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

Five fair six-sided dice are rolled. What is the probability that at least three of the five dice show the same value?

Answer: $\frac{113}{648}$


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

Given that there are 4 tiers of horses, with Tian Ji's top-tier horse being better than King Qi's middle-tier horse, worse than King Qi's top-tier horse, Tian Ji's middle-tier horse being better than King Qi's bottom-tier horse, worse than King Qi's middle-tier horse, and Tian Ji's bottom-tier horse being worse than King Qi's bottom-tier horse, determine the probability that Tian Ji's selected horse wins the race.

Answer: $\frac{1}{3}$


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

Let \( A \) be the set of real numbers \( x \) satisfying the inequality \( x^{2} + x - 110 < 0 \) and \( B \) be the set of real numbers \( x \) satisfying the inequality \( x^{2} + 10x - 96 < 0 \). Suppose that the set of integer solutions of the inequality \( x^{2} + ax + b < 0 \) is exactly the set of integers contained in \( A \cap B \). Find the maximum value of \( \lfloor |a - b| \rfloor \).

Answer: $71$


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

Given the number 826,000,000, express it in scientific notation.

Answer: $8.26\times 10^{8}$


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>.

Let $G$ be the centroid of triangle $PQR.$ If $GP^2 + GQ^2 + GR^2 = 22,$ then find $PQ^2 + PR^2 + QR^2.$

Answer: $66$


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

If \( a \) is the smallest cubic number divisible by 810, find the value of \( a \).

Answer: $729000$


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

Suppose  $ n$  is a product of four distinct primes  $ a,b,c,d$  such that:
 $ (i)$   $ a\plus{}c\equal{}d;$
 $ (ii)$   $ a(a\plus{}b\plus{}c\plus{}d)\equal{}c(d\minus{}b);$
 $ (iii)$   $ 1\plus{}bc\plus{}d\equal{}bd$ .
Determine  $ n$ .

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>.

Xiaoming saw a tractor pulling a rope slowly on the road. Xiaoming decided to measure the length of the rope. If Xiaoming walks in the same direction as the tractor, it takes 140 steps to walk from one end of the rope to the other end; if Xiaoming walks in the opposite direction of the tractor, it takes 20 steps. The speed of the tractor and Xiaoming remain constant, and Xiaoming can walk 1 meter per step. What is the length of the rope in meters?

Answer: $35$


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

Let  $a$  be the sum of the numbers: $99 \times 0.9$  $999 \times 0.9$  $9999 \times 0.9$  $\vdots$  $999\cdots 9 \times 0.9$ where the final number in the list is  $0.9$  times a number written as a string of  $101$  digits all equal to  $9$ .
Find the sum of the digits in the number  $a$ .

Answer: $891$


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

The set $\{[x]+[2x]+[3x] \mid x \in \mathbf{R}\} \bigcap \{1, 2, \cdots, 100\}$ contains how many elements, where $[x]$ represents the greatest integer less than or equal to $x$.

Answer: $67$


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

A factory has a fixed daily cost of 20,000 yuan, and the maximum daily production capacity is 360 units. The cost increases by 100 yuan for each unit produced. The revenue function for producing $x$ units of product per day is $R(x) = -\frac{1}{2}x^2 + 400x$. Let $L(x)$ and $P(x)$ represent the daily profit and average profit (average profit = $\frac{\text{total profit}}{\text{total quantity}}$) for producing $x$ units of product, respectively.
(1) At what daily production quantity $x$ does the profit $L(x)$ reach its maximum value, and what is this maximum value?
(2) At what daily production quantity $x$ does the average profit $P(x)$ reach its maximum value, and what is this maximum value?
(3) Due to an economic crisis, the factory had to lay off workers, causing the maximum daily production capacity to drop to 160 units. At what daily production quantity $x$ does the average profit $P(x)$ reach its maximum value, and what is this maximum value?

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>.

What is the largest \( x \) such that \( x^2 \) divides \( 24 \cdot 35 \cdot 46 \cdot 57 \)?

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>.

Several consecutive natural numbers are written on the board. Exactly 52% of them are even. How many even numbers are written on the board?

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>.

A table consisting of 1861 rows and 1861 columns is filled with natural numbers from 1 to 1861 such that each row contains all numbers from 1 to 1861. Find the sum of the numbers on the diagonal that connects the top left and bottom right corners of the table if the filling of the table is symmetric with respect to this diagonal.

