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

Wang Hong's father deposited 20,000 yuan in the bank for a fixed term of three years, with an annual interest rate of 3.33%. How much money, including the principal and interest, can Wang Hong's father withdraw at the end of the term?

Answer: $21998$


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

A function \(f(x)\) is defined for all real numbers \(x\). For all non-zero values \(x\), we have
\[3f\left(x\right) + f\left(\frac{1}{x}\right) = 15x + 8.\]
Let \(S\) denote the sum of all of the values of \(x\) for which \(f(x) = 2004\). Compute the integer nearest to \(S\).

Answer: $356$


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

Find the largest positive integer $m$ such that an $m \times m$ square can be exactly divided into 7 rectangles with pairwise disjoint interiors, and the lengths of the 14 sides of these 7 rectangles are $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14$.

Answer: $22$


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

For integers a, b, c, and d the polynomial  $p(x) =$   $ax^3 + bx^2 + cx + d$  satisﬁes  $p(5) + p(25) = 1906$ . Find the minimum possible value for  $|p(15)|$ .

Answer: $47$


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

Calculate the value of $8\cos ^{2}25^{\circ}-\tan 40^{\circ}-4$.

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>.

Determine the volume in cubic inches of a right, rectangular prism where the areas of the side, front, and bottom faces are 20 square inches, 12 square inches, and 8 square inches, respectively.

Answer: $8\sqrt{30}$


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>.

A particle begins at a point P on the parabola y = x^2 - 2x - 8 where the y-coordinate is 8. It rolls along the parabola to the nearest point Q where the y-coordinate is -8. Calculate the horizontal distance traveled by the particle, defined as the absolute difference between the x-coordinates of P and Q.

Answer: $\sqrt{17} - 1$


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>.

At a bus station, there are three buses departing to a school between 6:30 AM and 7:30 AM each day. The ticket prices for the buses are the same, but the comfort levels vary. Xiao Jie, a student, observes before boarding. When the first bus arrives, he does not get on but carefully observes its comfort level. If the comfort level of the second bus is better than the first, he will board the second bus; otherwise, he will take the third bus. Given that the comfort levels of the buses can be classified as high, medium, and low, the probability of Xiao Jie boarding a bus with a high comfort level is ______.

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


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

In a finite sequence of real numbers, the sum of any 7 consecutive terms is negative while the sum of any 11 consecutive terms is positive. What is the maximum number of terms in such a sequence?

Answer: $16$


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

Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 15x^2 + 50x - 60$. It is given that there exist real numbers $A$, $B$, and $C$ such that \[\dfrac{1}{s^3 - 15s^2 + 50s - 60} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}\]for all $s\not\in\{p,q,r\}$. What is $\tfrac1A+\tfrac1B+\tfrac1C$?
A) 133
B) 134
C) 135
D) 136
E) 137

Answer: $135$


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

In $\triangle PQR,$ $PQ=PR=30$ and $QR=28.$ Points $M, N,$ and $O$ are located on sides $\overline{PQ},$ $\overline{QR},$ and $\overline{PR},$ respectively, such that $\overline{MN}$ and $\overline{NO}$ are parallel to $\overline{PR}$ and $\overline{PQ},$ respectively. What is the perimeter of parallelogram $PMNO$?

Answer: $60$


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

A particular integer is the smallest multiple of 72, each of whose digits is either 0 or 1. How many digits does this integer have?

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>.

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

Answer: $18$


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

Given that $| \overrightarrow{a}|=3 \sqrt {2}$, $| \overrightarrow{b}|=4$, $\overrightarrow{m}= \overrightarrow{a}+ \overrightarrow{b}$, $\overrightarrow{n}= \overrightarrow{a}+λ \overrightarrow{b}$, $ < \overrightarrow{a}, \overrightarrow{b} > =135^{\circ}$, if $\overrightarrow{m} \perp \overrightarrow{n}$, find the value of $λ$.

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


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

Calculate the integrals:
1) \(\int_{0}^{\frac{\pi}{2}} \sin ^{3} x \, dx\);
2) \(\int_{0}^{\ln 2} \sqrt{e^{x}-1} \, dx\);
3) \(\int_{-a}^{a} x^{2} \sqrt{a^{2}-x^{2}} \, dx\);
4) \(\int_{1}^{2} \frac{\sqrt{x^{2}-1}}{x} \, dx\).

Answer: $\sqrt{3} - \frac{\pi}{3}$


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

Given that point P is a point on the graph of the function f(x)=e^(2x), find the minimum distance from point P to the line l:y=2x.

