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

If real numbers $\alpha, \beta, \gamma$ form a geometric sequence with a common ratio of 2, and $\sin \alpha, \sin \beta, \sin \gamma$ also form a geometric sequence, then $\cos \alpha = $ ?

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>.

What is the smallest number, \( n \), which is the product of 3 distinct primes where the mean of all its factors is not an integer?

Answer: $130$


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

Let triangle $ABC$ be equilateral with each side measuring 18 inches. Let $O$ be the intersection point of medians $AP$ and $CQ$ of triangle $ABC$. If $OQ$ is 6 inches, then $AQ$, in inches, is:
A) $6$ inches
B) $12$ inches
C) $15$ inches
D) $18$ inches
E) $21$ inches

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>.

If the inequality \( \ln(ax) + ax \leq x + e^x \) holds for all \( x \) and given real numbers \( a \), what is the maximum value of \( a \)?

Answer: $e$


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

In a selection of $2$ algebra questions and $3$ geometry questions, one question is randomly selected each time without replacement. Let $A=$"selecting an algebra question first" and $B=$"selecting a geometry question second". Find $P\left(AB\right)=\_\_\_\_\_\_$ and $P\left(B|A\right)=\_\_\_\_\_\_$.

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


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>.

Seven people stand in a row. (Write out the necessary process, and use numbers for the answers)
(1) How many ways can person A and person B stand next to each other?
(2) How many ways can person A and person B stand not next to each other?
(3) How many ways can person A, person B, and person C stand so that no two of them are next to each other?
(4) How many ways can person A, person B, and person C stand so that at most two of them are not next to each other?

Answer: $4320$


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

Given the price of Product A was set at 70 yuan per piece in the first year, with an annual sales volume of 118,000 pieces, starting from the second year, the price per piece increased by $$\frac {70\cdot x\%}{1-x\%}$$ yuan due to a management fee, and the annual sales volume decreased by $10^4x$ pieces, calculate the maximum value of x such that the management fee collected in the second year is not less than 1.4 million yuan.

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>.

Suppose two arithmetic sequences $\{a\_n\}$ and $\{b\_n\}$ have the sum of their first n terms as $S\_n$ and $T\_n$, respectively. If $a\_3 = 2b\_3$ and $\frac{S\_n}{T\_n} = \frac{3n - t}{2n + 2}$ for any $n \in \mathbb{N}^*$, find the value of $\frac{a\_9}{b\_6 + b\_8} + \frac{a\_5}{b\_3 + b\_11}$.

Answer: $\frac{12}{7}$


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>.

Define a function \( f(x) \) on \( \mathbb{R} \) that satisfies \( f(-x) = -f(x) \), \( f(x-2) = f(x+2) \), and when \( x \in (-1,0) \), \( f(x) = 2^x + \frac{1}{5} \). Then, evaluate \( f(\log_{2} 20) \).

Answer: $-1$


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

In the drawing, there is a grid consisting of 25 small equilateral triangles.

How many rhombuses can be formed from two adjacent small triangles?

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>.

50 students with blond hair, brown hair, and red hair are sitting around a round table. It is known that in any group of students sitting consecutively, there is at least one student with brown hair between any two students with blond hair, and at least one student with red hair between any two students with brown hair. What is the minimum number of students with red hair that can be sitting at this table?

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>.

A circle is inscribed in a circular sector with an arc that measures $60^{\circ}$. Find the ratio of the area of this circle to the area of the sector.

Answer: $2/3$


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

Given the complex number $z$ that satisfies $z= \frac {1-i}{i+1}$, find the value of $|1+z|$.

Answer: $\sqrt{2}$


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

In the expanded country of Mathlandia, all automobile license plates have five symbols. The first two must be vowels (A, E, I, O, U), the next two must be different non-vowels among the 21 non-vowels in the alphabet, and the fifth must be a digit (0 through 9). Determine the probability that the plate will read "AIE19"

Answer: $\frac{1}{105,000}$


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

The value of \(987 + 113 - 1000\) is

Answer: $100$


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

A positive integer  $N$  is divided in  $n$  parts inversely proportional to the numbers  $2, 6, 12, 20, \ldots$ The smallest part is equal to  $\frac{1}{400} N$ . Find the value of  $n$ .

Answer: $20$


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

Let \( x, y, z \) be positive integers such that
\[ (x+y)(y+z) = 2016 \]
\[ (x+y)(z+x) = 1080 \]
Determine the smallest possible value for \( x+y+z \).

