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

\[
\angle A = 90^\circ, \, AB = AC
\]

\( M \) and \( N \) are the midpoints of \( AB \) and \( AC \) respectively. \( D \) is an arbitrary point on the segment \( MN \) (excluding points \( M \) and \( N \)). The extensions of \( BD \) and \( CD \) intersect \( AC \) and \( AB \) at points \( F \) and \( E \) respectively. If

\[
\frac{1}{BE} + \frac{1}{CF} = \frac{3}{4}
\]

then find the length of \( BC \).

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>.

The numbers from 1 to 8 are placed at the vertices of a cube such that the sum of the numbers at any three vertices lying on the same face is at least 10. What is the minimum possible sum of the numbers at the vertices of one face?

Answer: $16$


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

The number of children in the families $A$, $B$, $C$, $D$, and $E$ are as shown in the table below:

|         | $A$ | $B$ | $C$ | $D$ | $E$ |
|---------|-----|-----|-----|-----|-----|
| Boys    | $0$ | $1$ | $0$ | $1$ | $1$ |
| Girls   | $0$ | $0$ | $1$ | $1$ | $2$ |

$(1)$ If a child is randomly selected from these children and it is known that the selected child is a girl, find the probability that the girl is from family $E$.
$(2)$ If three families are selected randomly from these $5$ families, let $X$ be the number of families where girls outnumber boys. Find the probability distribution and the expected value of $X$.

Answer: $\frac{6}{5}$


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

Three positive integers differ from each other by at most 6. The product of these three integers is 2808. What is the smallest integer among them?

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>.

Let \( x_{i} \in \{\sqrt{2}-1, \sqrt{2}+1\} \), where \( i = 1, 2, 3, \dots, 2012 \). Define
\[ S = x_{1} x_{2} + x_{3} x_{4} + x_{5} x_{6} + \cdots + x_{2000} x_{2010} + x_{2011} x_{2012}. \]

How many different positive integer values can \( S \) attain?

Answer: $504$


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

Medians $\overline{DP}$ and $\overline{EQ}$ of $\triangle DEF$ intersect at an angle of $60^\circ$. If $DP = 21$ and $EQ = 27$, determine the length of side $DE$.

Answer: $2\sqrt{67}$


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

20 points were marked inside a square and connected with non-intersecting segments to each other and to the vertices of the square, such that the square was divided into triangles. How many triangles were formed?

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>.

To test the quality of a certain product, it was decided to use the random number table method to draw 5 samples from 300 products for inspection. The products are numbered from 000, 001, 002, ..., to 299. The following are the 7th and 8th rows of the random number table. If we start from the 5th number in the 7th row and read to the right, the second sample number among the 5 obtained is:

7th row: 84 42 17 53 31 57 24 55 06 88 77 04 74 47 67 21 76 33 50 25 83 92 12 06 76
8th row: 63 01 63 78 59 16 95 55 67 19 98 10 50 71 75 12 86 73 58 07 44 39 52 38 79

Answer: $057$


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

Let the complex numbers \( z, -z, z^2 - z + 1, z^2 + z + 1 \) correspond to the points \( A, B, C, D \) respectively in the complex plane. Given that \( |z| = 2 \) and the quadrilateral \( ABCD \) is a rhombus, and let such \( z = a + bi \) ( \( \mathrm{i} \) is the imaginary unit, \( a, b \in \mathbb{R} \) ), then \( |a| + |b| = \quad \) (provided by Fu Lexin)

Answer: $\frac{\sqrt{7}+3}{2}$


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>.

How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once?

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>.

Given the function \( f(x)=\sin \omega x+\cos \omega x \) where \( \omega > 0 \) and \( x \in \mathbb{R} \), if the function \( f(x) \) is monotonically increasing on the interval \( (-\omega, \omega) \) and the graph of the function \( y=f(x) \) is symmetric with respect to the line \( x=\omega \), determine the value of \( \omega \).

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


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

In a novel that consists of 530 pages, each page number is printed once, starting from page 1 up to page 530. How many more 3's are printed than 7's throughout the book?