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


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>.

A cylindrical can has a circumference of 24 inches and a height of 7 inches. A spiral strip is painted on the can such that it winds around the can precisely once, reaching from the bottom to the top. However, instead of reaching directly above where it started, it ends 3 inches horizontally to the right. What is the length of the spiral strip?

Answer: $\sqrt{778}$


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

The minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{30}\cdot 5^3}$ as a decimal.

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>.

Determine the smallest natural number  $n$  for which there exist distinct nonzero naturals  $a, b, c$ , such that  $n=a+b+c$  and  $(a + b)(b + c)(c + a)$  is a perfect cube.

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>.

Point $P$ moves on the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{25} = 1$, and points $A$ and $B$ move respectively on the circles $x^2 + (y-4)^2 = 16$ and $x^2 + (y+4)^2 = 4$. The maximum value of $PA + PB$ is \_\_\_\_\_\_.

Answer: $16$


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

A space probe travels $5,555,555,555,555$ kilometers in its journey towards a distant star. After reaching its destination, it has traveled $3,333,333,333,333$ kilometers from its last refueling point. How many kilometers did the probe travel before the last refueling point?
A) $2,111,111,111,111$
B) $2,222,222,222,222$
C) $2,333,333,333,333$
D) $2,444,444,444,444$
E) $2,555,555,555,555$

Answer: $2,222,222,222,222$


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

Find the minimum value of the expression:
\[(15 - x)(8 - x)(15 + x)(8 + x).\]

Answer: $-6480.25$


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

An equilateral triangle of side length $12$ is completely filled by non-overlapping equilateral triangles of side length $2$.

Answer: $36$


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

In the Cartesian coordinate system $xOy$, the curve $C_{1}$ is defined by $\begin{cases} x=-2+\cos \alpha \\ y=-1+\sin \alpha \end{cases}$ (where $\alpha$ is a parameter). In the polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the curve $C_{2}$ is defined by $\rho\cos \theta-3=0$. Point $P$ is a moving point on the curve $C_{1}$.

$(1)$ Find the maximum distance from point $P$ to the curve $C_{2}$;

$(2)$ If the curve $C_{3}$: $\theta= \frac {\pi}{4}$ intersects curve $C_{1}$ at points $A$ and $B$, find the area of $\triangle ABC_{1}$.

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>.

If $\theta \in \left[\dfrac{\pi}{4}, \dfrac{\pi}{2}\right]$ and $\sin 2\theta = \dfrac{3\sqrt{7}}{8}$, find the value of $\sin \theta$.

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


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

Find the greatest common divisor of 4,004 and 10,010.

Answer: $2002$


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

Given that complex numbers $a,$ $b,$ and $c$ are zeros of a polynomial $P(z) = z^3 + qz + r,$ and $|a|^2 + |b|^2 + |c|^2 = 300$. The points corresponding to $a,$ $b,$ and $c$ on the complex plane are the vertices of a right triangle. Find the square of the length of the hypotenuse, $h^2$, given that the triangle's centroid is at the origin.

Answer: $450$


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

Let \[P(x) = (3x^4 - 39x^3 + ax^2 + bx + c)(4x^4 - 96x^3 + dx^2 + ex + f),\] where $a, b, c, d, e, f$ are real numbers. Suppose that the set of all complex roots of $P(x)$ is $\{1, 2, 2, 3, 3, 4, 6\}.$ Find $P(7).$

Answer: $86400$


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

How many distinct arrangements of the letters in the word "balloon" are there?

Answer: $1260$


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

Given a right-angled triangle, one of whose acute angles is $\alpha$. Find the ratio of the radii of the circumscribed and inscribed circles and determine for which value of $\alpha$ this ratio will be the smallest.

Answer: $\sqrt{2} + 1$


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

The five books "Poetry," "Documents," "Rites," "Changes," and "Spring and Autumn" all have different numbers of pages. The differences in the number of pages between the books are as follows:
1. "Poetry" and "Documents" differ by 24 pages.
2. "Documents" and "Rites" differ by 17 pages.
3. "Rites" and "Changes" differ by 27 pages.
4. "Changes" and "Spring and Autumn" differ by 19 pages.
5. "Spring and Autumn" and "Poetry" differ by 15 pages.

Determine the difference in the number of pages between the book with the most pages and the book with the least pages.

Answer: $34$


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

When arranging the performance order of 6 singers, requiring that both singers B and C are either before or after singer A, calculate the total number of different arrangements.

