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 child builds towers with $2$ red cubes, $3$ blue cubes, and $4$ green cubes, determine the number of different towers with a height of $8$ cubes that can be built, with one cube left out.

Answer: $1,260$


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>.

Xiao Ming forgot the last two digits of his WeChat login password. He only remembers that the last digit is one of the letters \\(A\\), \\(a\\), \\(B\\), or \\(b\\), and the other digit is one of the numbers \\(4\\), \\(5\\), or \\(6\\). The probability that Xiao Ming can successfully log in with one attempt is \_\_\_\_\_\_.

Answer: $\dfrac{1}{12}$


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 that points A and B are on the x-axis, and the two circles with centers at A and B intersect at points M $(3a-b, 5)$ and N $(9, 2a+3b)$, find the value of $a^{b}$.

Answer: $\frac{1}{8}$


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 strip of size $1 \times 10$ is divided into unit squares. The numbers $1, 2, \ldots, 10$ are written in these squares. First, the number 1 is written in one of the squares, then the number 2 is written in one of the neighboring squares, then the number 3 is written in one of the squares neighboring those already occupied, and so on (the choice of the first square and the choice of neighbor at each step are arbitrary). In how many ways can this be done?

Answer: $512$


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>.

Define $g$ by $g(x) = 3x + 2$. If $g(x) = f^{-1}(x) - 5$ and $f^{-1}(x)$ is the inverse of the function $f(x) = cx + d$, find $7c + 3d$.

Answer: $-\frac{14}{3}$


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>.

Suppose  $50x$  is divisible by 100 and  $kx$  is not divisible by 100 for all  $k=1,2,\cdots, 49$  Find number of solutions for  $x$  when  $x$  takes values  $1,2,\cdots 100$ .
[list=1]
[*] 20
[*] 25
[*] 15
[*] 50
[/list]

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>.

Suppose a cube has a side length of $8$. Its vertices are alternately colored black and green. What is the volume of the tetrahedron whose vertices are the green vertices of the cube?

Answer: $\frac{512}{3}$


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>.

What is the area of the region defined by the equation $x^2 + y^2 - 10 = 4y - 10x + 4$?

Answer: $43\pi$


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>.

If two lines \( l \) and \( m \) have equations \( y = -2x + 8 \), and \( y = -3x + 9 \), what is the probability that a point randomly selected in the 1st quadrant and below \( l \) will fall between \( l \) and \( m \)? Express your answer as a decimal to the nearest hundredth.

Answer: $0.16$


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>.

Two fair, eight-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 12?

Answer: $\frac{49}{64}$


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>.

Complex numbers \( p, q, r \) form an equilateral triangle with a side length of 24 in the complex plane. If \( |p + q + r| = 48 \), find \( |pq + pr + qr| \).

Answer: $768$


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 equation of line $l$ is $y=x+4$, and the parametric equation of circle $C$ is $\begin{cases} x=2\cos \theta \\ y=2+2\sin \theta \end{cases}$ (where $\theta$ is the parameter), with the origin as the pole and the positive half-axis of $x$ as the polar axis. Establish a polar coordinate system.
- (I) Find the polar coordinates of the intersection points of line $l$ and circle $C$.
- (II) If $P$ is a moving point on circle $C$, find the maximum value of the distance $d$ from $P$ to line $l$.

Answer: $\sqrt{2}+2$


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>.

A seafood wholesaler purchased 1000 kg of crucian carp at a price of 10 yuan/kg and cultured them in a lake (assuming that the weight of each crucian carp remains unchanged during the culture period). According to market estimates, the market price of crucian carp after lake culture will increase by 1 yuan/kg for each day of culture. During the culture period (up to a maximum of 20 days), an average of 10 kg of crucian carp will suffer from oxygen deficiency and float to the surface daily. It is assumed that the oxygen-deficient floating crucian carp can be sold out at a price of 5 yuan/kg.
(1) If the seafood wholesaler cultured the crucian carp for x days and then sold all the living crucian carp along with the oxygen-deficient floating crucian carp, a profit of 8500 yuan could be made. How many days should the crucian carp be cultured?
(2) If various expenses of 450 yuan are incurred for each day during the culture period, what is the maximum profit that the seafood wholesaler can make?

Answer: $6000$


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 \( AB \) and \( AD \) of the square \( ABCD \), points \( E \) and \( F \) are marked such that \( BE : EA = AF : FD = 2022 : 2023 \). The segments \( EC \) and \( FC \) intersect the diagonal of the square \( BD \) at points \( G \) and \( H \) respectively. Find \( \frac{GH}{BD} \).

Answer: $\frac{12271519}{36814556}$


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>.