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>.

Given the function $f(x) = \sin^2x + 2\sqrt{3}\sin x\cos x + \sin(x + \frac{\pi}{4})\sin(x - \frac{\pi}{4})$, if $x = x_0 (0 \leq x_0 \leq \frac{\pi}{2})$ is a zero of the function $f(x)$, then $\cos 2x_0 = \_\_\_\_\_\_$.

Answer: $\frac{3\sqrt{5} + 1}{8}$


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

Given rectangle $ABCD$ with $E$ the midpoint of diagonal $BD$. Point $E$ is connected to a point $F$ on $DA$ so that $DF=\frac{1}{4}DA$. Find the ratio of the area of $\triangle DFE$ to the area of quadrilateral $ABEF$.

Answer: $\frac{1}{7}$


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>.

Let set \( A = \{a \mid a = 3k + 2, k \leqslant 2000, k \in \mathbf{N}_{+}\} \) and set \( B = \{b \mid b = 4k - 1, k \leqslant 2000, k \in \mathbf{N}_{+}\} \). How many elements are in \( A \cap B \)?

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>.

Simplify and evaluate

(Ⅰ) Evaluate \\( \dfrac{ \sqrt{3}\sin (- \dfrac{20}{3}\pi)}{\tan \dfrac{11}{3}\pi}-\cos \dfrac{13}{4}\pi\cdot\tan (- \dfrac{35}{4}\pi) \).

(Ⅱ) Evaluate: \\( \dfrac{\sqrt{1-2\sin {10}^{\circ }\cos {10}^{\circ }}}{\cos {10}^{\circ }-\sqrt{1-{\cos }^{2}{170}^{\circ }}} \)

(Ⅲ) If \\( \sin \theta, \cos \theta \) are the roots of the equation \\( 2{x}^{2}-x+a=0 \) (where \\( a \) is a constant) and \\( \theta \in (0,\pi) \), find the value of \\( \cos \theta - \sin \theta \).

Answer: $- \dfrac{ \sqrt{7}}{2}$


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>.

A positive integer  $n$  between  $1$  and  $N=2007^{2007}$  inclusive is selected at random.  If  $a$  and  $b$  are natural numbers such that  $a/b$  is the probability that  $N$  and  $n^3-36n$  are relatively prime, find the value of  $a+b$ .

Answer: $1109$


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

Given the function $f(x) = A\sin(x + \varphi)$ ($A > 0$, $0 < \varphi < \pi$) has a maximum value of 1, and its graph passes through point M ($\frac{\pi}{3}$, $\frac{1}{2}$), then $f(\frac{3\pi}{4})$ = \_\_\_\_\_\_.

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>.

Alloy $A$ of two metals has a mass of 6 kg, with the first metal being twice as abundant as the second metal. When placed in a container of water, it exerts a force of $30\ \mathrm{N}$ on the bottom. Alloy $B$ of the same metals has a mass of 3 kg, with the first metal being five times less abundant than the second metal. When placed in a container of water, it exerts a force of $10\ \mathrm{N}$ on the bottom. What force (in newtons) will the third alloy, obtained by combining the original alloys, exert on the bottom?

Answer: $40$


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

In a triangle, the base is 12 units; one of the angles at this base is $120^{\circ}$; the side opposite this angle is 28 units.

Find the third side.

Answer: $20$


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

Given $(x^2+1)(x-2)^8 = a + a_1(x-1) + a_2(x-1)^2 + \ldots + a_{10}(x-1)^{10}$, find the value of $a_1 + a_2 + \ldots + a_{10}$.

Answer: $-2$


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

Let \( A B C \) be a triangle. The midpoints of the sides \( B C, A C \), and \( A B \) are denoted by \( D, E \), and \( F \) respectively.

The two medians \( A D \) and \( B E \) are perpendicular to each other and have lengths \( \overline{A D} = 18 \) and \( \overline{B E} = 13.5 \).

Calculate the length of the third median \( CF \) of this triangle.

Answer: $22.5$


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

Given the function $f(x)=a\sin x - \sqrt{3}\cos x$, one of its graphs has an axis of symmetry at $x=-\frac{\pi}{6}$, and $f(x_1) - f(x_2) = -4$, calculate the minimum value of $|x_1+x_2|$.

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


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

The minimum value of the sum of the distances from a point in space to the vertices of a regular tetrahedron with edge length 1 is:

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>.