Answer: $480$


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

The robotics club has 30 members: 12 boys and 18 girls. A 6-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl?

Answer: $\frac{574,287}{593,775}$


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

In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $b=2\left(a\cos B-c\right)$. Find:

$(1)$ The value of angle $A$;

$(2)$ If $a\cos C=\sqrt{3}$ and $b=1$, find the value of $c$.

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


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

In a regular octagon, find the ratio of the length of the shortest diagonal to the longest diagonal. Express your answer as a common fraction in simplest radical form.

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


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

Given that square $ABCE$ has side lengths $AF = 3FE$ and $CD = 3DE$, calculate the ratio of the area of $\triangle AFD$ to the area of square $ABCE$.

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


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 that there are 4 qualified and 2 defective products, determine the probability of finding the last defective product exactly on the fourth inspection when selectins products one at a time and not returning them after each selection.

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


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

Choose one of the following conditions from (1) $a\sin \left(B+C\right)+c\sin C-b\sin B=2a\sin C\sin B$, (2) $\frac{cosB}{cosC}+\frac{b}{c-\sqrt{2}a}=0$, (3) $2a^{2}=(a^{2}+b^{2}-c^{2})(1+\tan C)$, and fill in the blank in the question below, and answer the corresponding questions.
Given $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, and satisfying ____.
$(1)$ Find the measure of angle $B$;
$(2)$ If the area of $\triangle ABC$ is $3$ and $a=2\sqrt{2}$, find the value of $\sin A\sin C$.

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


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

For  $n$  measured in degrees, let  $T(n) = \cos^2(30^\circ -n) - \cos(30^\circ -n)\cos(30^\circ +n) +\cos^2(30^\circ +n)$ .  Evaluate   $$  4\sum^{30}_{n=1} n \cdot T(n). $$

Answer: $1395$


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

Simplify \[\frac{1}{\dfrac{3}{\sqrt{5}+2} + \dfrac{4}{\sqrt{7}-2}}.\]

Answer: $\frac{9\sqrt{5} + 4\sqrt{7} + 10}{(9\sqrt{5} + 4\sqrt{7})^2 - 100}$


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

Given a quadratic function $f(x) = ax^2 + bx + c$ (where $a$, $b$, and $c$ are constants). If the solution set of the inequality $f(x) \geq 2ax + b$ is $\mathbb{R}$ (the set of all real numbers), then the maximum value of $\frac{b^2}{a^2 + c^2}$ is __________.

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


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

In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $c=2$ and $b=\sqrt{2}a$. Find the maximum value of the area of $\triangle ABC$.

Answer: $2\sqrt{2}$


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

Two right triangles, $ABC$ and $ACD$, are joined as shown. Squares are drawn on four of the sides. The areas of three of the squares are 25, 49, and 64 square units. What is the number of square units in the area of the fourth square?

Note that the diagram is not provided, but imagine it similarly to the reference where:
- $AB$ and $CD$ have squares on them (inside the triangles $ABC$ and $ACD$ respectively),
- $AC$ is a common hypotenuse for triangles $ABC$ and $ACD$,
- A square is drawn on $AD$.

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>.

Simplify $(9 \times 10^{12}) \div (3 \times 10^4) + (2 \times 10^8) \div (4 \times 10^2)$.

Answer: $300,500,000$


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

A circle with center \( Q \) and radius 2 rolls around the inside of a right triangle \( DEF \) with side lengths 9, 12, and 15, always remaining tangent to at least one side of the triangle. When \( Q \) first returns to its original position, through what distance has \( Q \) traveled?

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 power function $f(x) = kx^a$ whose graph passes through the point $\left( \frac{1}{3}, 81 \right)$, find the value of $k + a$.

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>.

If $z=1+i$, then $|{iz+3\overline{z}}|=\_\_\_\_\_\_$.

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>.

In the sequence ${a_{n}}$, $a_{1}=1$, $a_{n+2}+(-1)^{n}a_{n}=1$. Let $s_{n}$ be the sum of the first $n$ terms of the sequence ${a_{n}}$. Find $s_{100}$ = \_\_\_\_\_\_.

Answer: $1300$


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

Suppose that for a positive integer \( n \), \( 2^n + 1 \) is a prime number. What remainder can this prime have when divided by 240?

Answer: $17$


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

In the Cartesian coordinate system $(xOy)$, the parametric equations of line $l$ are given by $\begin{cases}x=1+\frac{\sqrt{2}}{2}t\\y=\frac{\sqrt{2}}{2}t\end{cases}$ (where $t$ is the parameter), and in the polar coordinate system with the origin $O$ as the pole and the $x$-axis as the polar axis, the polar equation of curve $C$ is $\rho=4\sin\theta$.