Bob buys four burgers and three sodas for $\$5.00$, and Carol buys three burgers and four sodas for $\$5.40$. How many cents does a soda cost?

Answer: $94$


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>.

All positive integers whose digits add up to 12 are listed in increasing order. What is the eleventh number in that list?

Answer: $147$


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 circles with centers  $A$  and  $B$  intersect at points  $X$  and  $Y$ . The minor arc  $\angle{XY}=120$  degrees with respect to circle  $A$ , and  $\angle{XY}=60$  degrees with respect to circle  $B$ . If  $XY=2$ , find the area shared by the two circles.

Answer: $\frac{10\pi - 12\sqrt{3}}{9}$


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>.

In a diagram, the grid is composed of 1x1 squares. What is the area of the shaded region if the overall width of the grid is 15 units and its height is 5 units? Some parts are shaded in the following manner: A horizontal stretch from the left edge (6 units wide) that expands 3 units upward from the bottom, and another stretch that begins 6 units from the left and lasts for 9 units horizontally, extending from the 3 units height to the top of the grid.

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>.

An underground line has  $26$  stops, including the first and the final one, and all the stops are numbered from  $1$  to  $26$  according to their order. Inside the train, for each pair  $(x,y)$  with  $1\leq x < y \leq 26$  there is exactly one passenger that goes from the  $x$ -th stop to the  $y$ -th one. If every passenger wants to take a seat during his journey, find the minimum number of seats that must be available on the train.

*Proposed by **FedeX333X***

Answer: $25$


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 certain school is actively preparing for the "Sunshine Sports" activity and has decided to purchase a batch of basketballs and soccer balls totaling $30$ items. At a sports equipment store, each basketball costs $80$ yuan, and each soccer ball costs $60$ yuan. During the purchase period at the school, there is a promotion for soccer balls at 20% off. Let $m\left(0 \lt m \lt 30\right)$ be the number of basketballs the school wants to purchase, and let $w$ be the total cost of purchasing basketballs and soccer balls.
$(1)$ The analytical expression of the function between $w$ and $m$ is ______;
$(2)$ If the school requires the number of basketballs purchased to be at least twice the number of soccer balls, then the school will spend the least amount when purchasing ______ basketballs, and the minimum value of $w$ is ______ yuan.

Answer: $2080$


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 certain stationery store stipulates that if one purchases 250 or more exercise books at a time (including 250 books), they can pay at the wholesale price; if one purchases fewer than 250 books, they must pay at the retail price. Li, a teacher, went to the store to buy exercise books for the 8th-grade students. If he buys 1 book for each 8th-grade student, he must pay $240$ yuan at the retail price; if he buys 60 more books, he can pay at the wholesale price and will need to pay $260$ yuan.
$(1)$ Find the range of the number of 8th-grade students in the school.
$(2)$ If the amount needed to buy 288 books at the wholesale price is the same as the amount needed to buy 240 books at the retail price, find the number of 8th-grade students in the school.

Answer: $200$


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 composite number that has no prime factors less than 15.

Answer: $289$


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 numbers from 1 to 200, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability it is neither a perfect square, a perfect cube, nor a sixth power? Express your answer as a common fraction.

Answer: $\frac{183}{200}$


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 line \( y = -\frac{2}{3}x + 6 \) crosses the \( x \)-axis at \( P \) and the \( y \)-axis at \( Q \). Point \( T(r,s) \) is on the line segment \( PQ \). If the area of \( \triangle POQ \) is four times the area of \( \triangle TOP \), what is the value of \( r+s \)?

Answer: $8.25$


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 local community group sells 180 event tickets for a total of $2652. Some tickets are sold at full price, while others are sold at a discounted rate of half price. Determine the total revenue generated from the full-price tickets.
A) $960
B) $984
C) $1008
D) $1032

Answer: $984$


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 E: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}$$\=1 (a>b>0) passing through point P ($$\sqrt {3}$$, $$\frac {1}{2}$$) with its left focus at F ($$-\sqrt {3}$$, 0).

1. Find the equation of ellipse E.
2. If A is the right vertex of ellipse E, and the line passing through point F with a slope of $$\frac {1}{2}$$ intersects ellipse E at points M and N, find the area of △AMN.

Answer: $\frac {2 \sqrt {5}+ \sqrt {15}}{4}$


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>.

In $\triangle ABC$, $\tan A= \frac {3}{4}$ and $\tan (A-B)=- \frac {1}{3}$, find the value of $\tan C$.