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>.

Given two moving points \( A\left(x_{1}, y_{1}\right) \) and \( B\left(x_{2}, y_{2}\right) \) on the parabola \( x^{2}=4 y \) (where \( y_{1} + y_{2} = 2 \) and \( y_{1} \neq y_{2} \))), if the perpendicular bisector of line segment \( AB \) intersects the \( y \)-axis at point \( C \), then the maximum value of the area of triangle \( \triangle ABC \) is ________

Answer: $\frac{16 \sqrt{6}}{9}$


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 zeros are at the end of the product $s(1) \cdot s(2) \cdot \ldots \cdot s(100)$, where $s(n)$ denotes the sum of the digits of the natural number $n$?

Answer: $19$


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

Tessa the hyper-ant has a 2019-dimensional hypercube. For a real number \( k \), she calls a placement of nonzero real numbers on the \( 2^{2019} \) vertices of the hypercube \( k \)-harmonic if for any vertex, the sum of all 2019 numbers that are edge-adjacent to this vertex is equal to \( k \) times the number on this vertex. Let \( S \) be the set of all possible values of \( k \) such that there exists a \( k \)-harmonic placement. Find \( \sum_{k \in S}|k| \).

Answer: $2040200$


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

Given a set of data $x_{1}$, $x_{2}$, …, $x_{20}$, if the expectation of this set of data is $3$ and the variance is $3$, determine the expectation and variance of $2x_{1}+3$, $2x_{2}+3$, …, $2x_{20}+3$.

Answer: $12$


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

Given \( a=\underset{2016 \uparrow}{55 \cdots 5} \), what is the remainder when \( a \) is divided by 84?

Answer: $63$


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

Let the line \( p \) be the perpendicular bisector of points \( A = (20, 12) \) and \( B = (-4, 3) \). Determine the point \( C = (x, y) \) where line \( p \) meets segment \( AB \), and calculate \( 3x - 5y \).

Answer: $-13.5$


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>.

There are 13 weights arranged in a row on a table, ordered by mass (the lightest on the left, the heaviest on the right). It is known that the mass of each weight is an integer number of grams, the masses of any two adjacent weights differ by no more than 5 grams, and the total mass of the weights does not exceed 2019 grams. Find the maximum possible mass of the heaviest weight under these conditions.

Answer: $185$


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

The quartic equation \( x^{4} + a x^{3} + b x^{2} + a x + 1 = 0 \) has a real root. Find the minimum value of \( a^{2} + b^{2} \).

Answer: $4/5$


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

Given that \( P \) is any point on the arc \( \overparen{AD} \) of the circumscribed circle of square \( ABCD \), find the value of \( \frac{PA + PC}{PB} \).

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>.

What is the shortest distance from the origin to the circle defined by \( x^2 - 30x + y^2 - 8y + 325 = 0 \)?

Answer: $\sqrt{241} - 2\sqrt{21}$


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

Find the smallest fraction which, when divided by each of the fractions \(\frac{21}{25}\) and \(\frac{14}{15}\), results in natural numbers.

Answer: $42/5$


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

Given that the positive integer \( a \) has 15 factors and the positive integer \( b \) has 20 factors, and \( a + b \) is a perfect square, find the smallest possible value of \( a + b \) that meets these conditions.

Answer: $576$


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

Three people, A, B, and C, are taking an elevator from the 1st floor to the 3rd to 7th floors of a mall. Each floor can accommodate at most 2 people getting off the elevator. How many ways are there for them to get off the elevator?

Answer: $120$


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

Five million times eight million equals

Answer: $40,000,000,000,000$


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>.

Let \( P \) be an arbitrary point on the graph of the function \( y = x + \frac{2}{x} \) (for \( x > 0 \)). From point \( P \), perpendiculars are drawn to the line \( y = x \) and to the y-axis, with the feet of these perpendiculars labeled as \( A \) and \( B \), respectively. Find the value of \( \overrightarrow{P A} \cdot \overrightarrow{P B} \).

