{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Car $A$ departs from Station $J$ towards Station $Y$, while cars $B$ and $C$ depart from Station $Y$ towards Station $J$ simultaneously, and move in opposite directions towards car $A$. Car $A$ meets car $B$ first, then 20 minutes later it meets car $C$. Given the speeds of cars $A$, $B$, and $C$ are $90 \text{ km/h}$, $80 \text{ km/h}$, and $60 \text{ km/h}$ respectively, find the distance between stations $J$ and $Y$ in $\text{ km}$.

Answer: $425$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Aerith timed herself solving a contest and noted the time both as days:hours:minutes:seconds and in seconds. For example, if she spent 1,000,000 seconds, she recorded it as 11:13:46:40 and 1,000,000 seconds. Bob subtracts these numbers, ignoring punctuation. In this case, he computes:

\[ 11134640 - 1000000 = 10134640 \]

What is the largest number that always must divide his result?

Answer: $40$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A racer departs from point \( A \) along the highway, maintaining a constant speed of \( a \) km/h. After 30 minutes, a second racer starts from the same point with a constant speed of \( 1.25a \) km/h. How many minutes after the start of the first racer was a third racer sent from the same point, given that the third racer developed a speed of \( 1.5a \) km/h and simultaneously with the second racer caught up with the first racer?

Answer: $50$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

To shift the graph of the function $y=\sin \left(2x- \frac{\pi}{4}\right)$ to the graph of the function $y = \sin(2x)$, determine the horizontal shift required.

Answer: $\frac{\pi}{8}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Express $7.\overline{316}$ as a common fraction in lowest terms.

Answer: $\frac{7309}{999}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

The sum of the first 1000 terms of a geometric sequence is 300. The sum of the first 2000 terms is 570. Find the sum of the first 3000 terms.

Answer: $813$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

For each permutation $b_1, b_2, b_3, \ldots, b_8$ of the integers $1, 2, 3, \ldots, 8$, form the sum
\[|b_1 - b_2| + |b_3 - b_4| + |b_5 - b_6| + |b_7 - b_8|.\]
The average value of all such sums can be written in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Answer: $13$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the ellipse C: $$\frac {x^{2}}{4}+y^{2}=1,$$
(1) If a line $l$ passes through point Q $(1, \frac {1}{2})$ and intersects the ellipse C at points A and B, find the equation of line $l$ such that Q is the midpoint of AB;
(2) Given a fixed point M $(0, 2)$, and P is any point on the ellipse C, find the maximum value of the segment PM.

Answer: $\frac {2 \sqrt {21}}{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given sets $A=\{1, 2, 3, 4\}$, $B=\{5, 6, 7\}$, and $C=\{8, 9\}$. Now, take any two sets from these three sets, and then pick one element from each of the two selected sets to form a new set with two elements. The total number of distinct sets that can be formed is $\_\_\_\_\_\_\_\_$.

Answer: $26$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

What is the maximum area that a quadrilateral with side lengths of 1, 4, 7, and 8 can have?

Answer: $18$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A cubical cake with edge length 3 inches is iced on the sides and the top. It is cut vertically into four pieces, such that one of the cut starts at the midpoint of the top edge and ends at a corner on the opposite edge. The piece whose top is triangular contains an area $A$ and is labeled as triangle $C$. Calculate the volume $v$ and the total surface area $a$ of icing covering this triangular piece. Compute $v+a$.
A) $\frac{24}{5}$
B) $\frac{32}{5}$
C) $8+\sqrt{5}$
D) $5+\frac{16\sqrt{5}}{5}$
E) $22.5$

Answer: $22.5$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{\begin{array}{l}x=2t+1,\\ y=2t\end{array}\right.$ (where $t$ is a 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 curve $C$ is $\rho ^{2}-4\rho \sin \theta +3=0$.
$(1)$ Find the rectangular coordinate equation of the line $l$ and the general equation of the curve $C$;
$(2)$ A tangent line to the curve $C$ passes through a point $A$ on the line $l$, and the point of tangency is $B$. Find the minimum value of $|AB|$.

Answer: $\frac{\sqrt{14}}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

\[ y = x + \cos(2x) \] in the interval \((0, \pi / 4)\).

Answer: $\frac{\pi}{12} + \frac{\sqrt{3}}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A circle with a radius of 6 cm is tangent to three sides of a rectangle. The area of the circle is half the area of the rectangle. What is the length of the longer side of the rectangle, in centimeters? Express your answer in terms of $\pi$.

