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>.

Choose any four distinct digits $w, x, y, z$ and form the four-digit numbers $wxyz$ and $zyxw$. What is the greatest common divisor of the numbers of the form $wxyz + zyxw + wxyz \cdot zyxw$?

Answer: $11$


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

What is the area and perimeter of the smallest square that can contain a circle with a radius of 6?

Answer: $48$


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 wrote an even number in binary. By removing the trailing $0$ from this binary representation, we obtain the ternary representation of the same number. Determine the number!

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>.

The integers \( r \) and \( k \) are randomly selected, where \(-5 < r < 10\) and \(0 < k < 10\). What is the probability that the division \( r \div k \) results in \( r \) being a square number? Express your answer as a common fraction.

Answer: $\frac{8}{63}$


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

Use the bisection method to find an approximate zero of the function $f(x) = \log x + x - 3$, given that approximate solutions (accurate to 0.1) are $\log 2.5 \approx 0.398$, $\log 2.75 \approx 0.439$, and $\log 2.5625 \approx 0.409$.

Answer: $2.6$


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

Express $0.3\overline{45}$ as a common fraction.

Answer: $\frac{83}{110}$


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

Compute $\tan 40^\circ + 4 \sin 40^\circ.$

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>.

Find a constant \( A > 0 \) such that for all real \( x \geqslant A \), we have

\[ x^{3} + x + 1 \leq \frac{x^{4}}{1000000} \]

Answer: $1000000$


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 function $g$ is defined by $g(z) = (3 + 2i) z^2 + \beta z + \delta$ for all complex numbers $z$, where $\beta$ and $\delta$ are complex numbers and $i^2 = -1$. Suppose that $g(1)$ and $g(-i)$ are both real. What is the smallest possible value of $|\beta| + |\delta|$?

Answer: $2\sqrt{2}$


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

Given a sequence $\{a_n\}$ where each term is a positive number and satisfies the relationship $a_{n+1}^2 = ta_n^2 +(t-1)a_na_{n+1}$, where $n\in \mathbb{N}^*$.

(1) If $a_2 - a_1 = 8$, $a_3 = a$, and the sequence $\{a_n\}$ is unique:
   ① Find the value of $a$.
   ② Let another sequence $\{b_n\}$ satisfy $b_n = \frac{na_n}{4(2n+1)2^n}$. Is there a positive integer $m, n$ ($1 < m < n$) such that $b_1, b_m, b_n$ form a geometric sequence? If it exists, find all possible values of $m$ and $n$; if it does not exist, explain why.

(2) If $a_{2k} + a_{2k-1} + \ldots + a_{k+1} - (a_k + a_{k-1} + \ldots + a_1) = 8$, with $k \in \mathbb{N}^*$, determine the minimum value of $a_{2k+1} + a_{2k+2} + \ldots + a_{3k}$.

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>.

From vertex  $A$  of an equilateral triangle  $ABC$ , a ray  $Ax$  intersects  $BC$  at point  $D$ . Let  $E$  be a point on  $Ax$  such that  $BA =BE$ . Calculate  $\angle AEC$ .

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>.

In the diagram, the numbers 1 to 10 are placed around a circle. Sandy crosses out 1, then 4, and then 7. Continuing in a clockwise direction, she crosses out every third number of those remaining, until only two numbers are left. The sum of these two numbers 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 the function $f(x)=1-2\sin ^{2}(x+ \frac {π}{8})+2\sin (x+ \frac {π}{8})\cos (x+ \frac {π}{8})$.
(1) Find the smallest positive period and the monotonically increasing interval of $f(x)$;
(2) Find the maximum and minimum values of $f(x)$ on the interval $[- \frac {π}{4}, \frac {3π}{8}]$.

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>.

A bag contains 4 red, 3 blue, and 6 yellow marbles. What is the probability, expressed as a decimal, of drawing one red marble followed by one blue marble without replacement?

Answer: $0.076923$


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>.

Xiaoqiang conducts an experiment while drinking a beverage. He inserts a chopstick vertically into the bottom of the cup and measures the wetted part, which is exactly 8 centimeters. He then turns the chopstick around and inserts the other end straight into the bottom of the cup. He finds that the dry part of the chopstick is exactly half of the wetted part. How long is the chopstick?

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 the curve $C$: $y^{2}=4x$ with a focus at point $F$, a line $l$ passes through point $F$ and intersects curve $C$ at points $P$ and $Q$. If the relationship $\overrightarrow{FP}+2\overrightarrow{FQ}=\overrightarrow{0}$ holds, calculate the area of triangle $OPQ$.

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


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>.

Given that there are two alloys with different percentages of copper, with alloy A weighing 40 kg and alloy B weighing 60 kg, a piece of equal weight is cut from each of these two alloys, and each cut piece is then melted together with the remaining part of the other alloy, determine the weight of the alloy cut.

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>.