Answer: $\frac {79}{3}$


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 math teacher requires Noelle to do one homework assignment for each of the first four homework points she wants to earn; for each of the next four homework points, she needs to do two homework assignments; and so on, so that to earn the $n^{\text{th}}$ homework point, she has to do $\lceil n\div4 \rceil$ homework assignments. Noelle must earn at least 80% of 20 homework points for this semester. What is the smallest number of homework assignments necessary to earn these points?

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>.

Given that $O$ is the coordinate origin, the complex numbers $z_1$ and $z_2$ correspond to the vectors $\overrightarrow{OZ_1}$ and $\overrightarrow{OZ_2}$, respectively. $\bar{z_1}$ is the complex conjugate of $z_1$. The vectors are represented as $\overrightarrow{OZ_1} = (10 - a^2, \frac{1}{a + 5})$ and $\overrightarrow{OZ_2} = (2a - 5, 2 - a)$, where $a \in \mathbb{R}$, and $(z_2 - z_1)$ is a purely imaginary number.

(1) Determine the quadrant in which the point corresponding to the complex number $\bar{z_1}$ lies in the complex plane.
(2) Calculate $|z_1 \cdot z_2|$.

Answer: $\frac{\sqrt{130}}{8}$


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>.

How many different routes are there from point $A$ to point $B$ in a 3x3 grid (where you can only move to the right or down along the drawn segments)?

[asy]
unitsize(0.09inch);
draw((0,0)--(15,0)--(15,15)--(0,15)--cycle);
draw((5,0)--(5,15));
draw((10,0)--(10,15));
draw((0,5)--(15,5));
draw((0,10)--(15,10));
dot((0,15));
dot((15,0));
label("$A$",(0,15),NW);
label("$B$",(15,0),SE);
[/asy]

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>.

Multiply $555.55$ by $\frac{1}{3}$ and then subtract $333.33$. Express the result as a decimal to the nearest hundredth.

Answer: $-148.15$


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>.

Alice and Bob each arrive at a gathering at a random time between 12:00 noon and 1:00 PM. If Alice arrives after Bob, what is the probability that Bob arrived before 12:45 PM?

Answer: $0.5625$


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 point P is one of the intersections of the hyperbola with foci A(-√10,0), B(√10,0) and a real axis length of 2√2, and the circle x^2 + y^2 = 10, calculate the value of |PA| + |PB|.

Answer: $6\sqrt{2}$


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>.

Let $A = \{x \mid x^2 - ax + a^2 - 19 = 0\}$, $B = \{x \mid x^2 - 5x + 6 = 0\}$, and $C = \{x \mid x^2 + 2x - 8 = 0\}$.
(1) If $A = B$, find the value of $a$;
(2) If $B \cap A \neq \emptyset$ and $C \cap A = \emptyset$, find the value of $a$.

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>.

A sample size of 100 is divided into 10 groups with a class interval of 10. In the corresponding frequency distribution histogram, a certain rectangle has a height of 0.03. What is the frequency of that group?

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>.

Consider a square arrangement of tiles comprising 12 black and 23 white square tiles. A border consisting of an alternating pattern of black and white tiles is added around the square. The border follows the sequence: black, white, black, white, and so on. What is the ratio of black tiles to white tiles in the newly extended pattern?
A) $\frac{25}{37}$
B) $\frac{26}{36}$
C) $\frac{26}{37}$
D) $\frac{27}{37}$

Answer: $\frac{26}{37}$


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 line passing through the point P(3/2, 1/2) intersects the ellipse x^2/6 + y^2/2 = 1 at points A and B, satisfying PA + PB = 0. If M is any point on the line AB and O is the origin, find the minimum value of |OM|.

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>.

Given the ratio of the legs of a right triangle is $3: 4$, determine the ratio of the corresponding segments of the hypotenuse created by dropping a perpendicular from the opposite vertex of the right angle onto the hypotenuse.

Answer: $\frac{16}{9}$


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>.

What is the diameter of the circle inscribed in triangle $DEF$ if $DE = 13,$ $DF = 8,$ and $EF = 15$? Express your answer in simplest radical form.

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


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>.

Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never immediately followed by $D$, and $D$ is never immediately followed by $A$. How many eight-letter good words are there?

Answer: $8748$


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 shape was cut out from a regular hexagon as shown in the picture. The marked points on both the perimeter and inside the hexagon divide the respective line segments into quarters.

What is the ratio of the areas of the original hexagon to the cut-out shape?

Answer: $4:1$


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>.

The opposite of the real number $2023$ is

Answer: $-2023$


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  $ABC$  be a triangle with  $AB=13$ ,  $BC=14$ , and  $CA=15$ . Points  $P$ ,  $Q$ , and  $R$  are chosen on segments  $BC$ ,  $CA$ , and  $AB$ , respectively, such that triangles  $AQR$ ,  $BPR$ ,  $CPQ$  have the same perimeter, which is  $\frac{4}{5}$  of the perimeter of  $PQR$ . What is the perimeter of  $PQR$ ?