For transportation between points located hundreds of kilometers apart on the Earth's surface, people of the future will likely dig straight tunnels through which capsules will travel frictionlessly under the influence of Earth's gravity. Let points \( A, B, \) and \( C \) lie on the same meridian, with the surface distance from \( A \) to \( B \) related to the surface distance from \( B \) to \( C \) in the ratio \( m : n \). A capsule travels through the tunnel \( AB \) in approximately 42 minutes. Estimate the travel time through tunnel \( AC \). Provide the answer in minutes.

Answer: $42$


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

If three different natural numbers $a$, $b$ and $c$ each have exactly four natural-number factors, how many factors does $a^3b^4c^5$ have?

Answer: $2080$


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

Let  $ m\equal{}\left(abab\right)$  and  $ n\equal{}\left(cdcd\right)$  be four-digit numbers in decimal system. If  $ m\plus{}n$  is a perfect square, what is the largest value of  $ a\cdot b\cdot c\cdot d$ ?

Answer: $600$


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

$n$ mushroom gatherers went to the forest and brought back a total of 338 mushrooms (it is possible that some of the gatherers did not bring back any mushrooms). A boy named Petya, upon learning this, stated, "Some two of them must have brought an identical number of mushrooms!" For what smallest $n$ will Petya certainly be correct? Provide a justification for your answer.

Answer: $27$


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

Given that point \( P \) lies on the hyperbola \( \frac{x^2}{16} - \frac{y^2}{9} = 1 \), and the distance from \( P \) to the right directrix of this hyperbola is the arithmetic mean of the distances from \( P \) to the two foci of the hyperbola, find the x-coordinate of \( P \).

Answer: $-\frac{64}{5}$


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>.

Find the equation of the directrix of the parabola \( y = \frac{x^2 - 8x + 12}{16} \).

Answer: $y = -\frac{1}{2}$


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>.

The graph of $r = \sin \theta$ is also seen in a circular form but pertains to a vertical alignment. Find the smallest value of $t$ so that when $r = \sin \theta$ is plotted for $0 \le \theta \le t,$ the resulting graph forms a complete circle.

Answer: $2\pi$


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

Express 0.0000006 in scientific notation.

Answer: $6 \times 10^{-7}$


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>.

Determine $S$, the sum of all the real coefficients of the expansion of $(1+ix)^{2020}$, and find $\log_2(S)$.

Answer: $1010$


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

Two numbers \( x \) and \( y \) satisfy the equation \( 26x^2 + 23xy - 3y^2 - 19 = 0 \) and are respectively the sixth and eleventh terms of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression.

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>.

Given that the coefficients $p$ and $q$ are integers and the roots $\alpha_{1}$ and $\alpha_{2}$ are irrational, a quadratic trinomial $x^{2} + px + q$ is called an irrational quadratic trinomial. Determine the minimum sum of the absolute values of the roots among all irrational quadratic trinomials.

Answer: $\sqrt{5}$


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

Given the functions $f(x)=x^{2}+ax+3$, $g(x)=(6+a)\cdot 2^{x-1}$.

(I) If $f(1)=f(3)$, find the value of the real number $a$;

(II) Under the condition of (I), determine the monotonicity of the function $F(x)=\frac{2}{1+g(x)}$ and provide a proof;

(III) When $x \in [-2,2]$, $f(x) \geqslant a$, ($a \notin (-4,4)$) always holds, find the minimum value of the real number $a$.

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>.

Given an equilateral triangle $DEF$ with $DE = DF = EF = 8$ units and a circle with radius $4$ units tangent to line $DE$ at $E$ and line $DF$ at $F$, calculate the area of the circle passing through vertices $D$, $E$, and $F$.

Answer: $\frac{64\pi}{3}$


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

Given the equation \(x + 11y + 11z = n\) where \(n \in \mathbf{Z}_{+}\), there are 16,653 sets of positive integer solutions \((x, y, z)\). Find the minimum value of \(n\).

Answer: $2014$


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

The recruits were standing in a row, one behind the other, facing the same direction. Among them were three brothers: Peter, Nicholas, and Denis. There were 50 people ahead of Peter, 100 ahead of Nicholas, and 170 ahead of Denis. Upon the command "About face!", everyone turned to face the opposite direction. It turned out that in front of one of the brothers, there were now four times as many people as there were in front of another brother. How many recruits, including the brothers, could there be? List all possible variants.

Answer: $211$


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

Select 4 out of 7 different books to distribute to 4 students, one book per student, with the restriction that books A and B cannot be given to student C, and calculate the number of different distribution methods.