Answer: $6\pi$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let  $n$  be a positive integer such that  $12n^2+12n+11$  is a  $4$ -digit number with all  $4$  digits equal. Determine the value of  $n$ .

Answer: $21$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle $ABC$, it is known that $AB = 14$, $BC = 6$, and $AC = 10$. The angle bisectors $BD$ and $CE$ intersect at point $O$. Find $OD$.

Answer: $\sqrt{7}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

An author writes a book containing 60,000 words in 100 hours. For the first 20% of the time, the author writes with 50% increased productivity. How many words per hour did the author write on average?

Answer: $600$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let \(a\) be a positive real number. Find the value of \(a\) such that the definite integral
\[
\int_{a}^{a^2} \frac{\mathrm{d} x}{x+\sqrt{x}}
\]
achieves its smallest possible value.

Answer: $3 - 2\sqrt{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Calculate:
$(1)\frac{2}{5}-\frac{1}{5}\times \left(-5\right)+\frac{3}{5}$;
$(2)-2^{2}-\left(-3\right)^{3}\div 3\times \frac{1}{3}$.

Answer: $-1$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In a math competition consisting of problems $A$, $B$, and $C$, among the 39 participants, each person answered at least one problem correctly. Among the people who answered $A$ correctly, the number of people who only answered $A$ is 5 more than the number of people who answered other problems as well. Among the people who did not answer $A$ correctly, the number of people who answered $B$ is twice the number of people who answered $C$. Additionally, the number of people who only answered $A$ equals the sum of the number of people who only answered $B$ and the number of people who only answered $C$. What is the maximum number of people who answered $A$ correctly?

Answer: $23$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

When the product of $\underbrace{\frac{2}{3} \times \frac{2}{3} \times \cdots \times \frac{2}{3}}_{10}$ is written as a decimal, what are the first two decimal places?

Answer: $0.01$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In a convex 1950-sided polygon, all the diagonals are drawn, dividing it into smaller polygons. Consider the polygon with the greatest number of sides among these smaller polygons. What is the maximum number of sides it can have?

Answer: $1949$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

How many 7-digit numbers divisible by 9 are there, where the second to last digit is 5?

Answer: $100000$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let $p$, $q$, $r$, and $s$ be the roots of $x^4 - 24x^3 + 50x^2 - 26x + 7 = 0$. Compute \[(p+q)^2 + (q+r)^2 + (r+s)^2 + (s+p)^2 + (p+r)^2 + (q+s)^2.\]

Answer: $1052$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle $\triangle ABC$, sides a, b, and c are opposite to angles A, B, and C, respectively. Given $\vec{m} = (a-b, c)$ and $\vec{n} = (a-c, a+b)$, and that $\vec{m}$ and $\vec{n}$ are collinear, find the value of $2\sin(\pi+B) - 4\cos(-B)$.

Answer: $-\sqrt{3} - 2$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In rectangle $ABCD$, $AB = 6$ and $BC = 4$. Points $F$ and $G$ are on $\overline{CD}$ such that $DF = 2$ and $GC = 1.5$. Lines $AF$ and $BG$ intersect at $E$. Calculate the area of $\triangle AEB$.

Answer: $\frac{144}{7}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given a square with a side length of $1$, if two points are randomly selected with equal probability from the center and the vertices of the square, what is the probability that the distance between these two points is $\frac{\sqrt{2}}{2}$?

Answer: $\frac{2}{5}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

The city administration of a certain city launched a campaign that allows the exchange of four empty 1-liter bottles for one full 1-liter bottle of milk. How many liters of milk can a person obtain if they have 43 empty 1-liter bottles by making several of these exchanges?

Answer: $14$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Jennifer wants to enclose her rectangular vegetable garden using 160 feet of fencing. She has decided that one side of the garden should be exactly 30 feet long. What is the maximum area that she can enclose, assuming the sides of the rectangle are natural numbers?

Answer: $1500$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given an odd function $f(x)$ defined on $R$ satisfying $f(x+1)=f(1-x)$, when $0\leqslant x\leqslant 1$, $f(x)=x^{3}$, then $f(2019)=\_\_\_\_\_\_$.

Answer: $-1$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Compute $\tan \left(\operatorname{arccot} \frac{3}{5}\right)$ and verify if $\sin \left(\operatorname{arccot} \frac{3}{5}\right) > \frac{1}{2}$.

Answer: $\frac{5}{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that \( \mathrm{a}, \mathrm{b}, \mathrm{c} \) are three natural numbers, and the least common multiple (LCM) of \( \mathrm{a} \) and \( \mathrm{b} \) is 60, and the LCM of \( \mathrm{a} \) and \( \mathrm{c} \) is 270, find the LCM of \( \mathrm{b} \) and \( \mathrm{c} \).