(1) Find the Cartesian equation of line $l$ and the polar equation of curve $C$.

(2) Let $M$ be a moving point on curve $C$, and $P$ be the midpoint of $OM$. Find the minimum distance from point $P$ to line $l$.

Answer: $\sqrt{2}-1$


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

Given that $\operatorname{log}_{8}(p) = \operatorname{log}_{12}(q) = \operatorname{log}_{18}(p-q)$, calculate the value of $\frac{q}{p}$.

Answer: $\frac{\sqrt{5} - 1}{2}$


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

Let  $ABC$  be an acute-angled triangle and  $P$  be a point in its interior. Let  $P_A,P_B$  and  $P_c$  be the images of  $P$  under reflection in the sides  $BC,CA$ , and  $AB$ , respectively. If  $P$  is the orthocentre of the triangle  $P_AP_BP_C$  and if the largest angle of the triangle that can be formed by the line segments $ PA, PB$ . and  $PC$  is  $x^o$ , determine the value of  $x$ .

Answer: $120$


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

Express as a fraction in lowest terms: $0.\overline{23} + 0.\overline{056} + 0.\overline{004}$.

Answer: $\frac{28917}{98901}$


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

Given that all edges of a tetrahedron have a length of $\sqrt{2}$, and the four vertices are on the same sphere, calculate the surface area of this sphere.

Answer: $3 \pi$


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

How many sequences of 8 digits $x_1, x_2, \ldots, x_8$ can we form, given the condition that no two adjacent $x_i$ have the same parity and the sequence starts with an odd digit? Leading zeroes are allowed.

Answer: $390625$


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

Given the function $f(x)=\cos^4x-2\sin x\cos x-\sin^4x.$
$(1)$ Find the smallest positive period of $f(x)$.
$(2)$ When $x\in\left[0, \frac{\pi}{2}\right]$, find the minimum value of $f(x)$ and the set of $x$ values for which this minimum is achieved.

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 the increasing sequence of positive integers \(a_{1}, a_{2}, a_{3}, \cdots\) satisfies the recurrence relation \(a_{n+2} = a_{n} + a_{n+1}\) for \(n \geq 1\), and \(a_{7} = 120\), find the value of \(a_{8}\).

Answer: $194$


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

Let  $ABCD$  be a rectangle with sides  $AB,BC,CD$  and  $DA$ . Let  $K,L$  be the midpoints of the sides  $BC,DA$  respectivily. The perpendicular from  $B$  to  $AK$  hits  $CL$  at  $M$ . Find  $$ \frac{[ABKM]}{[ABCL]} $$

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>.

Let the function be $$f(x)=1-2\sin^{2}x-\cos(2x+ \frac {\pi}{3})$$
(1) Find the smallest positive period of the function $f(x)$.
(2) For triangle ABC, the sides $a$, $b$, $c$ are opposite to the angles $A$, $B$, $C$, respectively. Given $b=5$, and $$f\left( \frac {B}{2}\right)=1$$, find the maximum area of triangle ABC.

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


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 $α∈(\frac{\pi}{2},π)$, $\sin α =\frac{\sqrt{5}}{5}$.

(Ⅰ) Find the value of $\tan\left( \frac{\pi}{4}+2α \right)$;

(Ⅱ) Find the value of $\cos\left( \frac{5\pi}{6}-2α \right)$.

Answer: $-\frac{3\sqrt{3}+4}{10}$


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>.

Evaluate the determinant of the given matrix:
\[
\begin{vmatrix} \sin \theta \sin \phi & \sin \theta \cos \phi & \cos \theta \\ \cos \phi & -\sin \phi & 0 \\ -\cos \theta \sin \phi & -\cos \theta \cos \phi & \sin \theta \end{vmatrix}
\]

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>.

For each integer $n$ greater than 1, let $G(n)$ be the number of solutions of the equation $\sin x = \sin (n+1)x$ on the interval $[0, 2\pi]$. Calculate $\sum_{n=2}^{100} G(n)$.

Answer: $10296$


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

Rectangle \(WXYZ\) is divided into four smaller rectangles. The perimeters of three of these smaller rectangles are 11, 16, and 19. The perimeter of the fourth rectangle lies between 11 and 19. What is the length of the perimeter of \(WXYZ\)?
Options:
A) 28
B) 30
C) 32
D) 38
E) 40

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>.