$a,b,c,d,e$  are equal to  $1,2,3,4,5$  in some order, such that no two of  $a,b,c,d,e$  are equal to the same integer. Given that  $b \leq d, c \geq a,a \leq e,b \geq e,$  and that  $d\neq5,$  determine the value of  $a^b+c^d+e.$

Answer: $628$


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 chorus performance, there are 6 female singers (including 1 lead singer) and 2 male singers arranged in two rows.
(1) If there are 4 people per row, how many different arrangements are possible?
(2) If the lead singer stands in the front row and the male singers stand in the back row, with again 4 people per row, how many different arrangements are possible?

Answer: $5760$


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, and  $K$  and  $L$  be points on  $AB$  such that  $\angle ACK = \angle KCL = \angle LCB$ . Let  $M$  be a point in  $BC$  such that  $\angle MKC = \angle BKM$ . If  $ML$  is the angle bisector of  $\angle KMB$ , find  $\angle MLC$ .

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>.

When $\frac{1}{1001}$ is expressed as a decimal, what is the sum of the first 50 digits after the decimal point?

Answer: $216$


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

The bottoms of two vertical poles are 20 feet apart on a flat ground. One pole is 8 feet tall and the other is 18 feet tall. Simultaneously, the ground between the poles is sloped, with the base of the taller pole being 2 feet higher than the base of the shorter pole due to the slope. Calculate the length in feet of a wire stretched from the top of the shorter pole to the top of the taller pole.

Answer: $\sqrt{544}$


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 sum shown, each of the letters \( D, O, G, C, A \), and \( T \) represents a different digit.
$$
\begin{array}{r}
D O G \\
+C A T \\
\hline 1000
\end{array}
$$

What is the value of \( D + O + G + C + A + T \)?

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>.

One cube has each of its faces covered by one face of an identical cube, making a solid as shown. The volume of the solid is \(875 \ \text{cm}^3\). What, in \(\text{cm}^2\), is the surface area of the solid?

A) 750

B) 800

C) 875

D) 900

E) 1050

Answer: $750$


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 area of $\triangle ABC$ is $S$, and $\overrightarrow{AB} \cdot \overrightarrow{AC} = S$.
(I) Find the value of $\tan 2A$;
(II) If $\cos C = \frac{3}{5}$, and $|\overrightarrow{AC} - \overrightarrow{AB}| = 2$, find the area of $\triangle ABC$.

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


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 $a$ and $b$ are elements of the set ${ 1,2,3,4,5,6 }$ and $|a-b| \leqslant 1$, calculate the probability that any two people playing this game form a "friendly pair".

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


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 polynomial $-5x^2-xy^4+2^6xy+3$ has     terms, and the degree of this polynomial is     . The coefficient of the highest degree term is     .

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>.

What is the remainder when \(2,468,135,790\) is divided by \(99\)?

Answer: $54$


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 \( Q \) be a point chosen uniformly at random in the interior of the unit square with vertices at \( (0,0), (1,0), (1,1) \), and \( (0,1) \). Find the probability that the slope of the line determined by \( Q \) and the point \( \left(\frac{3}{4}, \frac{1}{4} \right) \) is less than or equal to \( -1 \).

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>.

How many $3$-digit positive integers have digits whose product equals $36$?

Answer: $21$


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

Solve the equations:
(1) $(x-2)^2=25$;
(2) $x^2+4x+3=0$;
(3) $2x^2+4x-1=0$.

Answer: $\frac{-2-\sqrt{6}}{2}$


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 sequence $\{a_{n}\}$ satisfies $a_{1}+\frac{1}{2}{a_2}+\frac{1}{3}{a_3}+…+\frac{1}{n}{a_n}={a_{n+1}}-1$, $n\in N^{*}$, and $a_{1}=1$.
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;
$(2)$ Let $S_{n}=a_{1}\cdot a_{n}+a_{2}\cdot a_{n-1}+a_{3}\cdot a_{n-2}+\ldots +a_{n}\cdot a_{1}$, $b_{n}=\frac{1}{{3{S_n}}}$, the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $T_{n}$, find the smallest positive integer $m$ such that $T_{n} \lt \frac{m}{{2024}}$ holds for any $n\in N*$.
$(Reference formula: $1^{2}+2^{2}+3^{2}+\ldots +n^{2}=\frac{{n({n+1})({2n+1})}}{6}$, $n\in N*)$

Answer: $1012$


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

Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$  Find the value of $p + q + r + s.$

Answer: $1716$


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>.

Ten positive integers include the numbers 3, 5, 8, 9, and 11. What is the largest possible value of the median of this list of ten positive integers?

Answer: $11$


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

A point  $P$  is given on the curve  $x^4+y^4=1$ . Find the maximum distance from the point  $P$  to the origin.

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>.

Six people stand in a row, with exactly two people between A and B. Calculate the number of different ways for them to stand.

Answer: $144$


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 sector with central angle $60^\circ$ is cut from a circle of radius 10. Determine both the radius of the circle circumscribed about the sector and the inradius of the triangle formed by the sector.

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


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>.

Given the curve $C$: $\begin{cases}x=2\cos \alpha \\ y= \sqrt{3}\sin \alpha\end{cases}$ ($\alpha$ is a parameter) and the fixed point $A(0, \sqrt{3})$, $F_1$ and $F_2$ are the left and right foci of this curve, respectively. Establish a polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis.