Answer: $1732591$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a=2b$. Also, $\sin A$, $\sin C$, $\sin B$ form an arithmetic sequence.

$(I)$ Find the value of $\cos (B+C)$;

$(II)$ If the area of $\triangle ABC$ is $\frac{8\sqrt{15}}{3}$, find the value of $c$.

Answer: $4 \sqrt {2}$


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

Let  $ n$  be a natural number. A cube of edge  $ n$  may be divided in 1996 cubes whose edges length are also natural numbers. Find the minimum possible value for  $ n$ .

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>.

If you want to form a rectangular prism with a surface area of 52, what is the minimum number of small cubes with edge length 1 needed?

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 shooter fires at a target until the first hit, with a hit rate of 0.6 for each shot. If there are 4 bullets in total, what is the expected number of remaining bullets $\xi$?

Answer: $2.376$


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

The altitudes of an acute-angled triangle \( ABC \) drawn from vertices \( B \) and \( C \) are 7 and 9, respectively, and the median \( AM \) is 8. Points \( P \) and \( Q \) are symmetric to point \( M \) with respect to sides \( AC \) and \( AB \), respectively. Find the perimeter of the quadrilateral \( APMQ \).

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>.

In triangle $\triangle ABC$, it is known that $AB=2$, $AC=3$, $\angle A=60^{\circ}$, and $\overrightarrow{BM}=\frac{1}{3}\overrightarrow{BC}$, $\overrightarrow{AN}=\overrightarrow{NB}$. Find $\overrightarrow{AC}•\overrightarrow{NM}$.

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


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

In the country of Anchuria, a unified state exam takes place. The probability of guessing the correct answer to each question on the exam is 0.25. In 2011, to receive a certificate, one needed to answer correctly 3 questions out of 20. In 2012, the School Management of Anchuria decided that 3 questions were too few. Now, one needs to correctly answer 6 questions out of 40. The question is, if one knows nothing and simply guesses the answers, in which year is the probability of receiving an Anchurian certificate higher - in 2011 or in 2012?

Answer: $2012$


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

Let the function $f(x)=\ln x-\frac{1}{2} ax^{2}-bx$.

$(1)$ When $a=b=\frac{1}{2}$, find the maximum value of the function $f(x)$;

$(2)$ Let $F(x)=f(x)+\frac{1}{2} x^{2}+bx+\frac{a}{x} (0 < x\leqslant 3)$. If the slope $k$ of the tangent line at any point $P(x_{0},y_{0})$ on its graph is always less than or equal to $\frac{1}{2}$, find the range of the real number $a$;

$(3)$ When $a=0$, $b=-1$, the equation $x^{2}=2mf(x)$ (where $m > 0$) has a unique real solution, find the value of $m$.

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>.

Eight hockey teams are competing against each other in a single round to advance to the semifinals. What is the minimum number of points that guarantees a team advances to the semifinals?

Answer: $11$


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

Given two circles $x^{2}+y^{2}=4$ and $x^{2}+y^{2}-2y-6=0$, find the length of their common chord.

Answer: $2\sqrt{3}$


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

Find the distance from the point \( M_{0} \) to the plane passing through the three points \( M_{1}, M_{2}, M_{3} \).

\( M_{1}(1, 3, 0) \)

\( M_{2}(4, -1, 2) \)

\( M_{3}(3, 0, 1) \)

\( M_{0}(4, 3, 0) \)

Answer: $\sqrt{6}$


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>.

Sixty students went on a trip to the zoo. Upon returning to school, it turned out that 55 of them forgot gloves at the zoo, 52 forgot scarves, and 50 managed to forget hats. Find the smallest number of the most scatterbrained students - those who lost all three items.

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>.

The older brother and the younger brother each bought several apples. The older brother said to the younger brother, "If I give you one apple, we will have the same number of apples." The younger brother thought for a moment and said to the older brother, "If I give you one apple, the number of apples you have will be twice as much as mine." How many apples did they buy in total?

Answer: $12$


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

If $2 - \sin^{2}(x + 2y - 1) = \frac{x^{2} + y^{2} - 2(x + 1)(y - 1)}{x - y + 1}$, then the minimum value of the product $xy$ is $\qquad$ .

Answer: $1/9$


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

The four zeros of the polynomial \(x^4 + px^2 + qx - 144\) are distinct real numbers in arithmetic progression. Compute the value of \(p.\)

Answer: $-40$


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

Given the hyperbola \( C_{1}: 2x^{2} - y^{2} = 1 \) and the ellipse \( C_{2}: 4x^{2} + y^{2} = 1 \), let \( M \) and \( N \) be moving points on the hyperbola \( C_{1} \) and the ellipse \( C_{2} \) respectively, with \( O \) as the origin. If \( O M \) is perpendicular to \( O N \), find the distance from point \( O \) to the line \( M N \).