What percent of the palindromes between 1000 and 2000 contain at least one 3 or 5, except in the first digit?

Answer: $36\%$


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

The third and twentieth terms of an arithmetic sequence are 10 and 65, respectively. What is the thirty-second term?

Answer: $103.8235294118$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 $a_{n+1}=(2|\sin \frac{n\pi}{2}|-1)a_{n}+n$, then the sum of the first $100$ terms of the sequence $\{a_n\}$ is __________.

Answer: $2550$


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

Calculate $1,000,000,000,000 - 888,777,888,777$.

Answer: $111,222,111,223$


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

Find \( n > m \geq 1 \) such that the last two digits of the decimal representation of \( 42^{n} \) are the same as those of \( 42^{m} \), and such that the sum \( m+n \) is minimal.

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 that $a \in \mathbb{R}$, if the real part and the imaginary part of the complex number $\frac{a + i}{1 + i}$ (where $i$ is the imaginary unit) are equal, then $\_\_\_\_\_\_$, $| \overline{z}| = \_\_\_\_\_\_$.

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


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

How many whole numbers between $200$ and $500$ contain the digit $3$?

Answer: $138$


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

Determine the number of all positive integers which cannot be written in the form  $80k + 3m$ , where  $k,m \in N = \{0,1,2,...,\}$

Answer: $79$


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

Given sets \( A = \{1, 2, 3\} \) and \( B = \{4, 5, 6, 7\} \), with a mapping \( f: A \rightarrow B \), if for \( x \in A \), the expression \( x + f(x) + x \cdot f(x) \) is odd, then the number of such mappings \( f \) is:

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>.

Find the biggest real number  $ k$  such that for each right-angled triangle with sides  $ a$ ,  $ b$ ,  $ c$ , we have
\[ a^{3}\plus{}b^{3}\plus{}c^{3}\geq k\left(a\plus{}b\plus{}c\right)^{3}.\]

Answer: $\frac{3\sqrt{2} - 4}{2}$


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>.

It is known that the equation $ |x - 1| + |x - 2| +... + |x - 2001| = a$  has exactly one solution. Find  $a$ .

Answer: $1001000$


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

A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 8 units has its center at $(18, 0)$. A line tangent to both circles intersects the $x$-axis at point $(x, 0)$ to the right of the origin. Find the value of $x$. Express your answer as a common fraction.

Answer: $\frac{54}{11}$


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>.

Five years ago, Tim was three times as old as his sister Sarah, and three years before that, Tim was five times as old as Sarah. Determine the number of years it will take for the ratio of their ages to be 3 : 2.

Answer: $13$


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

The minimum positive period of $y=\tan(4x+ \frac{\pi}{3})$ is $\pi$.

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


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>.

Which one satisfies $n^{29} \equiv 7 \pmod {65}$?

Answer: $37$


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

In triangle ABC, $a-b=4$, $a+c=2b$, and the largest angle is $120^\circ$. Find the perimeter of this triangle.

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 triangle $ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to the interior angles $A$, $B$, and $C$, respectively, and $a\cos C+\left(2b+c\right)\cos A=0$.
$(1)$ Find the value of angle $A$.
$(2)$ If $D$ is the midpoint of segment $BC$ and $AD=\frac{7}{2}$, $AC=3$, find the area of triangle $ABC$.

Answer: $6\sqrt{3}$


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>.

Regular hexagon $ABCDEF$ has an area of $n$. Let $m$ be the area of triangle $ACE$. What is $\tfrac{m}{n}?$
A) $\frac{1}{2}$
B) $\frac{2}{3}$
C) $\frac{3}{4}$
D) $\frac{1}{3}$
E) $\frac{3}{2}$

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 the function $f(x)=\cos x\cdot \sin \left(x+\frac{\pi }{3}\right)-\sqrt{3}\cos ^{2}x+\frac{\sqrt{3}}{4}$, where $x\in R$.

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

(2) Find the maximum and minimum values of $f(x)$ on the closed interval $\left[-\frac{\pi }{4},\frac{\pi }{4}\right]$.

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>.

From a batch of parts, 50 are drawn, and then 40 out of these 50 are inspected. It is found that there are 38 qualified products. Calculate the pass rate of this batch of products as a percentage.

Answer: $95\%$


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

For each integer \( n \geq 2 \), let \( A(n) \) be the area of the region in the coordinate plane defined by the inequalities \( 1 \leq x \leq n \) and \( 0 \leq y \leq x \left\lfloor \log_2{x} \right\rfloor \), where \( \left\lfloor \log_2{x} \right\rfloor \) is the greatest integer not exceeding \( \log_2{x} \). Find the number of values of \( n \) with \( 2 \leq n \leq 100 \) for which \( A(n) \) is an integer.