$(1)$ Find the polar equation of the line $AF_2$;

$(2)$ A line passing through point $F_1$ and perpendicular to the line $AF_2$ intersects this conic curve at points $M$ and $N$. Find the value of $||MF_1|-|NF_1||$.

Answer: $\frac{12\sqrt{3}}{13}$


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>.

【Information Extraction】
In some cases, it is not necessary to calculate the result to remove the absolute value symbol. For example: $|6+7|=6+7$, $|6-7|=7-6$, $|7-6|=7-6$, $|-6-7|=6+7$.
【Initial Experience】
$(1)$ According to the above rules, write the following expressions in the form without absolute value symbols (do not calculate the result):
① $|7-21|=$______;
② $|{-\frac{1}{2}+\frac{4}{5}}|=$______;
③ $|{\frac{7}{{17}}-\frac{7}{{18}}}|=\_\_\_\_\_\_.$
【Extended Application】
$(2)$ Calculate:
④ $|{\frac{1}{5}-\frac{{150}}{{557}}}|+|{\frac{{150}}{{557}}-\frac{1}{2}}|-|{-\frac{1}{2}}|$;
⑤ $|{\frac{1}{3}-\frac{1}{2}}|+|{\frac{1}{4}-\frac{1}{3}}|+|{\frac{1}{5}-\frac{1}{4}}|+…+|{\frac{1}{{2021}}-\frac{1}{{2020}}}|+|{\frac{1}{{2022}}-\frac{1}{{2021}}}|$.

Answer: $\frac{505}{1011}$


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 $S_{n}$ be the sum of the first $n$ terms of a geometric sequence $\{a_{n}\}$, and $2S_{3}=7a_{2}$. Determine the value of $\frac{{S}_{5}}{{a}_{2}}$.

Answer: $\frac{31}{8}$


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 positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 -> 231), the resulting number is exactly double the original number. Find the smallest possible nefelibata number.

Answer: $105263157894736842$


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 $a\in (\frac{\pi }{2},\pi )$ and $\sin \alpha =\frac{1}{3}$,

(1) Find the value of $\sin 2\alpha$;

(2) If $\sin (\alpha +\beta )=-\frac{3}{5}$, $\beta \in (0,\frac{\pi }{2})$, find the value of $\sin \beta$.

Answer: $\frac{6\sqrt{2}+4}{15}$


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

The three vertices of an inscribed triangle in a circle divide the circumference into three arcs of lengths $3$, $4$, and $5$. The area of this triangle is:

Answer: $\frac{9}{\pi^{2}}(\sqrt{3}+3)$


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

Last year, Michael took 8 math tests and received 8 different scores, each an integer between 85 and 95, inclusive. After each test, he noticed that the average of his test scores was an integer. His score on the eighth test was 90. What was his score on the seventh test?

Answer: $90$


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

At a school cafeteria, Sam wants to buy a lunch consisting of one main course, one beverage, and one snack. The table below lists Sam's options available in the cafeteria. How many different lunch combinations can Sam choose from?

\begin{tabular}{ |c | c | c | }
\hline \textbf{Main Courses} & \textbf{Beverages} & \textbf{Snacks} \\ \hline
Burger & Water & Apple \\ \hline
Pasta & Soda & Banana \\ \hline
Salad & Juice & \\ \hline
Tacos & & \\ \hline
\end{tabular}

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>.

Jia, Yi, Bing, Ding, and Wu sit around a circular table to play cards. Jia has a fixed seat. If Yi and Ding cannot sit next to each other, how many different seating arrangements are possible?

Answer: $12$


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

Given that the function $y=f(x)$ is an odd function defined on $R$, when $x\leqslant 0$, $f(x)=2x+x^{2}$. If there exist positive numbers $a$ and $b$ such that when $x\in[a,b]$, the range of $f(x)$ is $[\frac{1}{b}, \frac{1}{a}]$, find the value of $a+b$.

Answer: $\frac{3+ \sqrt{5}}{2}$


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>.

The sum to infinity of the terms of an infinite geometric progression is 10. The sum of the first two terms is 7. Compute the first term of the progression.

Answer: $10\left(1 + \sqrt{\frac{3}{10}}\right)$


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 the arithmetic sequences ${a_n}$ and ${b_n}$ are $S_n$ and $T_n$ respectively. If $$\frac {S_{n}}{T_{n}}= \frac {2n+1}{3n+2}$$, find the value of $$\frac {a_{3}+a_{11}+a_{19}}{b_{7}+b_{15}}$$.

Answer: $\frac{129}{130}$


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>.

Following the concept of a healthy, low-carbon lifestyle, an increasing number of people are renting bicycles for cycling tours. A particular bicycle rental point charges no fee for rentals that do not exceed two hours, and for rentals that exceed two hours, the charging standard is 2 yuan per hour (with fractions of an hour calculated as a full hour). Suppose two individuals, A and B, each rent a bicycle once. The probability that A and B return their bicycles within two hours is $\frac{1}{4}$ and $\frac{1}{2}$, respectively. The probability that they return their bicycles between two and three hours is $\frac{1}{2}$ and $\frac{1}{4}$, respectively. Neither A nor B will rent a bicycle for more than four hours.