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


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>.

Find the maximum possible area of a section passing through the diagonal $XY$ of an arbitrary parallelepiped with edges $a \leq b \leq c$. The section is a parallelogram $Z X T Y$ with vertices lying on opposite edges of the parallelepiped. The area of the parallelogram equals the product of the length of the diagonal $XY$ and the distance from point $Z$ to $XY$.

The largest area of a possible section, considering diagonal sections, is determined, with the given sides $a$, $b$, and $c$ in order of $a \le b \le c$. The areas of potential sections are:

$$
S_{1}=a \sqrt{b^{2}+c^{2}}, S_{2}=b \sqrt{a^{2}+c^{2}}, \text { and } S_{3}=c \sqrt{b^{2}+a^{2}} .
$$

Given the condition $a \leq b \leq c$, it is apparent that $S_{1} \leq S_{3}$ and $S_{2} \leq S_{3}$. Therefore, the maximum area section passes through the longest edge.

Thus, the maximum area section is $10 \sqrt{4^{2}+3^{2}}=50$.

Answer: $50$


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

For what real value of $u$ is $\frac{-15+\sqrt{205}}{8}$ a root of $4x^2 + 15x + u$?

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


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

Given that $P$ is a point inside rectangle $ABCD$, the distances from $P$ to the vertices of the rectangle are $PA = 5$ inches, $PD = 12$ inches, and $PC = 13$ inches. Find $PB$, which is $x$ inches.

Answer: $5\sqrt{2}$


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 $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. Given that $a=4$, $b=6$, and $C=60^\circ$:
1. Calculate $\overrightarrow{BC} \cdot \overrightarrow{CA}$;
2. Find the projection of $\overrightarrow{CA}$ onto $\overrightarrow{BC}$.

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>.

Point \(C\) divides diameter \(AB\) in the ratio \(AC:BC = 2:1\). A point \(P\) is selected on the circle. Determine the possible values that the ratio \(\tan \angle PAC: \tan \angle APC\) can take. Specify the smallest such value.

Answer: $1/2$


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

A one-meter gas pipe has rusted through in two places. Determine the probability that all three resulting parts can be used as connectors to gas stoves, given that according to regulations, the stove must not be closer than 25 cm to the main gas pipe.

Answer: $1/16$


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

The center of a balloon is observed by two ground observers at angles of elevation of $45^{\circ}$ and $22.5^{\circ}$, respectively. The first observer is to the south, and the second one is to the northwest of the point directly under the balloon. The distance between the two observers is 1600 meters. How high is the balloon floating above the horizontal ground?

Answer: $500$


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

In the unit cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, points $E, F, G$ are the midpoints of edges $A A_{1}, C_{1} D_{1}$, and $D_{1} A_{1}$, respectively. Find the distance from point $B_{1}$ to the plane $E F G$.

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


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

Determine the number of three-element subsets of the set \(\{1, 2, 3, 4, \ldots, 120\}\) for which the sum of the three elements is a multiple of 3.

Answer: $93640$


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

Given the function $f(x)=x^{3}+ax^{2}+bx+a^{2}-1$ has an extremum of $9$ at $x=1$, find the value of $f(2)$.

Answer: $17$


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

Given the sequence \( a_{1}, a_{2}, \cdots, a_{n}, \cdots \) that satisfies \( a_{1}=a_{2}=1, a_{3}=2 \), and for any natural number \( n \), \( a_{n} a_{n+1} a_{n+2} \neq 1 \). Furthermore, it is given that \( a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{1} + a_{n+1} + a_{n+2} + a_{n+3} \). Find the value of \( a_{1} + a_{2} + \cdots + a_{100} \).

Answer: $200$


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

It is known that there are four different venues $A$, $B$, $C$, $D$ at the Flower Expo. Person A and person B each choose 2 venues to visit. The probability that exactly one venue is the same in their choices is ____.

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>.

From the numbers \(1,2, \cdots, 14\), select \(a_{1}, a_{2}, a_{3}\) in ascending order such that \(a_{2} - a_{1} \geq 3\) and \(a_{3} - a_{2} \geq 3\). How many different ways are there to select the numbers satisfying these conditions?

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>.