Answer: $99$


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

Let \( x[n] \) denote \( x \) raised to the power of \( x \), repeated \( n \) times. What is the minimum value of \( n \) such that \( 9[9] < 3[n] \)?

(For example, \( 3[2] = 3^3 = 27 \); \( 2[3] = 2^{2^2} = 16 \).)

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>.

Determine the largest constant  $K\geq 0$  such that   $$ \frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2 $$   holds for all positive real numbers  $a,b,c$  such that  $ab+bc+ca=abc$ .

*Proposed by Orif Ibrogimov (Czech Technical University of Prague).*

Answer: $18$


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

Suppose $x$ and $y$ are positive real numbers such that $x^2 - 2xy + 3y^2 = 9$. Find the maximum possible value of $x^2 + 2xy + 3y^2$.

Answer: $18 + 9\sqrt{3}$


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>.

Given that $\alpha$ is an acute angle and satisfies $\cos(\alpha+\frac{\pi}{4})=\frac{\sqrt{3}}{3}$.
$(1)$ Find the value of $\sin(\alpha+\frac{7\pi}{12})$.
$(2)$ Find the value of $\cos(2\alpha+\frac{\pi}{6})$.

Answer: $\frac{2\sqrt{6}-1}{6}$


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

Evaluate $|\omega^2 + 7\omega + 40|$ if $\omega = 4 + 3i$.

Answer: $15\sqrt{34}$


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

Given that the base edge length of a regular square pyramid is $2$, and the side edge length is $\sqrt{6}$, determine the volume of the pyramid.

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


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

Given that z and w are complex numbers with a modulus of 1, and 1 ≤ |z + w| ≤ √2, find the minimum value of |z - w|.

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>.

Let be the set  $ \mathcal{C} =\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f''\bigg|_{[0,1]} \right.\quad\exists x_1,x_2\in [0,1]\quad x_1\neq x_2\wedge \left( f\left(
x_1 \right) = f\left( x_2 \right) =0\vee  f\left(
x_1 \right) = f'\left( x_1 \right) = 0\right)  \wedge f''<1 \right\} , $ and  $ f^*\in\mathcal{C} $  such that  $ \int_0^1\left| f^*(x) \right| dx =\sup_{f\in\mathcal{C}} \int_0^1\left| f(x) \right| dx .  $ Find  $ \int_0^1\left| f^*(x) \right| dx $  and describe  $ f^*. $

Answer: $1/12$


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 there are a total of 25 students in an art class, including 10 boys and 15 girls. In the final art exhibition, the average number of artworks per boy is 25 with a variance of 1, and the average number of artworks per girl is 30 with a variance of 2. Find the variance of the number of artworks per student among these 25 students.

Answer: $\frac{38}{5}$


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>.

Among the non-empty subsets of the set \( A = \{1, 2, \cdots, 10\} \), how many subsets have the sum of their elements being a multiple of 10?

Answer: $103$


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

The arithmetic mean of eleven numbers is 32. If three numbers $x$, $y$, and $z$ are added to this list, the mean of the fourteen-member list becomes 45. What is the mean of $x$, $y$, and $z$?

Answer: $\frac{278}{3}$


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

A zookeeper distributes a pile of peaches among several monkeys. If each monkey gets 6 peaches, there are 57 peaches left. If each monkey gets 9 peaches, 5 monkeys get none, and one monkey gets only 3 peaches. How many peaches are there in total?

Answer: $273$


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

Find $537_{8} - 261_{8}$. Verify by adding the result to $261_{8}$ and checking if it matches $537_{8}$.

Answer: $256_8$


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>.

Find the least positive integer  $ a$  such that  $ 2001$  divides  $ 55^n\plus{}a \cdot 32^n$  for some odd  $ n$ .

Answer: $436$


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

Aaron takes a square sheet of paper, with one corner labeled  $A$ . Point  $P$  is chosen at random inside of the square and Aaron folds the paper so that points  $A$  and  $P$  coincide. He cuts the sheet along the crease and discards the piece containing  $A$ . Let  $p$  be the probability that the remaining piece is a pentagon. Find the integer nearest to  $100p$ .

*Proposed by Aaron Lin*

Answer: $25$


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

A lattice point is a point whose coordinates are integers. How many lattice points are on the boundary or inside the region bounded by \( y = |x| \) and \( y = -x^2 + 8 \)?