(I) Calculate the probability that the bicycle rental fees paid by A and B are the same.
(II) Let $\xi$ be the random variable representing the sum of the bicycle rental fees paid by A and B. Find the distribution of $\xi$ and its mathematical expectation $E_{\xi}$.

Answer: $\frac{7}{2}$


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

Given that $E$ and $F$ are a chord of the circle $C: x^{2}+y^{2}-2x-4y+1=0$, and $CE\bot CF$, $P$ is the midpoint of $EF$. When the chord $EF$ moves on the circle $C$, there exist two points $A$ and $B$ on the line $l: x-y-3=0$ such that $∠APB≥\frac{π}{2}$ always holds. Determine the minimum value of the length of segment $AB$.

Answer: $4\sqrt{2}$


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 function $f(x)=x^3+ax^2+bx+1$, its derivative $f'(x)$ satisfies $f'(1)=2a$, $f'(2)=-b$, where $a$ and $b$ are constants.

(I) Find the values of $a$ and $b$;

(II) Let $g(x)=f'(x)e^{-x}$, find the extreme values of the function $g(x)$.

Answer: $15e^{-3}$


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

Let  $A$  be the greatest possible value of a product of positive integers that sums to  $2014$ .  Compute the sum of all bases and exponents in the prime factorization of  $A$ .  For example, if  $A=7\cdot 11^5$ , the answer would be  $7+11+5=23$ .

Answer: $677$


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>.

Into how many parts can a plane be divided by four lines? Consider all possible cases and make a drawing for each case.

Answer: $11$


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

Given a circle $⊙O$ with equation $x^{2}+y^{2}=6$, let $P$ be a moving point on the circle. Draw a line $PM$ perpendicular to the x-axis at point $M$, and let $N$ be a point on $PM$ such that $\overrightarrow{PM}= \sqrt {2} \overrightarrow{NM}$.

(I) Find the equation for the trajectory of point $N$, denoted as curve $C$.

(II) Given points $A(2,1)$, $B(3,0)$, if a line passing through point $B$ intersects curve $C$ at points $D$ and $E$, determine whether the sum of slopes $k_{AD}+k_{AE}$ is a constant value. If it is, find this value; if not, explain why.

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>.

How many distinct numbers can you get by multiplying two or more distinct members of the set $\{1, 2, 3, 7, 13\}$, or taking any number to the power of another (excluding power 1) from the same set?

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>.

A right cylinder with a height of 5 inches has a radius of 3 inches. A cone with a height of 3 inches and the same radius is placed on top of the cylinder. What is the total surface area of the combined object, in square inches? Express your answer in terms of \(\pi\).

Answer: $30\pi + 9\sqrt{2}\pi$


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>.

The NIMO problem writers have invented a new chess piece called the *Oriented Knight*.  This new chess piece has a limited number of moves: it can either move two squares to the right and one square upward or two squares upward and one square to the right.  How many ways can the knight move from the bottom-left square to the top-right square of a  $16\times 16$  chess board?

*Proposed by Tony Kim and David Altizio*

Answer: $252$


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 land of Chaina, people pay each other in the form of links from chains. Fiona, originating from Chaina, has an open chain with  $2018$  links. In order to pay for things, she decides to break up the chain by choosing a number of links and cutting them out one by one, each time creating  $2$  or  $3$  new chains. For example, if she cuts the  $1111$ th link out of her chain first, then she will have  $3$  chains, of lengths  $1110$ ,  $1$ , and  $907$ . What is the least number of links she needs to remove in order to be able to pay for anything costing from  $1$  to  $2018$  links using some combination of her chains?

*2018 CCA Math Bonanza Individual Round #10*

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 two vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ with an angle of $\frac {2\pi}{3}$ between them, $|\overrightarrow {a}|=2$, $|\overrightarrow {b}|=3$, let $\overrightarrow {m}=3\overrightarrow {a}-2\overrightarrow {b}$ and $\overrightarrow {n}=2\overrightarrow {a}+k\overrightarrow {b}$:
1. If $\overrightarrow {m} \perp \overrightarrow {n}$, find the value of the real number $k$;
2. Discuss whether there exists a real number $k$ such that $\overrightarrow {m} \| \overrightarrow {n}$, and explain the reasoning.

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>.

Kevin Kangaroo starts at 0 on a number line and aims to reach the point 2, but his hopping strategy alternates. On odd-numbered hops, he covers $\frac{1}{2}$ of the distance to his goal, while on even-numbered hops, he covers $\frac{1}{4}$ of the distance remaining to the goal. Determine the total distance Kevin hops after six hops. Express your answer as a fraction.

Answer: $\frac{485}{256}$


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

The distances from point \( P \), which lies inside an equilateral triangle, to its vertices are 3, 4, and 5. Find the area of the triangle.