Answer: $600$


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

In right triangle $ABC$ with $\angle C=90^{\circ}$ and $AC=4$, find the value of $\overrightarrow{AB} \cdot \overrightarrow{AC}$.

Answer: $16$


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

Given a linear function \( f(x) \). It is known that the distance between the points of intersection of the graphs \( y = x^2 + 1 \) and \( y = f(x) \) is \( 3\sqrt{2} \), and the distance between the points of intersection of the graphs \( y = x^2 \) and \( y = f(x) - 2 \) is \( \sqrt{10} \). Find the distance between the points of intersection of the graphs of the functions \( y = x^2 \) and \( y = f(x) \).

Answer: $\sqrt{26}$


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

Given that $x$ varies directly as the square of $y$, and $y$ varies directly as the cube root of $z$, determine the power $n$ such that $x$ varies as $x^n$.

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


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

Given a function $f(x)$ defined on $R$ such that $f(x) + x^{2}$ is an odd function and $f(x) + x^{3}$ is an even function, then $f(2)$ is ______.

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>.

Fill six numbers $1, 3, 5, 7, 9, 11$ into the circles (each circle containing only one number) so that the sum of three numbers on each side of the triangle equals 17. What is the sum of the three numbers in the circles at the vertices of the triangle?

Answer: $15$


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

Given that $α$ and $β$ belong to the interval $(0, \frac{π}{2})$, and they satisfy the equation $\tan(α + β) = 9\tan(β)$, find the maximum value of $\tan(α)$.

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


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

Given that $O$ is the center of the circumcircle of $\triangle ABC$, $D$ is the midpoint of side $BC$, and $BC=4$, and $\overrightarrow{AO} \cdot \overrightarrow{AD} = 6$, find the maximum value of the area of $\triangle ABC$.

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>.

In $\triangle ABC$, $D$ and $E$ are points on sides $BC$ and $AC$, respectively, such that $\frac{BD}{DC} = \frac{2}{3}$ and $\frac{AE}{EC} = \frac{3}{4}$. Find the value of $\frac{AF}{FD} \cdot \frac{BF}{FE}$.

Answer: $\frac{35}{12}$


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

Given the function $f(x)=x^{3}+3ax^{2}+bx+a^{2}$ has an extreme value of $0$ at $x=-1$, find the value of $a-b$.

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>.

Gavrila found that the front tires of the car last for 21,000 km, and the rear tires last for 28,000 km. Therefore, he decided to swap them at some point so that the car would travel the maximum possible distance. Find this maximum distance (in km).

Answer: $24000$


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

Given the function $f(x)=- \sqrt {2}\sin (2x+ \frac {\pi}{4})+6\sin x\cos x-2\cos ^{2}x+1$, where $x\in\mathbb{R}$.
(I) Find the smallest positive period of $f(x)$.
(II) Find the maximum and minimum values of $f(x)$ in the interval $\left[0, \frac {\pi}{2}\right]$.

Answer: $-2$


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

Given that a, b, and c are the sides opposite to angles A, B, and C respectively in triangle ABC, and c = 2, sinC(cosB - $\sqrt{3}$sinB) = sinA.
(1) Find the measure of angle C;
(2) If cosA = $\frac{2\sqrt{2}}{3}$, find the length of side b.

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


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

In the right triangle \( \triangle ABC \), \( \angle B = 90^\circ \). Point \( P \) is on the angle bisector of \( \angle A \) within \( \triangle ABC \). Point \( M \) (distinct from \( A \) and \( B \)) is a point on side \( AB \). The lines \( AP \), \( CP \), and \( MP \) intersect sides \( BC \), \( AB \), and \( AC \) at points \( D \), \( E \), and \( N \) respectively. Given that \( \angle MPB = \angle PCN \) and \( \angle NPC = \angle MBP \), find \( \frac{S_{\triangle APC}}{S_{ACDE}} \).

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 number is called ascending if each of its digits is greater than the digit to its left. For example, 2568 is ascending, and 175 is not. How many ascending numbers are there between 400 and 600?

Answer: $16$


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

Given a random variable $X \sim B(10, 0.6)$, calculate the values of $E(X)$ and $D(X)$.

Answer: $2.4$


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

Find a five-digit number that has the following property: when multiplied by 9, the result is a number represented by the same digits but in reverse order.