*2021 CCA Math Bonanza Individual Round #2*

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 a circle of radius 3, find the area of the region consisting of all line segments of length 6 that are tangent to the circle at their midpoints.
A) $3\pi$
B) $6\pi$
C) $9\pi$
D) $12\pi$
E) $15\pi$

Answer: $9\pi$


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>.

In triangle $\triangle ABC$, $a$, $b$, $c$ are the opposite sides of the internal angles $A$, $B$, $C$, respectively, and $\sin ^{2}A+\sin A\sin C+\sin ^{2}C+\cos ^{2}B=1$.
$(1)$ Find the measure of angle $B$;
$(2)$ If $a=5$, $b=7$, find $\sin C$.

Answer: $\frac{3\sqrt{3}}{14}$


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>.

How many distinct four-digit positive integers are there such that the product of their digits equals 18?

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>.

$(1)$ Given $x \gt 0$, $y \gt 0$, and $2x+3y=6$, find the maximum value of $xy$;<br/>$(2)$ Given $x＜\frac{1}{2}$, find the maximum value of $y=2x+\frac{4}{2x-1}$.

Answer: $-3$


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>.

For  $ n \in \mathbb{N}$ , let  $ f(n)\equal{}1^n\plus{}2^{n\minus{}1}\plus{}3^{n\minus{}2}\plus{}...\plus{}n^1$ . Determine the minimum value of:  $ \frac{f(n\plus{}1)}{f(n)}.$

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>.

Let \(ABCD\) be a trapezium with \(AD\) parallel to \(BC\) and \(\angle ADC = 90^\circ\). Given that \(M\) is the midpoint of \(AB\) with \(CM = \frac{13}{2} \text{ cm}\) and \(BC + CD + DA = 17 \text{ cm}\), find the area of the trapezium \(ABCD\) in \(\text{cm}^2\).

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>.

At a certain crosswalk, the pedestrian signal alternates between red and green lights, with the red light lasting for $40s$. If a pedestrian arrives at the crosswalk and encounters a red light, the probability that they need to wait at least $15s$ for the green light to appear is ______.

Answer: $\dfrac{5}{8}$


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>.

From the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, the probability of randomly selecting two different numbers such that both numbers are odd is $\_\_\_\_\_\_\_\_\_$, and the probability that the product of the two numbers is even is $\_\_\_\_\_\_\_\_\_$.

Answer: $\frac{13}{18}$


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>.

Given that $\log_{10}\sin x + \log_{10}\cos x= -2$ and that $\log_{10}(\sin x+\cos x)=\frac{1}{2}(\log_{10}m-2)$, find $m$.

Answer: $102$


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>.

Determine the coefficient of $x^{8}$ in the expansion of \\((x^{3}+ \frac{1}{2 \sqrt {x}})^{5}\\).

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


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>.

For any real number  $t$ , let  $\lfloor t \rfloor$  denote the largest integer  $\le t$ . Suppose that  $N$  is the greatest integer such that   $$ \left \lfloor \sqrt{\left \lfloor \sqrt{\left \lfloor \sqrt{N} \right \rfloor}\right \rfloor}\right \rfloor = 4 $$  Find the sum of digits of  $N$ .

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>.

Calculate \(3^{18} \div 27^2\) and multiply the result by 7. Write your answer as an integer.

Answer: $3720087$


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>.

China was the first country in the world to use negative numbers. Li Heng, in the book "Fa Jing" written during the Warring States period, already used negative numbers. If the year 500 BC is written as $-500$ years, then the year 2024 AD should be written as ______ years.

Answer: $+2024$


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 set of data $(1)$, $(a)$, $(3)$, $(6)$, $(7)$, its average is $4$, what is its variance?

Answer: $\frac{24}{5}$


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>.

Consider a $10\times10$ checkerboard with alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard?

Answer: $140$


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 \(a\), \(b\), and \(c\) be real numbers such that \(9a^2 + 4b^2 + 25c^2 = 4\). Find the maximum value of
\[6a + 3b + 10c.\]

Answer: $\sqrt{41}$


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 least possible sum of two bases, $c$ and $d$, such that the numeral $29$ in base $c$ represents the same number as $92$ in base $d$, where $c$ and $d$ are positive integers.

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 function $f(x)=\sqrt{3}\sin x\cos x-{\cos }^2x$.
$(1)$ Find the smallest positive period of $f(x)$;

$(2)$ If $f(x)=-1$, find the value of $\cos \left(\dfrac{2\pi }{3}-2x\right)$.