Answer: $9 + \frac{25\sqrt{3}}{4}$


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>.

Given the numbers \(-2, -1, 0, 1, 2\), arrange them in some order. Compute the difference between the largest and smallest possible values that can be obtained using the iterative average procedure.

Answer: $2.125$


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

Find the maximum value of the function $y=\frac{x}{{{e}^{x}}}$ on the interval $[0,2]$.

A) When $x=1$, $y=\frac{1}{e}$
B) When $x=2$, $y=\frac{2}{{{e}^{2}}}$
C) When $x=0$, $y=0$
D) When $x=\frac{1}{2}$, $y=\frac{1}{2\sqrt{e}}$

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


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

A diagonal from a vertex of a polygon forms a triangle with the two adjacent sides, so the number of triangles formed by the polygon is at least the number of sides of the polygon. However, the number of triangles cannot exceed the number of ways to choose 2 sides from the polygon, which is given by the combination formula $\binom{n}{2}=\frac{n(n-1)}{2}$. If the number of triangles is at most 2021, determine the number of sides of the polygon.

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>.

The diagram shows five circles of the same radius touching each other. A square is drawn so that its vertices are at the centres of the four outer circles. What is the ratio of the area of the shaded parts of the circles to the area of the unshaded parts of the circles?

Answer: $2:3$


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>.

Given $\tan ( \frac {π}{4}+x)=- \frac {1}{2}$, find the value of $\tan 2x$ (Part 1) and simplify the trigonometric expression $\sqrt { \frac {1+\sin x}{1-\sin x}}+ \sqrt { \frac {1-\sin x}{1+\sin x}}$ if $x$ is an angle in the second quadrant. Then, find its value (Part 2).

Answer: $2\sqrt {10}$


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

Round 1278365.7422389 to the nearest hundred.

Answer: $1278400$


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 non-empty subsets \( S \) of \( \{1, 2, 3, \ldots, 12\} \) have the following two properties?
1. No two consecutive integers belong to \( S \).
2. If \( S \) contains \( k \) elements, then \( S \) contains no number less than \( k \).

Answer: $128$


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 quadratic function $f(x)=x^{2}-x+k$, where $k\in\mathbb{Z}$, if the function $g(x)=f(x)-2$ has two distinct zeros in the interval $(-1, \frac{3}{2})$, find the minimum value of $\frac{[f(x)]^{2}+2}{f(x)}$.

Answer: $\frac{81}{28}$


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>.

Given vectors $\overrightarrow{a} = (1, 2)$, $\overrightarrow{b} = (x, 1)$,
1. If $\langle \overrightarrow{a}, \overrightarrow{b} \rangle$ forms an acute angle, find the range of $x$.
2. Find the value of $x$ when $(\overrightarrow{a}+2\overrightarrow{b}) \perp (2\overrightarrow{a}-\overrightarrow{b})$.

Answer: $\frac{7}{2}$


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

In triangle ABC, the lengths of the sides opposite to angles A, B, and C are a, b, and c, respectively. The area of triangle ABC is given by $\frac{\sqrt{3}}{6}b(b + c - a\cos C)$.

1. Find angle A.
2. If b = 1 and c = 3, find the value of $\cos(2C - \frac{\pi}{6})$.

Answer: $-\frac{4\sqrt{3}}{7}$


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>.

Which of the following is equal to \(\frac{4}{5}\)?

Answer: $0.8$


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>.

In triangle $DEF$, the side lengths are $DE = 15$, $EF = 20$, and $FD = 25$. A rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. Letting $WX = \lambda$, the area of $WXYZ$ can be expressed as the quadratic polynomial \[Area(WXYZ) = \gamma \lambda - \delta \lambda^2.\]
Then the coefficient $\gamma = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Answer: $16$


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

Let \( M = \{1, 2, \ldots, 10\} \), and let \( A_1, A_2, \ldots, A_n \) be distinct non-empty subsets of \( M \). For \( i \neq j \), the intersection \( A_i \cap A_j \) contains at most two elements. Find the maximum value of \( n \).

Answer: $175$


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

Given the function $f(x)=\cos^4x+2\sin x\cos x-\sin^4x$
$(1)$ Determine the parity, the smallest positive period, and the intervals of monotonic increase for the function $f(x)$.
$(2)$ When $x\in\left[0, \frac{\pi}{2}\right]$, find the maximum and minimum values of the function $f(x)$.

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 an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with left focus $F$ and a chord perpendicular to the major axis of length $6\sqrt{2}$, a line passing through point $P(2,1)$ with slope $-1$ intersects $C$ at points $A$ and $B$, where $P$ is the midpoint of $AB$. Find the maximum distance from a point $M$ on ellipse $C$ to focus $F$.