Answer: $-1$


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

Let $A$ be a set of ten distinct positive numbers (not necessarily integers). Determine the maximum possible number of arithmetic progressions consisting of three distinct numbers from the set $A$.

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>.

Determine the sum of all single-digit replacements for $z$ such that the number ${36{,}z72}$ is divisible by both 6 and 4.

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>.

Walter gets up at 6:30 a.m., catches the school bus at 7:30 a.m., has 6 classes that last 50 minutes each, has 30 minutes for lunch, and has 2 hours additional time at school. He takes the bus home and arrives at 4:00 p.m. Calculate the total number of minutes Walter spent on the bus.

Answer: $60$


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

Two candidates participated in an election with \(p+q\) voters. Candidate \(A\) received \(p\) votes and candidate \(B\) received \(q\) votes, with \(p > q\). During the vote counting, only one vote is registered at a time on a board. Let \(r\) be the probability that the number associated with candidate \(A\) on the board is always greater than the number associated with candidate \(B\) throughout the entire counting process.

a) Determine the value of \(r\) if \(p=3\) and \(q=2\).

b) Determine the value of \(r\) if \(p=1010\) and \(q=1009\).

Answer: $\frac{1}{2019}$


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

A series of numbers were written: \(100^{100}, 101^{101}, 102^{102}, \ldots, 234^{234}\) (i.e., the numbers of the form \(n^{n}\) for natural \(n\) from 100 to 234). How many of the numbers listed are perfect squares? (A perfect square is defined as the square of an integer.)

Answer: $71$


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

A student correctly added the two two-digit numbers on the left of the board and got the answer 137. What answer will she obtain if she adds the two four-digit numbers on the right of the board?

Answer: $13837$


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

Given \( f(x) = 2^x \) and \( g(x) = \log_{\sqrt{2}} (8x) \), find the value of \( x \) that satisfies \( f[g(x)] = g[f(x)] \).

Answer: $\frac{1 + \sqrt{385}}{64}$


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

Calculate the value of the following expressions:
(1) $$\sqrt[4]{(3-\pi )^{4}}$$ + (0.008) $$^{- \frac {1}{3}}$$ - (0.25) $$^{ \frac {1}{2}}$$ × ($$\frac {1}{ \sqrt {2}}$$)$$^{-4}$$;
(2) $\log_{3}$$\sqrt {27}$$ - $\log_{3}$$\sqrt {3}$$ - $\lg{625}$ - $\lg{4}$ + $\ln(e^{2})$ - $$\frac {4}{3}$$$\lg$$\sqrt {8}$$.

Answer: $-1$


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

If $\sqrt[3]{3}$ is approximated as $1.442$, calculate the value of $\sqrt[3]{3}-3\sqrt[3]{3}-98\sqrt[3]{3}$.

Answer: $-144.2$


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 vectors $\overrightarrow {m}=(\cos\alpha- \frac { \sqrt {2}}{3}, -1)$, $\overrightarrow {n}=(\sin\alpha, 1)$, and $\overrightarrow {m}$ is collinear with $\overrightarrow {n}$, and $\alpha\in[-\pi,0]$.
(Ⅰ) Find the value of $\sin\alpha+\cos\alpha$;
(Ⅱ) Find the value of $\frac {\sin2\alpha}{\sin\alpha-\cos\alpha}$.

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


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>.

At 9:00, a pedestrian set off on a journey. An hour later, a cyclist set off from the same starting point. At 10:30, the cyclist caught up with the pedestrian and continued ahead, but after some time, the bicycle broke down. After repairing the bike, the cyclist resumed the journey and caught up with the pedestrian again at 13:00. How many minutes did the repair take? (The pedestrian's speed is constant, and he moved without stopping; the cyclist's speed is also constant except for the repair interval.)

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>.

The TV station is broadcasting 5 different advertisements, among which there are 3 different commercial advertisements and 2 different Olympic promotional advertisements. The last advertisement must be an Olympic promotional advertisement, and the two Olympic promotional advertisements cannot be broadcast consecutively. Determine the number of different broadcasting methods.