The quadratic polynomial \( f(x) = a x^{2} + b x + c \) has exactly one root, and the quadratic polynomial \( f(3x + 2) - 2f(2x - 1) \) also has exactly one root. Find the root of the polynomial \( f(x) \).

Answer: $-7$


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

\[
\frac{\sin ^{2}\left(135^{\circ}-\alpha\right)-\sin ^{2}\left(210^{\circ}-\alpha\right)-\sin 195^{\circ} \cos \left(165^{\circ}-2 \alpha\right)}{\cos ^{2}\left(225^{\circ}+\alpha\right)-\cos ^{2}\left(210^{\circ}-\alpha\right)+\sin 15^{\circ} \sin \left(75^{\circ}-2 \alpha\right)}=-1
\]

Answer: $-1$


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

Given that $n\in N^{*}$, select $k(k\in N, k\geqslant 2)$ numbers $j\_1$, $j\_2$, $...$, $j\_k$ from the set ${1,2,3,...,n}$ such that they simultaneously satisfy the following two conditions: $①1\leqslant j\_1 < j\_2 < ...j\_k\leqslant n$; $②j_{i+1}-j_{i}\geqslant m(i=1,2,…,k-1)$. Then the array $(j\_1, j\_2, ..., j\_k)$ is called a combination of selecting $k$ elements with a minimum distance of $m$ from $n$ elements, denoted as $C_{ n }^{ (k,m) }$. For example, from the set ${1,2,3}$, we have $C_{ 3 }^{ (2,1) }=3$. For the given set ${1,2,3,4,5,6,7}$, find $C_{ 7 }^{ (3,2) }=$ \_\_\_\_\_\_.

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>.

In a school cafeteria line, there are 16 students alternating between boys and girls (starting with a boy, followed by a girl, then a boy, and so on). Any boy, followed immediately by a girl, can swap places with her. After some time, all the girls end up at the beginning of the line and all the boys are at the end. How many swaps were made?

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>.

Given that $f(x)$ is an even function defined on $\mathbb{R}$, and $g(x)$ is an odd function defined on $\mathbb{R}$ that passes through the point $(-1, 3)$ and $g(x) = f(x-1)$, find the value of $f(2007) + f(2008)$.

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>.

Points $A_{1}$ and $C_{1}$ are located on the sides $BC$ and $AB$ of triangle $ABC$. Segments $AA_{1}$ and $CC_{1}$ intersect at point $M$.

In what ratio does line $BM$ divide side $AC$, if $AC_{1}: C_{1}B = 2: 3$ and $BA_{1}: A_{1}C = 1: 2$?

Answer: $1:3$


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

A 20-quart container is fully filled with water. Five quarts are removed and replaced with pure antifreeze liquid. Then, five quarts of the mixture are removed and replaced with pure antifreeze. This process is repeated three more times (for a total of five times). Determine the fractional part of the final mixture that is water.

Answer: $\frac{243}{1024}$


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

A company needs to transport two types of products, $A$ and $B$, with the following volumes and masses per unit as shown in the table:

|          | Volume $(m^{3}/$unit) | Mass (tons$/$unit) |
|----------|-----------------------|--------------------|
| $A$ type | $0.8$                 | $0.5$              |
| $B$ type | $2$                   | $1$                |

1. Given a batch of products containing both $A$ and $B$ types with a total volume of $20m^{3}$ and a total mass of $10.5$ tons, find the number of units for each type of product.

2. A logistics company has trucks with a rated load of $3.5$ tons and a capacity of $6m^{3}$. The company offers two payment options:
   - Charging per truck: $600$ yuan per truck to transport goods to the destination.
   - Charging per ton: $200$ yuan per ton to transport goods to the destination.

   Determine how the company should choose to transport the products from part (1) in one or multiple shipments to minimize the shipping cost, and calculate the cost under the chosen method.

Answer: $2100$


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

If the product of 6 consecutive odd numbers is 135135, what is the sum of these 6 numbers? $\qquad$

Answer: $48$


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

Find the maximum value of the function
$$
f(x) = \sqrt{3} \sin 2x + 2 \sin x + 4 \sqrt{3} \cos x.
$$

Answer: $\frac{17}{2}$


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

Given a $24$-inch by $30$-inch pan of brownies, cut into pieces that measure $3$ inches by $4$ inches. Calculate the number of pieces of brownie the pan contains.

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>.

Find the smallest six-digit number that is divisible by 3, 7, and 13 without a remainder.