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


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

A pyramid is constructed using twenty cubical blocks: the first layer has 10 blocks arranged in a square, the second layer contains 6 blocks arranged in a larger square centered on the 10, the third layer has 3 blocks arranged in a triangle, and finally one block sits on top of the third layer. Each block in layers 2, 3, and 4 has a number assigned which is the sum of the blocks directly below it from the previous layer. If the blocks in the first layer are numbered consecutively from 1 to 10 in any order, what is the smallest possible number that could be assigned to the top block?
A) 45
B) 54
C) 63
D) 72
E) 81

Answer: $54$


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 school there are \( n \) students, each with a different student number. Each student number is a positive factor of \( 60^{60} \), and the H.C.F. of any two student numbers is not a student number in the school. Find the greatest possible value of \( n \).

Answer: $3721$


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 digit $B$ will make the number $527B$ divisible by $5$?

Answer: $0 \text{ or } 5$


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

(The full score for this question is 8 points) Arrange 3 male students and 2 female students in a row,
　　(1) The number of all different arrangements;
(2) The number of arrangements where exactly two male students are adjacent;
(3) The number of arrangements where male students are of different heights and are arranged from tallest to shortest from left to right.
[Answer all questions with numbers]

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 a connected simple graph \( G \) with a known number of edges \( e \), where each vertex has some number of pieces placed on it (each piece can only be placed on one vertex of \( G \)). The only operation allowed is when a vertex \( v \) has a number of pieces not less than the number of its adjacent vertices \( d \), you can choose \( d \) pieces from \( v \) and distribute them to the adjacent vertices such that each adjacent vertex gets one piece. If every vertex in \( G \) has a number of pieces less than the number of its adjacent vertices, no operations can be performed.

Find the minimum value of \( m \) such that there exists an initial placement of the pieces with a total of \( m \) pieces, allowing you to perform infinitely many operations starting from this placement.

Answer: $e$


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

A shooter has a probability of hitting the target of $0.8$ each time. Now, using the method of random simulation to estimate the probability that the shooter hits the target at least $3$ times out of $4$ shots: first, use a calculator to generate random integers between $0$ and $9$, where $0$, $1$ represent missing the target, and $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ represent hitting the target; since there are $4$ shots, every $4$ random numbers are grouped together to represent the results of $4$ shots. After randomly simulating, $20$ groups of random numbers were generated:
$5727$ $0293$  $7140$  $9857$  $0347$  $4373$  $8636$  $9647$  $1417$   $4698$
$0371$  $6233$  $2616$  $8045$  $6011$  $3661$  $9597$  $7424$  $6710$   $4281$
Based on this, estimate the probability that the shooter hits the target at least $3$ times out of $4$ shots.

Answer: $0.75$


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 "Kiwi" business owner has two types of "Kiwi" for promotion, type \\(A\\) and type \\(B\\). If you buy 2 pieces of type \\(A\\) "Kiwi" and 1 piece of type \\(B\\) "Kiwi", the total cost is 120 yuan; if you buy 3 pieces of type \\(A\\) "Kiwi" and 2 pieces of type \\(B\\) "Kiwi", the total cost is 205 yuan.

\\((1)\\) Let the price per piece of type \\(A\\) and type \\(B\\) "Kiwi" be \\(a\\) yuan and \\(b\\) yuan, respectively. Find the values of \\(a\\) and \\(b\\).

\\((2)\\) The cost price of each piece of type \\(B\\) "Kiwi" is 40 yuan. According to market research: if sold at the unit price found in \\((1)\\), the "Kiwi" business owner sells 100 pieces of type \\(B\\) "Kiwi" per day; if the selling price increases by 1 yuan, the daily sales volume of type \\(B\\) "Kiwi" decreases by 5 pieces. Find the selling price per piece that maximizes the daily profit of type \\(B\\) "Kiwi", and what is the maximum profit?

Answer: $1125$


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 $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a>0$, $b>0$) with its left focus $F$ and right vertex $A$, upper vertex $B$. If the distance from point $F$ to line $AB$ is $\frac{5\sqrt{14}}{14}b$, find the eccentricity of the ellipse.

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>.

Given a sequence $\{a_n\}$ where $a_n = n$, for each positive integer $k$, in between $a_k$ and $a_{k+1}$, insert $3^{k-1}$ twos (for example, between $a_1$ and $a_2$, insert three twos, between $a_2$ and $a_3$, insert $3^1$ twos, between $a_3$ and $a_4$, insert $3^2$ twos, etc.), to form a new sequence $\{d_n\}$. Let $S_n$ denote the sum of the first $n$ terms of the sequence $\{d_n\}$. Find the value of $S_{120}$.

Answer: $245$


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 arithmetic sequence $\{a_n\}$, $a_{10} < 0$, $a_{11} > 0$ and $a_{11} > |a_{10}|$. If the sum of the first $n$ terms of $\{a_n\}$, denoted as $S_n$, is less than $0$, the maximum value of $n$ is ____.

Answer: $19$


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, B, C, and D obtained the top four positions (without ties) in the school, and they made the following statements:
A: "I am neither first nor second."
B: "My position is adjacent to C's position."
C: "I am neither second nor third."
D: "My position is adjacent to B's position."

It is known that A, B, C, and D respectively obtained the places $A, B, C, D$. Determine the four-digit number $\overrightarrow{\mathrm{ABCD}}$.