Answer: $10989$


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

Two identical resistors $R_{0}$ are connected in series and connected to a DC voltage source. An ideal voltmeter is connected in parallel with one of the resistors. Its reading is $U=2 \text{V}$. If the voltmeter is replaced with an ideal ammeter, its reading will be $I=4 \text{A}$. Determine the value of $R_{0}$.

Answer: $0.5$


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>.

What is the sum of all three-digit and four-digit positive integers up to 2000?

Answer: $1996050$


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

Given a right prism $ABC-A_{1}B_{1}C_{1}$, where $AB=3$, $AC=4$, and $AB \perp AC$, $AA_{1}=2$, find the sum of the surface areas of the inscribed sphere and the circumscribed sphere of the prism.

Answer: $33\pi$


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

The product of all the positive integer divisors of \( 6^{16} \) equals \( 6^k \) for some integer \( k \). Determine the value of \( k \).

Answer: $2312$


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

There are 6 class officers, among which there are 3 boys and 3 girls.

(1) Now, 3 people are randomly selected to participate in the school's voluntary labor. Calculate the probability that at least 2 of the selected people are girls.

(2) If these 6 people stand in a row for a photo, where boy A can only stand at the left or right end, and girls B and C must stand next to each other, how many different arrangements are there?

Answer: $96$


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

In an isosceles right triangle $ABC$ with $\angle A = 90^{\circ}$ and $AB = AC = 2$, calculate the projection of the vector $\vec{AB}$ in the direction of $\vec{BC}$.

Answer: $-\sqrt{2}$


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

The sum of the first thirteen terms of an arithmetic progression is $50\%$ of the sum of the last thirteen terms of this progression. The sum of all terms of this progression, excluding the first three terms, is to the sum of all terms excluding the last three terms in the ratio $5:4$. Find the number of terms in this progression.

Answer: $22$


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

For a certain type of car, the purchase cost is $10$ ten thousand yuan, and the annual expenses for insurance, road maintenance, and car fees are about $0.9$ ten thousand yuan. The maintenance fee for the first year is $0.2$ ten thousand yuan, and it increases by $0.2$ ten thousand yuan each subsequent year. How many years of use would minimize the average cost of this car?

Answer: $10$


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

Given a sequence of real numbers \( a_{1}, a_{2}, a_{3}, \ldots \) that satisfy
1) \( a_{1}=\frac{1}{2} \), and
2) \( a_{1} + a_{2} + \ldots + a_{k} = k^{2} a_{k} \), for \( k \geq 2 \).
Determine the value of \( a_{100} \).

Answer: $\frac{1}{10100}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 graph of the parabola $y=ax^2 - 3x + 3 \ (a \neq 0)$ is symmetric with the graph of the parabola $y^2 = 2px \ (p > 0)$ with respect to the line $y = x + m$. Find the product of the real numbers $a$, $p$, and $m$.

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>.

Given that $\sin(\theta + 3\pi) = -\frac{2}{3}$, find the value of $\frac{\tan(-5\pi - \theta) \cdot \cos(\theta - 2\pi) \cdot \sin(-3\pi - \theta)}{\tan(\frac{7\pi}{2} + \theta) \cdot \sin(-4\pi + \theta) \cdot \cot(-\theta - \frac{\pi}{2})} + 2 \tan(6\pi - \theta) \cdot \cos(-\pi + \theta)$.

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>.

What is the remainder when $2024 \cdot 3047$ is divided by $800$?

Answer: $728$


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

Given that spinner S has numbers 1, 4, 3, spinner T has numbers 2, 4, 6, and spinner U has numbers 2, 3, 5, determine the probability that the sum of the numbers obtained from spinning each of the three spinners is an even number.

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


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>.

If \( p \) and \( q \) are positive integers and \(\frac{2008}{2009} < \frac{p}{q} < \frac{2009}{2010} \), what is the minimum value of \( p \)?

Answer: $4017$


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

There are  $2022$  grids in a row. Two people A and B play a game with these grids. At first, they mark each odd-numbered grid from the left with A's name, and each even-numbered grid from the left with B's name. Then, starting with the player A, they take turns performing the following action: $\bullet$  One should select two grids marked with his/her own name, such that these two grids are not adjacent, and all grids between them are marked with the opponent's name. Then, change the name in all the grids between those two grids to one's own name.

Two people take turns until one can not perform anymore. Find the largest positive integer  $m$  satisfying the following condition:  $\bullet$  No matter how B acts, A can always make sure that there are at least  $m$  grids marked with A's name.