Answer: $33$


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

In the Cartesian coordinate system, with the origin as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. The polar equation of curve C is $\rho - 2\cos\theta - 6\sin\theta + \frac{1}{\rho} = 0$, and the parametric equation of line l is $\begin{cases} x=3+ \frac{1}{2}t \\ y=3+ \frac{\sqrt{3}}{2}t \end{cases}$ (t is the parameter).
(1) Find the standard equation of curve C;
(2) If line l intersects curve C at points A and B, and the coordinates of point P are (3, 3), find the value of $|PA|+|PB|$.

Answer: $2\sqrt{6}$


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>.

Given that real numbers $a$ and $b$ satisfy $\frac{1}{a} + \frac{2}{b} = \sqrt{ab}$, calculate the minimum value of $ab$.

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>.

The Yellers are coached by Coach Loud. The Yellers have 15 players, but three of them, Max, Rex, and Tex, refuse to play together in any combination. How many starting lineups (of 5 players) can Coach Loud make, if the starting lineup can't contain any two of Max, Rex, and Tex together?

Answer: $2277$


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

A natural number  $n$  is said to be  $good$  if  $n$  is the sum or  $r$  consecutive positive integers, for some  $r \geq 2 $ . Find the number of good numbers in the set  $\{1,2 \dots , 100\}$ .

Answer: $93$


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

Given the function $f(x) = x^3 - 3x$,

(Ⅰ) Find the intervals of monotonicity for $f(x)$;

(Ⅱ) Find the maximum and minimum values of $f(x)$ in the interval $[-3,2]$.

Answer: $-18$


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

Triangle $ABC$ has vertices $A(0, 10)$, $B(3, 0)$, $C(9, 0)$. A horizontal line with equation $y=s$ intersects line segment $\overline{AB}$ at $P$ and line segment $\overline{AC}$ at $Q$, forming $\triangle APQ$ with area 18. Compute $s$.

Answer: $10 - 2\sqrt{15}$


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

Given the function $f(x)= \begin{cases} 2x-10, & x\leqslant 7 \\ \frac {1}{f(x-2)}, & x > 7 \end{cases}$, and the sequence ${a_{n}}={f(n)}$ where $n\in\mathbb{N}^{*}$, find the sum of the first 50 terms of the sequence ${a_{n}}$.

Answer: $\frac {225}{4}$


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

In $\triangle ABC$, the lengths of the sides opposite to angles A, B, and C are a, b, and c respectively. Given that a = 3, cosC = $- \frac{1}{15}$, and 5sin(B + C) = 3sin(A + C).
(1) Find the length of side c.
(2) Find the value of sin(B - $\frac{\pi}{3}$).

Answer: $\frac{2\sqrt{14} - 5\sqrt{3}}{18}$


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

The diagram shows a polygon made by removing six  $2\times 2$  squares from the sides of an  $8\times 12$  rectangle. Find the perimeter of this polygon.
![Image](https://cdn.artofproblemsolving.com/attachments/6/3/c23510c821c159d31aff0e6688edebc81e2737.png)

Answer: $52$


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

A frustum of a cone has a lower base radius of 8 inches, an upper base radius of 4 inches, and a height of 5 inches. Calculate its lateral surface area and total surface area.

Answer: $(80 + 12\sqrt{41})\pi$


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

In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $b\sin C + c\sin B = 4a\sin B\sin C$ and $b^2 + c^2 - a^2 = 8$. Find the area of $\triangle ABC$.

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


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

An equilateral triangle ABC has a side length of 4. A right isosceles triangle DBE, where $DB=EB=1$ and angle $D\hat{B}E = 90^\circ$, is cut from triangle ABC. Calculate the perimeter of the remaining quadrilateral.

Answer: $10 + \sqrt{2}$


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>.

In $\triangle ABC$, given $A(1,4)$, $B(4,1)$, $C(0,-4)$, find the minimum value of $\overrightarrow{PA} \cdot \overrightarrow{PB} + \overrightarrow{PB} \cdot \overrightarrow{PC} + \overrightarrow{PC} \cdot \overrightarrow{PA}$.

Answer: $- \dfrac {62}{3}$


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>.

Around a circular table, there are 18 girls seated, 11 dressed in blue and 7 dressed in red. Each girl is asked if the girl to her right is dressed in blue, and each one responds with either yes or no. It is known that a girl tells the truth only when both of her neighbors, the one to her right and the one to her left, are dressed in the same color. How many girls will respond yes? If there is more than one possibility, state all.