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


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>.

The South China tiger is a first-class protected animal in our country. To save the species from the brink of extinction, the country has established a South China tiger breeding base. Due to scientific artificial cultivation, the relationship between the number of South China tigers $y$ (individuals) and the breeding time $x$ (years) can be approximately described by $y=a\log_{2}(x+1)$. If there were 20 tigers in the first year of breeding (2012), then by 2015, it is predicted that there will be approximately how many tigers?

Answer: $46$


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 sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\overrightarrow{m}=(\sin B+\sin C,\sin A+\sin B)$, $\overrightarrow{n}=(\sin B-\sin C,\sin A)$, and $\overrightarrow{m}\perp \overrightarrow{n}$.

(1) Find the measure of angle $C$;

(2) If $\triangle ABC$ is an isosceles triangle and its circumcircle is a unit circle, find the perimeter $L$ of $\triangle ABC$.

Answer: $2+\sqrt{3}$


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 19 candy boxes arranged in a row, with the middle box containing $a$ candies. Moving to the right, each box contains $m$ more candies than the previous one; moving to the left, each box contains $n$ more candies than the previous one ($a$, $m$, and $n$ are all positive integers). If the total number of candies is 2010, then the sum of all possible values of $a$ is.

Answer: $105$


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  $a_{n+1} = \frac{4}{7}a_n + \frac{3}{7}a_{n-1}$  and  $a_0 = 1$ ,  $a_1 = 2$ . Find  $\lim_{n \to \infty} a_n$ .

Answer: $1.7$


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>.

Equilateral triangle $ABC$ has a side length of $\sqrt{144}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to triangle $ABC$, with $BD_1 = BD_2 = \sqrt{12}$. Additionally, $BD_1$ and $BD_2$ are placed such that $\angle ABD_1 = 30^\circ$ and $\angle ABD_2 = 150^\circ$. Determine the sum $\sum_{k=1}^4 (CE_k)^2$.

Answer: $576$


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 \(7 \cdot 9\frac{2}{5}\).

Answer: $65\frac{4}{5}$


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>.

Given that line $l$ intersects the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ at points $A$ and $B$, and the midpoint of segment $AB$ has coordinates $(6, 2)$, determine the slope of line $l$.

Answer: $\frac{4}{3}$


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>.

The circle centered at $(3,-2)$ and with radius $5$ intersects the circle centered at $(3,4)$ and with radius $3$ at two points $C$ and $D$. Find $(CD)^2$.

Answer: $\frac{224}{9}$


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>.

The distance on the map is 3.6 cm, and the actual distance is 1.2 mm. What is the scale of this map?

Answer: $30:1$


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 vectors $\overrightarrow{a}=(2\sin x,-\cos x)$ and $\overrightarrow{b}=(\sqrt{3}\cos x,2\cos x)$, and function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}+1$.

(I) Find the smallest positive period of function $f(x)$, and find the range of $f(x)$ when $x\in\left[\dfrac{\pi}{12},\dfrac{2\pi}{3}\right]$;

(II) Translate the graph of function $f(x)$ to the left by $\dfrac{\pi}{3}$ unit to obtain the graph of function $g(x)$. In triangle $ABC$, sides $a$, $b$, and $c$ are opposite to angles $A$, $B$, and $C$, respectively. If $g\left(\dfrac{A}{2}\right)=1$, $a=2$, and $b+c=4$, find the area of $\triangle ABC$.

Answer: $\sqrt{3}$


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>.

Real numbers $x_{1}, x_{2}, \cdots, x_{2001}$ satisfy $\sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right|=2001$. Let $y_{k}=\frac{1}{k} \sum_{i=1}^{k} x_{i}$ for $k=1,2, \cdots, 2001$. Find the maximum possible value of $\sum_{k=1}^{2000}\left|y_{k}-y_{k+1}\right|$.

Answer: $2000$


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 5 students are to be distributed into two groups, A and B, with at least one person in each group, and student A cannot be in group A, calculate the number of different distribution schemes.

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>.

Given the line $l: x+ \sqrt {2}y=4 \sqrt {2}$ and the ellipse $C: mx^{2}+ny^{2}=1$ ($n>m>0$) have exactly one common point $M[2 \sqrt {2},2]$.
(1) Find the equation of the ellipse $C$;
(2) Let the left and right vertices of the ellipse $C$ be $A$ and $B$, respectively, and $O$ be the origin. A moving point $Q$ satisfies $QB \perp AB$. Connect $AQ$ and intersect the ellipse at point $P$. Find the value of $\overrightarrow {OQ} \cdot \overrightarrow {OP}$.

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>.