Answer: $4213$


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 circle $O$, points $E$ and $F$ are on the same side of diameter $\overline{AB}$, $\angle AOE = 60^\circ$, and $\angle FOB = 90^\circ$. Calculate the ratio of the area of the smaller sector $EOF$ to the area of the circle.

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


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 Let \\(S_{n}\\) and \\(T_{n}\\) be the sums of the first \\(n\\) terms of the arithmetic sequences \\(\{a_{n}\}\\) and \\(\{b_{n}\}\\), respectively, and \\( \frac {S_{n}}{T_{n}}= \frac {n}{2n+1} (n∈N^{*})\\), determine the value of \\( \frac {a_{6}}{b_{6}}\\).

Answer: $\frac{11}{23}$


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  $I$  be the incenter of  $\triangle ABC$ , and  $O$  be the excenter corresponding to  $B$ . If  $|BI|=12$ ,  $|IO|=18$ , and  $|BC|=15$ , then what is  $|AB|$ ?

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 the function $f(x)=\sin 2x-2\cos^2x$ $(x\in\mathbb{R})$.
- (I) Find the value of $f\left( \frac{\pi}{3}\right)$;
- (II) When $x\in\left[0, \frac{\pi}{2}\right]$, find the maximum value of the function $f(x)$ and the corresponding value of $x$.

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


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

Given a sequence ${a_n}$ defined by $|a_{n+1}|+|a_n|=3$ for any positive integer $n$ and $a_1=2$, find the minimum value of the sum $S_{2019}$ of its first 2019 terms.

Answer: $-3025$


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 Elmer’s new car provides a 70% better fuel efficiency and uses a type of fuel that is 35% more expensive per liter, calculate the percentage by which Elmer will save money on fuel costs if he uses his new car for his journey.

Answer: $20.6\%$


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

A ray starting from point $A(-4,1)$ reflects off the line $l_{1}: x-y+3=0$ and the reflected ray passes through point $B(-3,2)$. Find the slope of the line containing the reflected ray.

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>.

Let  $a$  and  $b$  be positive integers such that all but  $2009$  positive integers are expressible in the form  $ma + nb$ , where  $m$  and  $n$  are nonnegative integers. If  $1776 $ is one of the numbers that is not expressible, find  $a + b$ .

Answer: $133$


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  $5$ -digit numbers  $N$  (in base  $10$ ) contain no digits greater than  $3$  and satisfy the equality  $\gcd(N,15)=\gcd(N,20)=1$ ? (The leading digit of  $N$  cannot be zero.)

*Based on a proposal by Yannick Yao*

Answer: $256$


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

Given $\sin\left(\frac{\pi}{3}+\frac{\alpha}{6}\right)=-\frac{3}{5}$, $\cos\left(\frac{\pi}{12}-\frac{\beta}{2}\right)=-\frac{12}{13}$, $-5\pi < \alpha < -2\pi$, $-\frac{11\pi}{6} < \beta < \frac{\pi}{6}$,

Find the value of $\sin \left(\frac{\alpha }{6}+\frac{\beta }{2}+\frac{\pi }{4}\right)$.

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


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

In rectangle $ABCD$, $AB = 4$ and $BC = 8$. The rectangle is folded so that points $B$ and $D$ coincide, forming the pentagon $ABEFC$. What is the length of segment $EF$? Express your answer in simplest radical form.

Answer: $4\sqrt{5}$


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>.

Three $1 \times 1 \times 1$ cubes are joined face to face in a single row and placed on a table. The cubes have a total of 11 exposed $1 \times 1$ faces. If sixty $1 \times 1 \times 1$ cubes are joined face to face in a single row and placed on a table, how many $1 \times 1$ faces are exposed?

Answer: $182$


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 probability that a light bulb lasts more than 1000 hours is 0.2. Determine the probability that 1 out of 3 light bulbs fails after 1000 hours of use.

Answer: $0.096$


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 an institute, there are truth-tellers, who always tell the truth, and liars, who always lie. One day, each of the employees made two statements:

1) There are fewer than ten people in the institute who work more than I do.
2) In the institute, at least one hundred people have a salary greater than mine.

It is known that the workload of all employees is different, and their salaries are also different. How many people work in the institute?

Answer: $110$


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 Asha's study times were 40, 60, 50, 70, 30, 55, 45 minutes each day of the week and Sasha's study times were 50, 70, 40, 100, 10, 55, 0 minutes each day, find the average number of additional minutes per day Sasha studied compared to Asha.

Answer: $-3.57$


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 four-digit positive integers $x$ satisfy $3874x + 481 \equiv 1205 \pmod{31}$?

Answer: $290$


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 \( A(2,0) \) be a fixed point on the plane, \( P\left(\sin \left(2 t-60^{\circ}\right), \cos \left(2 t-60^{\circ}\right)\right) \) be a moving point. When \( t \) changes from \( 15^{\circ} \) to \( 45^{\circ} \), the area swept by the line segment \( AP \) is ______.