Answer: $1011$


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

In an apartment building, each floor in every entrance has the same number of apartments (more than one). Additionally, each entrance has the same number of floors. The number of floors is more than the number of apartments per floor, but less than the number of entrances. How many floors are in the building if there are a total of $715$ apartments?

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>.

If the equation \( x^{2} - a|x| + a^{2} - 3 = 0 \) has a unique real solution, then \( a = \) ______.

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>.

In a cube $ABCD-A_{1}B_{1}C_{1}D_{1}$ with edge length 1, $E$ is the midpoint of $AB$, and $F$ is the midpoint of $CC_{1}$. Find the distance from $D$ to the plane passing through points $D_{1}$, $E$, and $F$.

Answer: $\frac{4 \sqrt{29}}{29}$


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>.

In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, where $b=2$.
$(1)$ If $A+C=120^{\circ}$ and $a=2c$, find the length of side $c$.
$(2)$ If $A-C=15^{\circ}$ and $a=\sqrt{2}c\sin A$, find the area of triangle $\triangle ABC$.

Answer: $3 - \sqrt{3}$


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 point \( P \) lies in the plane of triangle \( \triangle ABC \) and satisfies the condition \( PA - PB - PC = \overrightarrow{BC} \), determine the ratio of the area of \( \triangle ABP \) to the area of \( \triangle ABC \).

Answer: $2:1$


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

Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace and the second card is a King?

Answer: $\dfrac{4}{663}$


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

When $n$ is an odd number, $\frac{1+(-1)^{n}}{4}=$______; when $n$ is an even number, $\frac{1+(-1)^{n}}{4}=\_\_\_\_\_\_$.

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>.

$p(n) $  is a product of all digits of n.Calculate: $ p(1001) + p(1002) + ... + p(2011) $

Answer: $91125$


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

Given a triangular pyramid \( P-ABC \) with a base that is an equilateral triangle of side length \( 4 \sqrt{3} \), and with \( PA=3 \), \( PB=4 \), and \( PC=5 \). If \( O \) is the center of the triangle \( ABC \), find the length of \( PO \).

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


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 total investment of the Xiangshan Port Sea-Crossing Bridge, about 7.7 billion yuan, should be expressed in scientific notation.

Answer: $7.7 \times 10^9$


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>.

Rectangle \(PQRS\) is divided into 60 identical squares, as shown. The length of the diagonal of each of these squares is 2. The length of \(QS\) is closest to

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>.

Two distinct natural numbers end in 5 zeros and have exactly 42 divisors. Find their sum.

Answer: $700000$


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

When $n$ standard 8-sided dice are rolled, the probability of obtaining a sum of 3000 is greater than zero and is the same as the probability of obtaining a sum of S. Find the smallest possible value of S.

Answer: $375$


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

How many positive integers less than 500 are congruent to 7 (mod 13)?

Answer: $38$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 heights are given: \(h_{a} = \frac{1}{3}\), \(h_{b} = \frac{1}{4}\), \(h_{c} = \frac{1}{5}\). Find the ratio of the angle bisector \(CD\) to the circumradius.

Answer: $\frac{24\sqrt{2}}{35}$


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>.

Two positive integers are written on the blackboard, one being 2002 and the other being a number less than 2002. If the arithmetic mean of the two numbers is an integer $m$, then the following operation can be performed: one of the numbers is erased and replaced by $m$. What is the maximum number of times this operation can be performed?

Answer: $10$


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

$A$ and $B$ both live such that one lives $3 \text{ km}$ further from the city than the other. They both arrived in the city at the same time, with $A$ traveling by car and $B$ by truck. They were picked up by the vehicles halfway to their destinations. $A$ walks $1.5$ times faster than $B$, but the truck that picked up $B$ travels $1.5$ times faster than the car that picked up $A$. Additionally, the car travels twice as fast as $A$ walks. How far do $A$ and $B$ live from the city?

Answer: $16.5$


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 vectors $$\overrightarrow {m}=(\cos \frac {x}{3}, \sqrt {3}\cos \frac {x}{3})$$, $$\overrightarrow {n}=(\sin \frac {x}{3}, \cos \frac {x}{3})$$, and $$f(x)= \overrightarrow {m}\cdot \overrightarrow {n}$$.
 (Ⅰ) Find the monotonic intervals of the function $f(x)$;
(Ⅱ) If the graph of $f(x)$ is first translated to the left by $\varphi$ ($\varphi>0$) units, and then, keeping the ordinate unchanged, the abscissa is scaled to $\frac {1}{3}$ of its original, resulting in the graph of the function $g(x)$. If $g(x)$ is an even function, find the minimum value of $\varphi$.