Answer: $100191$


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

There are 1000 rooms in a row along a long corridor. Initially, the first room contains 1000 people, and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are simultaneous (so nobody moves more than once within a minute). After one hour, how many different rooms will have people in them?

Answer: $61$


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

A digit is inserted between the digits of a two-digit number to form a three-digit number. Some two-digit numbers, when a certain digit is inserted in between, become three-digit numbers that are $k$ times the original two-digit number (where $k$ is a positive integer). What is the maximum value of $k$?

Answer: $19$


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

In an isosceles triangle \(ABC\), the angle at the vertex \(B\) is \(20^\circ\). Points \(D\) and \(K\) are taken on the sides \(AB\) and \(BC\) respectively such that \(\angle KAC = 50^\circ\) and \(\angle DCA = 60^\circ\). Calculate \(\angle CDK\).

Answer: $30$


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

In the expansion of $((x^2+1)^2(x-1)^6)$, find the coefficient of the $x^3$ term.

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>.

Kolya was supposed to square a certain natural number for his homework. Instead, he mistakenly doubled the number and got a two-digit number, written with the same digits as the square of the number but in reverse order. What should be the correct answer?

Answer: $81$


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

Evaluate \( \frac{18}{4.9 \times 106} \).

Answer: $\frac{18}{519.4}$


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>.

A conveyor system produces on average 85% of first-class products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the frequency of first-class products from 0.85 in absolute magnitude does not exceed 0.01?

Answer: $11475$


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

Teams A and B each have 7 players who will compete in a Go tournament in a predetermined order. The match starts with player 1 from each team competing against each other. The loser is eliminated, and the winner next competes against the loser’s teammate. This process continues until all players of one team are eliminated, and the other team wins. Determine the total number of possible sequences of matches.

Answer: $3432$


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

Given \( z \in \mathbf{C} \) and \( z^{7} = 1 \) (where \( z \neq 1 \)), find the value of \( \cos \alpha + \cos 2 \alpha + \cos 4 \alpha \), where \(\alpha\) is the argument of \(z\).

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>.

Determine the value of the following sum:

$$
\log _{3}\left(1-\frac{1}{15}\right)+\log _{3}\left(1-\frac{1}{14}\right)+\log _{3}\left(1-\frac{1}{13}\right)+\cdots+\log _{3}\left(1-\frac{1}{8}\right)+\log _{3}\left(1-\frac{1}{7}\right)+\log _{3}\left(1-\frac{1}{6}\right)
$$

(Note that the sum includes a total of 10 terms.)

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>.

Let $Q$ be a point outside of circle $C$. A segment is drawn from $Q$, tangent to circle $C$ at point $R$, and a different secant from $Q$ intersects $C$ at points $D$ and $E$ such that $QD < QE$. If $QD = 5$ and the length of the tangent from $Q$ to $R$ ($QR$) is equal to $DE - QD$, calculate $QE$.

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


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

In bag A, there are 3 white balls and 2 red balls, while in bag B, there are 2 white balls and 4 red balls. If a bag is randomly chosen first, and then 2 balls are randomly drawn from that bag, the probability that the second ball drawn is white given that the first ball drawn is red is ______.

Answer: $\frac{17}{32}$


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>.

Given a finite sequence \(P = \left(p_{1}, p_{2}, \cdots, p_{n}\right)\), the Caesar sum (named after a mathematician Caesar) is defined as \(\frac{s_{1}+s_{2}+\cdots+s_{n}}{n}\), where \(s_{k} = p_{1} + p_{2} + \cdots + p_{k}\) for \(1 \leq k \leq n\). If a sequence of 99 terms \(\left(p_{1}, p_{2}, \cdots, p_{99}\right)\) has a Caesar sum of 1000, determine the Caesar sum of the 100-term sequence \(\left(1, p_{1}, p_{2}, \cdots, p_{99}\right)\).

Answer: $991$


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

First, a number \( a \) is randomly selected from the set \(\{1,2,3, \cdots, 99,100\}\), then a number \( b \) is randomly selected from the same set. Calculate the probability that the last digit of \(3^{a} + 7^{b}\) is 8.

Answer: $\frac{3}{16}$


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

Given the imaginary number \( z \) satisfies \( z^3 + 1 = 0 \), \( z \neq -1 \). Then \( \left( \frac{z}{z-1} \right)^{2018} + \left( \frac{1}{z-1} \right)^{2018} = \) .

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>.