Answer: $\frac{\pi}{6}$


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

For some positive integers $c$ and $d$, the product \[\log_c(c+1) \cdot \log_{c+1} (c+2) \dotsm \log_{d-2} (d-1)  \cdot\log_{d-1} d\]contains exactly $930$ terms, and its value is $3.$ Compute $c+d.$

Answer: $1010$


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 eve of the 2010 Guangzhou Asian Games, a 12-person tour group took a commemorative photo near a venue of the Asian Games. They initially stood in a formation with 4 people in the front row and 8 people in the back row. Now, the photographer plans to keep the order of the front row unchanged, and move 2 people from the back row to the front row, ensuring that the 2 moved people are not adjacent in the front row. The number of different ways to adjust their positions is _____. (Answer in numerals)

Answer: $560$


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 8-sided dice are tossed (each die has faces numbered 1 to 8). What is the probability that the sum of the numbers shown on the dice is either a prime or a multiple of 4?

Answer: $\frac{39}{64}$


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>.

Rohan wants to cut a piece of string into nine pieces of equal length. He marks his cutting points on the string. Jai wants to cut the same piece of string into only eight pieces of equal length. He marks his cutting points on the string. Yuvraj then cuts the string at all the cutting points that are marked. How many pieces of string does Yuvraj obtain?

A 15
B 16
C 17
D 18
E 19

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>.

A bag contains $5$ small balls of the same shape and size, with $2$ red balls and $3$ white balls. Three balls are randomly drawn from the bag.
$(1)$ Find the probability that exactly one red ball is drawn;
$(2)$ Let the random variable $X$ represent the number of red balls drawn. Find the distribution of the random variable $X$.

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


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

Given a sphere O with a radius of 2, a cone is inscribed in the sphere O. When the volume of the cone is maximized, find the radius of the sphere inscribed in the cone.

Answer: $\frac{4(\sqrt{3} - 1)}{3}$


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

In triangle $ABC$, $AB=15$ and $AC=8$. The angle bisector of $\angle A$ intersects $BC$ at point $D$, and the incenter $I$ of triangle $ABC$ is on the segment $AD$. Let the midpoint of $AD$ be $M$. Find the ratio of $IP$ to $PD$ where $P$ is the intersection of $AI$ and $BM$.

Answer: $1:1$


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>.

Find the maximum of the expression
$$
|| \ldots|| x_{1}-x_{2}\left|-x_{3}\right|-\ldots\left|-x_{2023}\right|,
$$
where \( x_{1}, x_{2}, \ldots, x_{2023} \) are distinct natural numbers between 1 and 2023.

Answer: $2022$


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 triangle with vertices at $(2, 1)$, $(4, 7)$, and $(7, 3)$. This triangle is reflected about the line $y=4$. Find the area of the union of the original and the reflected triangles.

Answer: $26$


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

In quadrilateral \( \square ABCD \), point \( M \) lies on diagonal \( BD \) with \( MD = 3BM \). Let \( AM \) intersect \( BC \) at point \( N \). Find the value of \( \frac{S_{\triangle MND}}{S_{\square ABCD}} \).

Answer: $1/8$


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>.

(1) Given a sequence $\{a_n\}$ that satisfies $a_1a_2…a_n=n+1$, find $a_3=$

(2) Let $\overrightarrow{e_1}, \overrightarrow{e_2}$ be unit vectors, where $\overrightarrow{a}=2\overrightarrow{e_1}+\overrightarrow{e_2}, \overrightarrow{b}=\overrightarrow{e_2}$, and $\overrightarrow{a} \cdot \overrightarrow{b}=2$, find $|\overrightarrow{a}|=$

(3) For an arithmetic sequence with the first term $-24$, starting from the 9th term, the terms become positive. Find the range of the common difference $d$.

(4) In the sequence $\{a_n\}$, if the point $(n,a_n)$ lies on a fixed line $l$ passing through the point $(5,3)$, find the sum of the first 9 terms of the sequence, $S_9=$

(5) In rectangle $ABCD$, where $AB=2AD=2$, if $P$ is a moving point on $DC$, find the minimum value of $\overrightarrow{PA} \cdot \overrightarrow{PB}-\overrightarrow{PA} \cdot \overrightarrow{BC}=$

(6) In $\triangle ABC$, where angle $A$ is the largest and angle $C$ is the smallest, and $A=2C$, $a+c=2b$, find the ratio $a:b:c=$

Answer: $6:5:4$


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>.

As shown in the diagram, \( D \), \( E \), and \( F \) are points on the sides \( BC \), \( CA \), and \( AB \) of \(\triangle ABC\), respectively, and \( AD \), \( BE \), \( CF \) intersect at point \( G \). Given that the areas of \(\triangle BDG\), \(\triangle CDG\), and \(\triangle AEG\) are 8, 6, and 14 respectively, find the area of \(\triangle ABC\).

Answer: $63$


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

How many divisors of $9!$ are multiples of $10$?

Answer: $70$


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

In triangle $DEF$, $DE = 6$, $EF = 8$, and $FD = 10$. Point $Q$ is randomly selected inside triangle $DEF$. What is the probability that $Q$ is closer to $D$ than it is to either $E$ or $F$?

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