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 sequence \( b_1, b_2, b_3, \dots \) is defined recursively by \( b_1 = 2, b_2 = 2, \) and for \( k \ge 3, \)
\[
b_k = \frac{1}{2} b_{k - 1} + \frac{1}{3} b_{k - 2}.
\]
Evaluate \( b_1 + b_2 + b_3 + \dotsb. \)

Answer: $18$


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

Let $x$, $y\in \mathbb{R}$, vectors $\overrightarrow{a}=(2,x)$, $\overrightarrow{b}=(y,1)$, $\overrightarrow{c}=(3,-3)$, and $\overrightarrow{a}⊥\overrightarrow{b}$, $\overrightarrow{b}∥\overrightarrow{c}$.
$(1)$ Find $|\overrightarrow{a}+\overrightarrow{b}|$;
$(2)$ Find the cosine value of the angle between vectors $\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}+2\overrightarrow{b}+\overrightarrow{c}$.

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


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 each of the four numbers $3, 4, 6,$ and $7$ replaces a $\square$, what is the largest possible sum of the fractions shown?

Answer: $\frac{23}{6}$


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

Calculate the angle $\theta$ for the sum
\[e^{3\pi i/60} + e^{11\pi i/60} + e^{19\pi i/60} + e^{27\pi i/60} + e^{35\pi i/60} + e^{43\pi i/60} + e^{51\pi i/60} + e^{59\pi i/60}\]
when expressed in the form of $r e^{i\theta}$, where $0 \leq \theta < 2\pi$.

Answer: $\dfrac{31\pi}{60}$


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

Given that $a$, $b$, $c$ form an arithmetic sequence in triangle $ABC$, $\angle B=30^{\circ}$, and the area of $\triangle ABC$ is $\frac{1}{2}$, determine the value of $b$.

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


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

In triangle $ABC$, the sides opposite to angles A, B, and C are denoted by $a$, $b$, and $c$ respectively, with $A+C=\frac{2\pi}{3}$ and $b=1$.
(1) If we let angle A be $x$ and define $f(x)=a+c$, find the range of $f(x)$ when triangle $ABC$ is an acute triangle;
(2) Determine the maximum area of triangle $ABC$.

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


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>.

In an equilateral triangle $ABC$ with side length of 10, a similar process of division by midpoints and shading of one of these triangles occurs. If this dividing and shading process is repeated indefinitely, and the first triangle to be shaded is the triangle involving vertex $A$, the total shaded area will converge towards?
A) $15\sqrt{3}$
B) $18\sqrt{3}$
C) $\frac{25\sqrt{3}}{3}$
D) $25$
E) $30\sqrt{3}$

Answer: $\frac{25\sqrt{3}}{3}$


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 variables $a$ and $b$ satisfying $b=-\frac{1}{2}a^2+3\ln(a)$ (with $a>0$), if point $Q(m,n)$ is on the line $y=2x+\frac{1}{2}$, calculate the minimum value of $(a-m)^2+(b-n)^2$.

Answer: $\frac{9}{5}$


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>.

Let $(x,y)$ satisfy the constraints $\begin{cases} 8x - y - 4 \leqslant 0 \\ x + y + 1 \geqslant 0 \\ y - 4x \leqslant 0 \end{cases}$, and the maximum value of the objective function $z = ax + by (a > 0, b > 0)$ is $2$. Find the minimum value of $\frac{1}{a} + \frac{1}{b}$.

Answer: $\frac{9}{2}$


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

There are 8 young people, among whom 5 are capable of doing English translation work, and 4 are capable of doing computer software design work (including one person who is capable of doing both tasks). Now, 5 young people are to be selected to undertake a task, with 3 people doing English translation work and 2 people doing software design work. The number of different ways to select them is ____.

Answer: $42$


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

The integer points $(x, y)$ in the first quadrant satisfy $x + y > 8$ and $x \leq y \leq 8$. Determine the number of such integer points $(x, y)$.

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>.

The sum of an infinite geometric series is 64 times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio?

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>.

Given the function g defined on the set of positive rational numbers by g(x \cdot y) = g(x) + g(y) for all positive rational numbers x and y, and g(n) = n^2 for every prime number n, calculate g(x) for x = \frac{25}{21}.

Answer: $-8$


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 a large library storage room, there are $1584$ boxes, each containing $45$ books. The library dean asks for these books to be repacked so that each new box contains $47$ books. How many books will be left over after repacking the books into as many full boxes of $47$ books each as possible?

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>.

In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle?

Answer: $50$