In triangle $ABC$, $AB = 6$, $AC = 8$, and $BC = 10$. The medians $AD$, $BE$, and $CF$ of triangle $ABC$ intersect at the centroid $G$. Let the projections of $G$ onto $BC$, $AC$, and $AB$ be $P$, $Q$, and $R$, respectively. Find $GP + GQ + GR$.

Answer: $\frac{94}{15}$


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>.

Given that 60% of all students in Ms. Hanson's class answered "Yes" to the question "Do you love science" at the beginning of the school year, 40% answered "No", 80% answered "Yes" and 20% answered "No" at the end of the school year, calculate the difference between the maximum and the minimum possible values of y%, the percentage of students that gave a different answer at the beginning and end of the school year.

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>.

In right triangle $ABC$, $\sin A = \frac{8}{17}$ and $\sin B = 1$. Find $\sin C$.

Answer: $\frac{15}{17}$


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>.

Three congruent circles of radius  $2$  are drawn in the plane so that each circle passes through the centers of the other two circles. The region common to all three circles has a boundary consisting of three congruent circular arcs. Let  $K$  be the area of the triangle whose vertices are the midpoints of those arcs. If  $K = \sqrt{a} - b$  for positive integers  $a, b$ , find  $100a+b$ .

*Proposed by Michael Tang*

Answer: $300$


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>.

Due to a snow and ice disaster, a citrus orchard suffered severe damage. To address this, experts proposed a rescue plan for the fruit trees, which needs to be implemented over two years and is independent each year. The plan estimates that in the first year, the probability of the citrus yield recovering to 1.0, 0.9, and 0.8 times the pre-disaster level is 0.2, 0.4, and 0.4, respectively. In the second year, the probability of the citrus yield reaching 1.5, 1.25, and 1.0 times the first year's yield is 0.3, 0.3, and 0.4, respectively. Calculate the probability that the citrus yield will exactly reach the pre-disaster level after two years.

Answer: $0.2$


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>.

What is the maximum number of consecutive positive integers starting from 10 that can be added together before the sum exceeds 500?

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>.

Let  $P$  be an interior point of triangle  $ABC$ . Let  $a,b,c$  be the sidelengths of triangle  $ABC$  and let  $p$  be it's semiperimeter. Find the maximum possible value of  $$  \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right) $$  taking into consideration all possible choices of triangle  $ABC$  and of point  $P$ .

by Elton Bojaxhiu, Albania

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


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>.

The sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is $S_n$, and the sequence $\{b_n\}$ is a geometric sequence, satisfying $a_1=3$, $b_1=1$, $b_2+S_2=10$, and $a_5-2b_2=a_3$. The sum of the first $n$ terms of the sequence $\left\{ \frac{a_n}{b_n} \right\}$ is $T_n$. If $T_n < M$ holds for all positive integers $n$, then the minimum value of $M$ is ______.

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>.

Given a tetrahedron $ABCD$, with $AD$ perpendicular to plane $BCD$, $BC$ perpendicular to $CD$, $AD=2$, $BD=4$, calculate the surface area of the circumscribed sphere of tetrahedron $ABCD$.

Answer: $20\pi$


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>.

Given that point M $(3n-2, 2n+7)$ is on the angle bisector of the second and fourth quadrants, then $n=$ .

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>.

Given that the terminal side of angle $\alpha$ passes through the point $(3a, 4a)$ ($a < 0$), then $\sin\alpha=$ ______, $\tan(\pi-2\alpha)=$ ______.

Answer: $\frac{24}{7}$


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>.

In the sequence $\{a_n\}$, $a_1 = 1$, $a_2 = 2$, and $a_{n+2} - a_n = 1 + (-1)^n$ $(n \in \mathbb{N}^*)$, then $S_{100} = \_\_\_\_\_\_\_\_$.

Answer: $2600$


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 $f(x)=2|x|+|2x-m|$ where $m>0$, and the graph of the function is symmetric about the line $x=1$.
$(Ⅰ)$ Find the minimum value of $f(x)$.
$(Ⅱ)$ Let $a$ and $b$ be positive numbers such that $a+b=m$. Find the minimum value of $\frac{1}{a}+\frac{4}{b}$.

Answer: $\frac{9}{4}$


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>.

Given that the terminal side of $\alpha$ passes through the point $(a, 2a)$ (where $a < 0$),
(1) Find the values of $\cos\alpha$ and $\tan\alpha$.
(2) Simplify and find the value of $$\frac {\sin(\pi-\alpha)\cos(2\pi-\alpha)\sin(-\alpha+ \frac {3\pi}{2})}{\tan(-\alpha-\pi)\sin(-\pi-\alpha)}$$.