There are several soldiers forming a rectangular formation with exactly eight columns. If adding 120 people or removing 120 people from the formation can both form a square formation, how many soldiers are there in the original rectangular formation?

Answer: $136$


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

A sphere is inscribed in a cone whose axial cross-section is an equilateral triangle. Find the volume of the cone if the volume of the sphere is \( \frac{32\pi}{3} \ \text{cm}^3 \).

Answer: $24 \pi$


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

Let $n = 2^{35}3^{17}$. How many positive integer divisors of $n^2$ are less than $n$ but do not divide $n$?

Answer: $594$


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

A  triangle with perimeter  $7$  has integer sidelengths. What is the maximum possible area of such a triangle?

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


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 $f(x)= \frac{1}{2^{x}+ \sqrt {2}}$, use the method for deriving the sum of the first $n$ terms of an arithmetic sequence to find the value of $f(-5)+f(-4)+…+f(0)+…+f(5)+f(6)$.

Answer: $3 \sqrt {2}$


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 an isosceles trapezoid, the bases are 40 and 24, and its diagonals are mutually perpendicular. Find the area of the trapezoid.

Answer: $1024$


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

In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $(2c-a)\cos B=b\cos A$.
(1) Find angle $B$;
(2) If $b=6$ and $c=2a$, 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>.

The vertical drops of six roller coasters at Fractal Fun Park are listed in the table below:
\begin{tabular}{|l|c|}
\hline
The Spiral Slide & 173 feet \\
\hline
The Velocity Vortex & 125 feet \\
\hline
The Cyclone & 150 feet \\
\hline
The Apex Jump & 310 feet \\
\hline
The Quantum Leap & 205 feet \\
\hline
The Zero Gravity & 180 feet \\
\hline
\end{tabular}
What is the positive difference between the mean and the median of these values?

Answer: $14$


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

A prime number of the form \( 2^{p} - 1 \) is called a Mersenne prime, where \( p \) is also a prime number. To date, the largest known Mersenne prime is \( 2^{82589933} - 1 \). Find the last two digits of this number.

Answer: $91$


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

In a right triangle where the ratio of the legs is 1:3, a perpendicular is dropped from the vertex of the right angle to the hypotenuse. Find the ratio of the segments created on the hypotenuse by this perpendicular.

Answer: $1:9$


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

Find a six-digit number $\overline{xy243z}$ that is divisible by 396.

Answer: $432432$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 \( 16 \) times the series that results if the first two terms of the original series are removed. What is the value of the series' common ratio?

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


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

In the diagram, \( P Q = 19 \), \( Q R = 18 \), and \( P R = 17 \). Point \( S \) is on \( P Q \), point \( T \) is on \( P R \), and point \( U \) is on \( S T \) such that \( Q S = S U \) and \( U T = T R \). The perimeter of \(\triangle P S T\) is equal to:

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>.

Given that $α$ and $β$ are acute angles, and $\cos(α+β)=\frac{3}{5}$, $\sin α=\frac{5}{13}$, find the value of $\cos β$.

Answer: $\frac{56}{65}$


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

Determine the degree of the polynomial resulting from $(5x^6 - 4x^5 + x^2 - 18)(2x^{12} + 6x^9 - 11x^6 + 10) - (x^3 + 4)^6$ when this expression is expanded and simplified.

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>.

Calculate the areas of the regions bounded by the curves given in polar coordinates.

$$
r=\cos 2 \phi
$$

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


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 division problem, the dividend is 12, and the divisor is a natural number less than 12. What is the sum of all possible different remainders?

Answer: $15$


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

Calculate the value of the expression \(\sin \frac{b \pi}{36}\), where \(b\) is the sum of all distinct numbers obtained from the number \(a = 987654321\) by cyclic permutations of its digits (in a cyclic permutation, all the digits of the number, except the last one, are shifted one place to the right, and the last digit moves to the first place).

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 unordered pairs of edges of a given octahedron determine a plane?

Answer: $66$


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

Bethany has 11 pound coins and some 20 pence coins and some 50 pence coins in her purse. The mean value of the coins is 52 pence. Which could not be the number of coins in the purse?
A) 35
B) 40
C) 50
D) 65
E) 95

Answer: $40$


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

If \( a, b, c \) are the three real roots of the equation
\[ x^{3} - x^{2} - x + m = 0, \]
then the minimum value of \( m \) is _____.

Answer: $-\frac{5}{27}$


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

Find the volume of the solid $T$ consisting of all points $(x, y, z)$ such that $|x| + |y| \leq 2$, $|x| + |z| \leq 2$, and $|y| + |z| \leq 2$.