Answer: $540$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that $\tan{\alpha} = 3$, calculate:

1. $$\frac{4\sin{\alpha} - \cos{\alpha}}{3\sin{\alpha} + 5\cos{\alpha}}$$
2. $$\sin{\alpha} \cdot \cos{\alpha}$$

Answer: $\frac{3}{10}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Starting with the display "1," calculate the fewest number of keystrokes needed to reach "240" using only the keys [+1] and [x2].

Answer: $10$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let $g$ be a function defined by $g(x) = \frac{px + q}{rx + s}$, where $p$, $q$, $r$ and $s$ are nonzero real numbers, and the function has the properties $g(31)=31$, $g(41)=41$, and $g(g(x))=x$ for all values except $\frac{-s}{r}$. Determine the unique number that is not in the range of $g$.

Answer: $36$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let \( x \neq y \), and suppose the two sequences \( x, a_{1}, a_{2}, a_{3}, y \) and \( b_{1}, x, b_{2}, b_{3}, y, b_{1} \) are both arithmetic sequences. Determine the value of \( \frac{b_{4}-b_{3}}{a_{2}-a_{1}} \).

Answer: $8/3$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

If \( p \) and \( q \) are prime numbers, the number of divisors \( d(a) \) of a natural number \( a = p^{\alpha} q^{\beta} \) is given by the formula
$$
d(a) = (\alpha+1)(\beta+1).
$$

For example, \( 12 = 2^2 \times 3^1 \), the number of divisors of 12 is
$$
d(12) = (2+1)(1+1) = 6,
$$

and the divisors are \( 1, 2, 3, 4, 6, \) and \( 12 \).

Using the given calculation formula, answer: Among the divisors of \( 20^{30} \) that are less than \( 20^{15} \), how many are not divisors of \( 20^{15} \)?

Answer: $450$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given $(1-2x)^7 = a + a_1x + a_2x^2 + \ldots + a_7x^7$, calculate the value of $a_2 + a_3 + a_4 + a_5 + a_6 + a_7$.

Answer: $12$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given positive integers \( a, b, c, \) and \( d \) such that \( a > b > c > d \) and \( a + b + c + d = 2004 \), as well as \( a^2 - b^2 + c^2 - d^2 = 2004 \), what is the minimum value of \( a \)?

Answer: $503$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find the smallest positive real number \( r \) with the following property: For every choice of 2023 unit vectors \( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_{2023} \in \mathbb{R}^2 \), a point \( \mathbf{p} \) can be found in the plane such that for each subset \( S \) of \(\{1, 2, \ldots, 2023\}\), the sum
\[ \sum_{i \in S} \mathbf{v}_i \]
lies inside the disc \(\left\{ \mathbf{x} \in \mathbb{R}^2 : \| \mathbf{x} - \mathbf{p} \| \leq r \right\}\).

Answer: $2023/2$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given three natural numbers 1, 2, 3, perform an operation by replacing one of these numbers with the sum of the other two. After performing this operation 9 times, what is the maximum possible value of the largest number among the resulting three natural numbers?

Answer: $233$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Calculate the following value expressed as a common fraction: $$\frac{1}{4^{1}}+\frac{1}{4^{2}}+\frac{1}{4^{3}}+\frac{1}{4^{4}}+\frac{1}{4^{5}}+\frac{1}{4^{6}}+\frac{1}{4^{7}}.$$

Answer: $\frac{16383}{49152}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $e = \dfrac{\sqrt{6}}{3}$, and the distance between the left focus and one endpoint of the minor axis is $\sqrt{3}$.

$(I)$ Find the standard equation of the ellipse;

$(II)$ Given the fixed point $E(-1, 0)$, if the line $y = kx + 2$ intersects the ellipse at points $A$ and $B$. Is there a real number $k$ such that the circle with diameter $AB$ passes through point $E$? Please explain your reasoning.

Answer: $\dfrac{7}{6}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let \( F_{1} \) and \( F_{2} \) be the two foci of an ellipse. A circle with center \( F_{2} \) is drawn, which passes through the center of the ellipse and intersects the ellipse at point \( M \). If the line \( ME_{1} \) is tangent to circle \( F_{2} \) at point \( M \), find the eccentricity \( e \) of the ellipse.

Answer: $\sqrt{3}-1$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the hyperbola $C:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ with its right vertex at point $A$, a circle $A$ is created with center at $A$ and radius $b$. Circle $A$ intersects with one of the asymptotes of hyperbola $C$ at points $M$ and $N$. If $\angle MAN = 60^{\circ}$, find the eccentricity of hyperbola $C$.