Answer: $\frac{1}{10}$


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>.

An academy has $200$ students and $8$ teachers. The class sizes are as follows: $80, 40, 40, 20, 10, 5, 3, 2$. Calculate the average number of students per class as seen by a randomly picked teacher, represented by $t$, and the average number of students per class from the perspective of a randomly selected student, denoted as $s$, and compute the value of $t-s$.

Answer: $-25.69$


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 two quadratic functions $y=x^{2}-2x+2$ and $y=-x^{2}+ax+b$ $(a > 0,b > 0)$, if their tangent lines at one of their intersection points are perpendicular to each other, find the maximum value of $ab$.

Answer: $\frac{25}{16}$


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>.

Seven distinct integers are picked at random from $\{1,2,3,\ldots,12\}$. What is the probability that, among those selected, the third smallest is $4$?

Answer: $\frac{35}{132}$


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 $\triangle PQR$ be a right triangle with angle $Q$ as the right angle. A circle with diameter $QR$ intersects side $PR$ at point $S$. If the area of $\triangle PQR$ is $192$ and $PR = 32$, what is the length of $QS$?

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>.

If a 5-digit number $\overline{x a x a x}$ is divisible by 15, calculate the sum of all such numbers.

Answer: $220200$


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 a point $(-4,a)$ lies on the terminal side of an angle of $600^{\circ}$, determine the value of $a$.

Answer: $-4 \sqrt{3}$


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 the triangular pyramid $P-ABC$, $PA \perp$ the base $ABC$, $AB=1$, $AC=2$, $\angle BAC=60^{\circ}$, the volume is $\frac{\sqrt{3}}{3}$, then the volume of the circumscribed sphere of the triangular pyramid is $\_\_\_\_\_\_\_\_\_\_.$

Answer: $\frac{8 \sqrt{2}}{3} \pi$


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>.

Given $a-b=4$ and $b+c=2$, determine the value of $a^2+b^2+c^2-ab+bc+ca$.

Answer: $28$


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 ratio of the volume of a cube with edge length four inches to the volume of a cube with edge length two feet? Additionally, calculate the ratio of their surface areas.

Answer: $\frac{1}{36}$


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>.

Any type of nature use affects at least one of the natural resources, including lithogenic base, soil, water, air, plant world, and animal world. Types that affect the same set of resources belong to the same type. Research has shown that types of nature use developed in the last 700 years can be divided into 23 types. How many types remain unused?

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>.

Given the function $f(x) = \sqrt{3}\cos x\sin x - \frac{1}{2}\cos 2x$.

(1) Find the smallest positive period of $f(x)$.

(2) Find the maximum and minimum values of $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$ and the corresponding values of $x$.

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>.

Let $a$, $n$, and $l$ be real numbers, and suppose that the roots of the equation \[x^4 - 10x^3 + ax^2 - nx + l = 0\] are four distinct positive integers. Compute $a + n + l.$

Answer: $109$


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 $$\frac {π}{11}$$cos $$\frac {2π}{11}$$cos $$\frac {3π}{11}$$cos $$\frac {4π}{11}$$cos $$\frac {5π}{11}$$\=\_\_\_\_\_\_．

Answer: $\frac {1}{32}$


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>.

Meredith drives 5 miles to the northeast, then 15 miles to the southeast, then 25 miles to the southwest, then 35 miles to the northwest, and finally 20 miles to the northeast. How many miles is Meredith from where she started?

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>.

James and his sister each spin a spinner once. The modified spinner has six congruent sectors numbered from 1 to 6. If the absolute difference of their numbers is 2 or less, James wins. Otherwise, his sister wins. What is the probability that James wins?

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


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>.

The sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is $S_n$, and the sequence $\{b_n\}$ is a geometric sequence, satisfying $a_1=3$, $b_1=1$, $b_2+S_2=10$, and $a_5-2b_2=a_3$. The sum of the first $n$ terms of the sequence $\left\{ \frac{a_n}{b_n} \right\}$ is $T_n$. If $T_n < M$ holds for all positive integers $n$, then the minimum value of $M$ is ______.

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>.

Within a triangular piece of paper, there are 100 points, along with the 3 vertices of the triangle, making it a total of 103 points, and no three of these points are collinear. If these points are used as vertices to create triangles, and the paper is cut into small triangles, then the number of such small triangles is ____.

Answer: $201$


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 $\triangle ABC$, if $BC=4$, $\cos B= \frac{1}{4}$, then $\sin B=$ _______, the minimum value of $\overrightarrow{AB} \cdot \overrightarrow{AC}$ is: _______.

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>.