Answer: $\frac{32}{3}$


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

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

Answer: $2 \sqrt {3}$


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

Given \( x, y, z > 0 \) and \( x + y + z = 1 \), find the maximum value of
$$
f(x, y, z) = \sum \frac{x(2y - z)}{1 + x + 3y}.
$$

Answer: $1/7$


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

Shift the graph of the function $y=3\sin (2x+ \frac {\pi}{6})$ to the graph of the function $y=3\cos 2x$ and determine the horizontal shift units.

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


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

Snow White entered a room where 30 chairs were arranged around a circular table. Some of the chairs were occupied by dwarfs. It turned out that Snow White could not sit in such a way that there was no one next to her. What is the minimum number of dwarfs that could have been at the table? (Explain how the dwarfs must have been seated and why there would be a chair with no one next to it if there were fewer dwarfs.)

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>.

Two students, A and B, are preparing to have a table tennis match during their physical education class. Assuming that the probability of A winning against B in each game is $\frac{1}{3}$, the match follows a best-of-three format (the first player to win two games wins the match). What is the probability of A winning the match?

Answer: $\frac{7}{27}$


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

Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$

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


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

Given a $3 \times 3$ grid (like a Tic-Tac-Toe board), four randomly selected cells have been randomly placed with four tokens.

Find the probability that among these four tokens, three are aligned in a row either vertically, horizontally, or diagonally.

Answer: $8/21$


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

The numeral $65$ in base $c$ represents the same number as $56$ in base $d$. Assuming that both $c$ and $d$ are positive integers, find the least possible value of $c+d$.

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>.

Sandhya must save 35 files onto disks, each with 1.44 MB space. 5 of the files take up 0.6 MB, 18 of the files take up 0.5 MB, and the rest take up 0.3 MB. Files cannot be split across disks. Calculate the smallest number of disks needed to store all 35 files.

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>.

In $\triangle ABC$, the sides opposite to angles $A, B, C$ are denoted as $a, b, c$, respectively, and $a=1, A=\frac{\pi}{6}$.

(Ⅰ) When $b=\sqrt{3}$, find the magnitude of angle $C$;
(Ⅱ) Find the maximum area of $\triangle ABC$.

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


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>.

How many consecutive "0"s are there at the end of the product \(5 \times 10 \times 15 \times 20 \times \cdots \times 2010 \times 2015\)?

Answer: $398$


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

Given $a= \int_{ 0 }^{ \pi }(\sin x-1+2\cos ^{2} \frac {x}{2})dx$, find the constant term in the expansion of $(a \sqrt {x}- \frac {1}{ \sqrt {x}})^{6}\cdot(x^{2}+2)$.

Answer: $-332$


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

Given the set \( M = \{1, 2, \cdots, 2017\} \) which consists of the first 2017 positive integers, if one element is removed from \( M \) such that the sum of the remaining elements is a perfect square, what is the removed element?

Answer: $1677$


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

Convert the binary number $110101_{(2)}$ to decimal.

Answer: $53$


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

Given: Circle $C$ passes through point $D(0,1)$, $E(-2,1)$, $F(-1,\sqrt{2})$, $P$ is any point on the line $l_{1}: y=x-2$, and the line $l_{2}: y=x+1$ intersects circle $C$ at points $A$ and $B$.
$(Ⅰ)$ Find the equation of circle $C$;
$(Ⅱ)$ Find the minimum value of $|PA|^{2}+|PB|^{2}$.

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>.

A paper equilateral triangle of side length 2 on a table has vertices labeled \(A\), \(B\), and \(C\). Let \(M\) be the point on the sheet of paper halfway between \(A\) and \(C\). Over time, point \(M\) is lifted upwards, folding the triangle along segment \(BM\), while \(A\), \(B\), and \(C\) remain on the table. This continues until \(A\) and \(C\) touch. Find the maximum volume of tetrahedron \(ABCM\) at any time during this process.

Answer: $\frac{\sqrt{3}}{6}$


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>.

If \(\frac{1}{4} + 4\left(\frac{1}{2013} + \frac{1}{x}\right) = \frac{7}{4}\), find the value of \(1872 + 48 \times \left(\frac{2013 x}{x + 2013}\right)\).

Answer: $2000$


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

How many non-similar quadrilaterals have angles whose degree measures are distinct positive integers in an arithmetic progression?