Answer: $\frac{2\sqrt{3}}{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A rectangular box has a total surface area of 166 square inches and the sum of the lengths of all its edges is 64 inches. Find the sum of the lengths in inches of all of its interior diagonals.

Answer: $12\sqrt{10}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given $f(x)=\cos x+\cos (x+ \frac {π}{2}).$
(1) Find $f( \frac {π}{12})$;
(2) Suppose $α$ and $β∈(- \frac {π}{2},0)$, $f(α+ \frac {3π}{4})=- \frac {3 \sqrt {2}}{5}$, $f( \frac {π}{4}-β)=- \frac {5 \sqrt {2}}{13}$, find $\cos (α+β)$.

Answer: $\frac {16}{65}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A point \( D \) is marked on the altitude \( BH \) of triangle \( ABC \). Line \( AD \) intersects side \( BC \) at point \( E \), and line \( CD \) intersects side \( AB \) at point \( F \). It is known that \( BH \) divides segment \( FE \) in the ratio \( 1:3 \), starting from point \( F \). Find the ratio \( FH:HE \).

Answer: $1:3$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let $ABCDE$ be a convex pentagon, and let $H_A,$ $H_B,$ $H_C,$ $H_D$ denote the centroids of triangles $BCD,$ $ACE,$ $ABD,$ and $ABC,$ respectively. Determine the ratio $\frac{[H_A H_B H_C H_D]}{[ABCDE]}.$

Answer: $\frac{1}{9}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Calculate $0.7$ less than the sum of $34.5$ and $15.2$.

Answer: $49$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Two cross sections of a right octagonal pyramid are obtained by cutting the pyramid with planes parallel to the octagonal base. The areas of the cross sections are $300\sqrt{2}$ square feet and $675\sqrt{2}$ square feet. The two planes are $10$ feet apart. How far from the apex of the pyramid is the larger cross section, in feet?

Answer: $30$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that $a > 0$, $b > 0$, and $2a+b=1$, find the maximum value of $2 \sqrt {ab}-4a^{2}-b^{2}$.

Answer: $\dfrac { \sqrt {2}-1}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that $\triangle ABC$ is an acute triangle, vector $\overrightarrow{m} = (\cos(A + \frac{\pi}{3}), \sin(A + \frac{\pi}{3}))$, $\overrightarrow{n} = (\cos B, \sin B)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.

(Ⅰ) Find the value of $A-B$;
(Ⅱ) If $\cos B = \frac{3}{5}$ and $AC = 8$, find the length of $BC$.

Answer: $4\sqrt{3} + 3$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In $\triangle ABC$, the three sides $a, b, c$ form an arithmetic sequence, and $\angle A = 3 \angle C$. Find $\cos \angle C$.

Answer: $\frac{1 + \sqrt{33}}{8}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A point is randomly thrown on the segment [3, 8] and let \( k \) be the resulting value. Find the probability that the roots of the equation \((k^{2}-2k-3)x^{2}+(3k-5)x+2=0\) satisfy the condition \( x_{1} \leq 2x_{2} \).

Answer: $4/15$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A square $EFGH$ is inscribed in the region bound by the parabola $y = x^2 - 8x + 16$ and the $x$-axis. Find the area of square $EFGH$.

[asy]
unitsize(0.8 cm);

real parab2 (real x) {
  return(x^2 - 8*x + 16);
}

pair E, F, G, H;
real x = -2 + 2*sqrt(3);

E = (4 - x,0);
F = (4 + x,0);
G = (4 + x,-2*x);
H = (4 - x,-2*x);

draw(graph(parab2,0.5,7.5));
draw(E--H--G--F);
draw((0,0)--(8,0));

label("$E$", E, N);
label("$F$", F, N);
label("$G$", G, SE);
label("$H$", H, SW);
[/asy]

Answer: $16$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let \( A B C D \) be a quadrilateral and \( P \) the intersection of \( (A C) \) and \( (B D) \). Assume that \( \widehat{C A D} = 50^\circ \), \( \widehat{B A C} = 70^\circ \), \( \widehat{D C A} = 40^\circ \), and \( \widehat{A C B} = 20^\circ \). Calculate the angle \( \widehat{C P D} \).

Answer: $70$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

How many numbers are in the list starting from $-48$, increasing by $7$ each time, up to and including $119$?