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


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

How many integers are there from 1 to 1,000,000 that are neither perfect squares, nor perfect cubes, nor fourth powers?

Answer: $998910$


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

Two different natural numbers end with 7 zeros and have exactly 72 divisors. Find their sum.

Answer: $70000000$


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

Two runners started simultaneously in the same direction from the same point on a circular track. The first runner, moving ahead, caught up with the second runner at the moment when the second runner had only run half a lap. From that moment, the second runner doubled their speed. Will the first runner catch up with the second runner again? If so, how many laps will the second runner complete by that time?

Answer: $2.5$


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

Given an ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with an eccentricity of $\frac{\sqrt{5}}{5}$, and the equation of the right directrix is $x=5$.

(1) Find the equation of the ellipse;
(2) A line $l$ with a slope of 1 passes through the right focus $F$ of the ellipse $C$ and intersects the ellipse at points $A$ and $B$. Let $P$ be a moving point on the ellipse. Find the maximum area of $\triangle PAB$.

Answer: $\frac{16\sqrt{10}}{9}$


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>.

How many integers $m$ are there from 1 to 1996, such that $\frac{m^{2}+7}{m+4}$ is not a reduced fraction?

Answer: $86$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 $a$, $b$, $c$ respectively. Given vectors $\overrightarrow{m}=(b,a-2c)$, $\overrightarrow{n}=(\cos A-2\cos C,\cos B)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
1. Find the value of $\frac{\sin C}{\sin A}$;
2. If $a=2, |m|=3 \sqrt {5}$, find the area of $\triangle ABC$, denoted as $S$.

Answer: $\frac{3\sqrt{15}}{4}$


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

Calculate the definite integral:

$$
\int_{0}^{\frac{\pi}{2}}\left(x^{2}-5x+6\right) \sin 3x \, dx
$$

Answer: $\frac{67 - 3\pi}{27}$


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>.

Given an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, if $a_{2}+a_{4}+3a_{7}+a_{9}=24$, calculate the value of $S_{11}$.

Answer: $44$


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

Given a regular tetrahedron A-BCD with an edge length of 1, and $\overrightarrow{AE} = 2\overrightarrow{EB}$, $\overrightarrow{AF} = 2\overrightarrow{FD}$, calculate $\overrightarrow{EF} \cdot \overrightarrow{DC}$.

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>.

In the cartesian coordinate system $(xOy)$, the parametric equations of line $l$ are given by: $ \begin{cases} x=1+t\cos \alpha \\ y=2+t\sin \alpha \end{cases} (t$ is the parameter, $0\leqslant \alpha < \pi)$, and the polar coordinate system is established with $O$ as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of curve $C$ is $\rho=6\sin \theta$.
(I)(i) When $\alpha=\dfrac{\pi}{4}$, write the ordinary equation of line $l$;
(ii) Write the cartesian equation of curve $C$;
(II) If point $P(1, 2)$, and curve $C$ intersects with line $l$ at points $A$ and $B$, find the minimum value of $\dfrac{1}{|PA|}+\dfrac{1}{|PB|}$.

Answer: $\dfrac{2\sqrt{7}}{7}$


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>.

Vasya has three cans of paint of different colors. In how many different ways can he paint a fence of 10 boards such that any two adjacent boards are of different colors, and all three colors are used?

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>.

The area of the enclosed shape formed by the line $y=0$, $x=e$, $y=2x$, and the curve $y= \frac {2}{x}$ is $\int_{1}^{e} \frac{2}{x} - 2x \,dx$.

Answer: $e^{2}-3$


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>.

There are 100 points marked on a circle, painted either red or blue. Some points are connected by segments, with each segment having one blue end and one red end. It is known that no two red points are connected to the same number of segments. What is the maximum possible number of red points?

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>.

Ancient astronaut theorist Nutter B. Butter claims that the Caloprians from planet Calop, 30 light years away and at rest with respect to the Earth, wiped out the dinosaurs. The iridium layer in the crust, he claims, indicates spaceships with the fuel necessary to travel at 30% of the speed of light here and back, and that their engines allowed them to instantaneously hop to this speed. He also says that Caloprians can only reproduce on their home planet. Call the minimum life span, in years, of a Caloprian, assuming some had to reach Earth to wipe out the dinosaurs,  $T$ . Assume that, once a Caloprian reaches Earth, they instantaneously wipe out the dinosaurs. Then,  $T$  can be expressed in the form  $m\sqrt{n}$ , where  $n$  is not divisible by the square of a prime. Find  $m+n$ .