Find the gradient of the function \( z = x^2 - xy + y^3 \) at the point \( A(1, -1) \) and the derivative in the direction of the vector \( \bar{a} = 3\bar{i} - 4\bar{j} \).

Answer: $\frac{1}{5}$


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>.

Solve for $x$ in the equation $\frac{4}{7} \cdot \frac{1}{9} \cdot x = 14$.

Answer: $220.5$


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>.

If point A $(3,1)$ lies on the line $mx+ny+1=0$, where $mn>0$, then the maximum value of $\frac {3}{m}+ \frac {1}{n}$ is \_\_\_\_\_.

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>.

The teacher and two boys and two girls stand in a row for a photo, with the requirement that the two girls must stand together and the teacher cannot stand at either end. Calculate the number of different arrangements.

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>.

Given a hyperbola \( H: x^{2}-y^{2}=1 \) with a point \( M \) in the first quadrant, and a line \( l \) tangent to the hyperbola \( H \) at point \( M \), intersecting the asymptotes of \( H \) at points \( P \) and \( Q \) (where \( P \) is in the first quadrant). If point \( R \) is on the same asymptote as \( Q \), then the minimum value of \( \overrightarrow{R P} \cdot \overrightarrow{R Q} \) is ______.

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>.

In triangle $\triangle ABC$, $a$, $b$, and $c$ are the opposite sides of angles $A$, $B$, and $C$ respectively, and $\dfrac{\cos B}{\cos C}=-\dfrac{b}{2a+c}$.
(1) Find the measure of angle $B$;
(2) If $b=\sqrt {13}$ and $a+c=4$, find the area of $\triangle ABC$.

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


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>.

Find the square root of $\dfrac{9!}{126}$.

Answer: $12.648$


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 $f(x)=\sin(2x- \frac{\pi}{6})$, determine the horizontal shift required to obtain the graph of the function $g(x)=\sin(2x)$.

Answer: $\frac{\pi}{12}$


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>.

We call a pair of natural numbers $(a, p)$ good if the number $a^{3} + p^{3}$ is divisible by $a^{2} - p^{2}$, and $a > p$.

(a) (1 point) Give any possible value of $a$ for which the pair $(a, 11)$ is good.

(b) (3 points) Find the number of good pairs, where $p$ is a prime number less than 16.

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 rectangle has its length increased by $30\%$ and its width increased by $15\%$. What is the percentage increase in the area of the rectangle?

Answer: $49.5\%$


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>.

Two distinct positive integers $a$ and $b$ are factors of 48. If $a\cdot b$ is not a factor of 48, what is the smallest possible value of $a\cdot b$?

Answer: $32$


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  $a$ ,  $b$ ,  $c$  be positive integers such that  $abc + bc + c = 2014$ . Find the minimum possible value of  $a + b + c$ .

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  $ABC$  be a triangle with  $AB=9$ ,  $BC=10$ ,  $CA=11$ , and orthocenter  $H$ .  Suppose point  $D$  is placed on  $\overline{BC}$  such that  $AH=HD$ .  Compute  $AD$ .

Answer: $\sqrt{102}$


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 was floating in a lake when the lake froze. The ball was removed, leaving a hole $32$ cm across at the top and $16$ cm deep. What was the radius of the ball (in centimeters)?

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 the function $f(x)=\frac{x}{ax+b}(a≠0)$, and its graph passes through the point $(-4,4)$, and is symmetric about the line $y=-x$, find the value of $a+b$.

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


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>.

Let \( a \), \( b \), and \( c \) be the roots of \( x^3 - x + 2 = 0 \). Find \( \frac{1}{a+2} + \frac{1}{b+2} + \frac{1}{c+2} \).

Answer: $\frac{11}{4}$


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>.

Suppose $\cos S = 0.5$ in a right triangle where $SP = 10$. What is $SR$?

[asy]
pair S,P,R;
S = (0,0);
P = (10,0);
R = (0,10*tan(acos(0.5)));
draw(S--P--R--S);
draw(rightanglemark(S,P,R,18));
label("$S$",S,SW);
label("$P$",P,SE);
label("$R$",R,N);
label("$10$",P/2,S);
[/asy]

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>.

Given that a three-digit positive integer (a_1 a_2 a_3) is said to be a convex number if it satisfies (a_1 < a_2 > a_3), determine the number of three-digit convex numbers.

Answer: $240$


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 are perpendicular and intersect at point $O$. Points $A$ and $B$ move along these two lines at a constant speed. When $A$ is at point $O$, $B$ is 500 yards away from point $O$. After 2 minutes, both points $A$ and $B$ are equidistant from $O$. After another 8 minutes, they are still equidistant from $O$. What is the ratio of the speed of $A$ to the speed of $B$?

Answer: $2: 3$