Answer: $24$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find the maximum value of the expression

$$
\left(x_{1}-x_{2}\right)^{2}+\left(x_{2}-x_{3}\right)^{2}+\ldots+\left(x_{2010}-x_{2011}\right)^{2}+\left(x_{2011}-x_{1}\right)^{2}
$$

where \(x_{1}, \ldots, x_{2011} \in [0, 1]\).

Answer: $2010$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Point O is the circumcenter of acute triangle ABC, with AB=6, AC=2. Determine the value of $\overrightarrow {AO}\cdot ( \overrightarrow {AB}+ \overrightarrow {AC})$.

Answer: $20$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Katya is passing time while her parents are at work. On a piece of paper, she absentmindedly drew Cheburashkas in two rows (at least one Cheburashka was drawn in each row).

Afterwards, she drew a Crocodile Gena between every two adjacent Cheburashkas in both rows. Then she drew an Old Lady Shapoklyak to the left of each Cheburashka. Finally, she drew a Krakozyabra between each pair of characters in the rows.

Upon closer inspection of the drawing, she realized that only the Krakozyabras turned out nicely, so she angrily erased all the other characters. In the end, her parents saw two rows of Krakozyabras: a total of 29.

How many Cheburashkas were erased?

Answer: $11$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A bag contains 13 balls of each of 4 different colors. How many balls must be taken out to ensure that among the balls taken out, there are at least 3 balls of different colors?

Answer: $27$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Xiaoming is riding a bicycle, while Xiaoming's father is walking. They start from locations $A$ and $B$ respectively, moving towards each other. After meeting, Xiaoming continues for another 18 minutes to reach $B$. It is known that Xiaoming's cycling speed is 4 times that of his father's walking speed, and it takes Xiaoming's father a certain number of minutes to walk from the meeting point to $A$. How long does Xiaoming's father need to walk from the meeting point to $A$?

Answer: $72$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given an ellipse with a chord passing through the focus and perpendicular to the major axis of length $\sqrt{2}$, and a distance from the focus to the corresponding directrix of $1$, determine the eccentricity of the ellipse.

Answer: $\frac{\sqrt{2}}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In $\triangle ABC$, $AC:CB = 3:4$. The external angle bisector of $\angle C$ intersects the extension of $BA$ at $P$ (with $A$ between $P$ and $B$). Find the ratio $PA:AB$.

Answer: $3:1$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

On every kilometer marker along the highway between the villages of Yolkino and Palkino, there is a post with a sign. On one side of the sign, the distance to Yolkino is indicated, and on the other side, the distance to Palkino is indicated. Borya noticed that the sum of all the digits on each sign equals 13. What is the distance from Yolkino to Palkino?

Answer: $49$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

An equilateral triangle is placed on top of a square with each side of the square equal to one side of the triangle, forming a pentagon. What percent of the area of the pentagon is the area of the equilateral triangle?

Answer: $\frac{4\sqrt{3} - 3}{13} \times 100\%$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Rationalize the denominator of $\frac{7}{3+\sqrt{8}}$. Write the simplified expression in the form $\frac{A\sqrt{B}+C}{D}$, where $A$, $B$, $C$, and $D$ are integers, $D$ is positive, and $B$ is not divisible by the square of any prime. Determine $A+B+C+D$ if the greatest common divisor of $A$, $C$, and $D$ is 1.

Answer: $10$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find the value of \( \cos \frac{\pi}{11} + \cos \frac{3\pi}{11} + \cos \frac{5\pi}{11} + \cos \frac{7\pi}{11} + \cos \frac{9\pi}{11} \).

Answer: $\frac{1}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A circle has its center at $(2,0)$ with a radius of 2, and another circle has its center at $(5,0)$ with a radius of 1. A line is tangent to both circles in the first quadrant. The $y$-intercept of this line is closest to:

Answer: $2 \sqrt{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Two lines passing through point \( M \), which lies outside the circle with center \( O \), touch the circle at points \( A \) and \( B \). Segment \( OM \) is divided in half by the circle. In what ratio is segment \( OM \) divided by line \( AB \)?

Answer: $1:3$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let \( A = \{1, 2, \cdots, 2004\} \) and \( f: A \rightarrow A \) be a bijection satisfying \( f^{[2004]}(x) = f(x) \), where \( f^{[2004]}(x) \) denotes applying \( f \) 2004 times to \( x \). How many such functions \( f \) are there?

Answer: $1 + 2004!$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

48 blacksmiths need to shoe 60 horses. Each blacksmith takes 5 minutes to make one horseshoe. What is the minimum time they should spend on the job? (Note: A horse cannot stand on two legs.)

Answer: $25$