*(B. Dejean, 6 points)*

Answer: $111$


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

When a five-digit number is multiplied by 9, the resulting number consists of the same digits in reverse order. Find the original number.

Answer: $10989$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 $11011001_2$ to base 4.

Answer: $3121_4$


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>.

The side \( AB \) of a regular hexagon \( ABCDEF \) is equal to \( \sqrt{3} \) and serves as a chord of a certain circle, while the other sides of the hexagon lie outside this circle. The length of the tangent \( CM \), drawn to the same circle from vertex \( C \), is 3. Find the diameter of the circle.

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>.

The number  $2021$  leaves a remainder of  $11$  when divided by a positive integer. Find the smallest such integer.

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>.

In $\triangle ABC$, $A=30^{\circ}$, $AB=\sqrt {3}$, $BC=1$, find the area of $\triangle ABC$.

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


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

Five standard, six-sided dice are to be rolled. If the product of their values is an even number, what is the probability that their sum is divisible by 3?

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>.

Of the numbers $\frac{8}{12}, \frac{5}{6}$, and $\frac{9}{12}$, which number is the arithmetic mean of the other two?

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


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

In a paper, a $4 \times 6$ grid was drawn, and then the diagonal from $A$ to $B$ was traced.

Observe that the diagonal $AB$ intersects the grid at 9 points.

If the grid were of size $12 \times 17$, at how many points would the diagonal $AB$ intersect the grid?

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>.

In the Chinese Basketball Association (CBA) Finals, a best-of-seven series is played, where the first team to win four games is declared the champion. Team A and Team B are evenly matched in the finals, with each team having a probability of $\frac{1}{2}$ of winning each game. According to historical data, the ticket revenue for the first game is 4 million yuan, and for each subsequent game, the revenue increases by 1 million yuan.

$(1)$ Find the probability that the total ticket revenue for the finals is exactly 30 million yuan.

$(2)$ Let $X$ be the total ticket revenue for the finals. Find the mathematical expectation $E(X)$ of $X$.

Answer: $3775$


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

How many times should two dice be rolled so that the probability of getting two sixes at least once is greater than $1/2$?

Answer: $25$


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

Let positive integers $a$, $b$, $c$, and $d$ be randomly and independently selected with replacement from the set $\{1, 2, 3,\dots, 2000\}$. Find the probability that the expression $ab+bc+cd+d+1$ is divisible by $4$.

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>.

Given that $α \in (0, \frac{π}{2})$, and $\sin (\frac{π}{6} - α) = -\frac{1}{3}$, find the value of $\cos α$.

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


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

Solve for the positive integer(s) \( n \) such that \( \phi\left(n^{2}\right) = 1000 \phi(n) \).

Answer: $1000$


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

Given that a bank has 320 employees with an average annual profit of 200,000 yuan per employee, and each remaining employee's profit increases by 20,000 yuan for every laid-off employee, while the bank must pay 60,000 yuan per year for each laid-off employee, and the number of employees required to operate normally is at least three-quarters of the current number, determine the optimal number of employees to lay off.

Answer: $80$


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

(x^2+1)(2x+1)^9=a_0+a_1(x+2)+a_2(x+2)^2+\ldots+a_{11}(x+2)^{11}, calculate the sum of the coefficients a_0 through a_11.

Answer: $-2$


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

Let \( n \) be a fixed integer, \( n \geq 2 \).

1. Determine the smallest constant \( c \) such that the inequality \[
\sum_{1 \leq i < j \leq n} x_i x_j (x_i^2 + x_j^2) \leq c \left( \sum_{i=1}^n x_i \right)^4
\] holds for all non-negative real numbers \( x_1, x_2, \cdots, x_n \).

2. For this constant \( c \), determine the necessary and sufficient conditions for equality to hold.

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


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

Solve the equation \(3 \cdot 4^{\log_{x} 2} - 46 \cdot 2^{\log_{x} 2 - 1} = 8\).

Answer: $\sqrt[3]{2}$


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

When I saw Eleonora, I found her very pretty. After a brief trivial conversation, I told her my age and asked how old she was. She answered:

- When you were as old as I am now, you were three times older than me. And when I will be three times older than I am now, together our ages will sum up to exactly a century.

How old is this capricious lady?

Answer: $15$