Answer: $29$


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

Find  $k$  where  $2^k$  is the largest power of  $2$  that divides the product \[2008\cdot 2009\cdot 2010\cdots 4014.\]

Answer: $2007$


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

Given the function $f(x)=\sin x\cos x- \sqrt {3}\cos ^{2}x.$
(I) Find the smallest positive period of $f(x)$;
(II) When $x\in[0, \frac {π}{2}]$, find the maximum and minimum values of $f(x)$.

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>.

Given that $b = 8$ and $n = 15$, calculate the number of positive factors of $b^n$ where both $b$ and $n$ are positive integers, with $n$ being 15. Determine if this choice of $b$ and $n$ maximizes the number of factors compared to similar calculations with other bases less than or equal to 15.

Answer: $46$


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

Given the function $f(x)= \frac{1}{x+1}$, point $O$ is the coordinate origin, point $A_{n}(n,f(n))(n∈N^{})$ where $N^{}$ represents the set of positive integers, vector $ \overrightarrow{i}=(0,1)$, and $θ_{n}$ is the angle between vector $ \overrightarrow{OA_{n}}$ and $ \overrightarrow{i}$, determine the value of $\frac{cosθ_{1}}{sinθ_{1}}+ \frac{cosθ_{2}}{sinθ_{2}}+…+\frac{cosθ_{2017}}{sinθ_{2017}}$.

Answer: $\frac{2017}{2018}$


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

For any real number $x$, the symbol $\lfloor x \rfloor$ represents the integer part of $x$, which is the greatest integer not exceeding $x$. This function, $\lfloor x \rfloor$, is called the "floor function". Calculate the sum $\lfloor \log_3 1 \rfloor + \lfloor \log_3 2 \rfloor + \lfloor \log_3 3 \rfloor + \lfloor \log_3 4 \rfloor + \ldots + \lfloor \log_3 243 \rfloor$.

Answer: $857$


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

Compute $\sin 870^\circ$ and $\cos 870^\circ$.

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


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

Vasya has three cans of paint of different colors. In how many different ways can he paint a fence consisting of 10 planks so that any two adjacent planks are different colors and he uses all three colors? Provide a justification for your answer.

Answer: $1530$


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

In the arithmetic sequence $\{a_n\}$, the common difference is $\frac{1}{2}$, and $a_1+a_3+a_5+\ldots+a_{99}=60$. Find the value of $a_2+a_4+a_6+\ldots+a_{100}$.

Answer: $85$


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

The following diagram shows a square where each side has four dots that divide the side into three equal segments. The shaded region has area 105. Find the area of the original square.

[center]![Image](https://snag.gy/r60Y7k.jpg)[/center]

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>.

Given that $x, y > 0$ and $\frac{1}{x} + \frac{1}{y} = 2$, find the minimum value of $x + 2y$.

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


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>.

Given the circle \(\Gamma: x^{2} + y^{2} = 1\) with two intersection points with the \(x\)-axis as \(A\) and \(B\) (from left to right), \(P\) is a moving point on circle \(\Gamma\). Line \(l\) passes through point \(P\) and is tangent to circle \(\Gamma\). A line perpendicular to \(l\) passes through point \(A\) and intersects line \(BP\) at point \(M\). Find the maximum distance from point \(M\) to the line \(x + 2y - 9 = 0\).

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


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

Last year, 10% of the net income from our school's ball was allocated to clubs for purchases, and the remaining part covered the rental cost of the sports field. This year, we cannot sell more tickets, and the rental cost remains the same, so increasing the share for the clubs can only be achieved by raising the ticket price. By what percentage should the ticket price be increased to make the clubs' share 20%?

Answer: $12.5$


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

Given a random variable $ξ$ follows a normal distribution $N(2,σ^{2})$, and $P(ξ \leqslant 4-a) = P(ξ \geqslant 2+3a)$, solve for the value of $a$.

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>.

Determine the volume of the region in space defined by
\[|x + y + z| + |x + y - z| \le 12\]
and \(x, y, z \ge 0.\)

Answer: $108$


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

A rock is dropped off a cliff of height  $ h $  As it falls, a camera takes several photographs, at random intervals. At each picture, I measure the distance the rock has fallen. Let the average (expected value) of all of these distances be  $ kh $ . If the number of photographs taken is huge, find  $ k $ . That is: what is the time-average of the distance traveled divided by  $ h $ , dividing by  $h$ ?

*Problem proposed by Ahaan Rungta*

Answer: $\dfrac{1}{3}$