Answer: $36$


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

Given the function $f(x)=\frac{1}{2}{f'}(1){x^2}+lnx+\frac{{f(1)}}{{3x}}$, find the value of ${f'}\left(2\right)$.

Answer: $\frac{33}{4}$


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>.

Beginner millionaire Bill buys a bouquet of 7 roses for $20. Then, he can sell a bouquet of 5 roses for $20 per bouquet. How many bouquets does he need to buy to "earn" a difference of $1000?

Answer: $125$


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

A circle is inscribed in a convex quadrilateral \(ABCD\) with its center at point \(O\), and \(AO=OC\). Additionally, \(BC=5\), \(CD=12\), and \(\angle DAB\) is a right angle.

Find the area of the quadrilateral \(ABCD\).

Answer: $60$


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

Given $\sin\theta= \frac {m-3}{m+5}$ and $\cos\theta= \frac {4-2m}{m+5}$ ($\frac {\pi}{2} < \theta < \pi$), calculate $\tan\theta$.

Answer: $- \frac {5}{12}$


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>.

Find the number of permutations \(a_1, a_2, \ldots, a_{10}\) of the numbers \(1, 2, \ldots, 10\) such that \(a_{i+1}\) is not less than \(a_i - 1\) for \(i = 1, 2, \ldots, 9\).

Answer: $512$


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

Given an ellipse with its foci on the x-axis and its lower vertex at D(0, -1), the eccentricity of the ellipse is $e = \frac{\sqrt{6}}{3}$. A line L passes through the point P(0, 2).
(Ⅰ) Find the standard equation of the ellipse.
(Ⅱ) If line L is tangent to the ellipse, find the equation of line L.
(Ⅲ) If line L intersects the ellipse at two distinct points M and N, find the maximum area of triangle DMN.

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


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

In a similar tournament setup, the top 6 bowlers have a playoff. First #6 bowls #5, and the loser gets the 6th prize. The winner then bowls #4, and the loser of this match gets the 5th prize. The process continues with the previous winner bowling the next highest ranked bowler until the final match, where the winner of this match gets the 1st prize and the loser gets the 2nd prize. How many different orders can bowlers #1 through #6 receive the prizes?

Answer: $32$


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

The sequence  $\{a_i\}_{i \ge 1}$  is defined by  $a_1 = 1$  and \[ a_n = \lfloor a_{n-1} + \sqrt{a_{n-1}} \rfloor \] for all  $n \ge 2$ . Compute the eighth perfect square in the sequence.

*Proposed by Lewis Chen*

Answer: $64$


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

In the Cartesian coordinate system $xOy$, the sum of distances from point $P$ to the two points $(0, -\sqrt{3})$ and $(0, \sqrt{3})$ equals $4$. Let the trajectory of point $P$ be $C$.

$(1)$ Write the equation of $C$;

$(2)$ Suppose the line $y=kx+1$ intersects $C$ at points $A$ and $B$. For what value of $k$ is $\overrightarrow{OA} \bot \overrightarrow{OB}$? What is the value of $|\overrightarrow{AB}|$ at this time?

Answer: $\frac{4\sqrt{65}}{17}$


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 sequence of two distinct numbers is extended in two ways: one to form a geometric progression and the other to form an arithmetic progression. The third term of the geometric progression coincides with the tenth term of the arithmetic progression. With which term of the arithmetic progression does the fourth term of the geometric progression coincide?

Answer: $74$


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

In \(\triangle ABC\), \(AC = AB = 25\) and \(BC = 40\). \(D\) is a point chosen on \(BC\). From \(D\), perpendiculars are drawn to meet \(AC\) at \(E\) and \(AB\) at \(F\). \(DE + DF\) equals:

Answer: $24$


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

Two riders simultaneously departed from points \( A \) and \( C \) towards point \( B \). Despite the fact that \( C \) was 20 km farther from \( B \) than \( A \) was from \( B \), both riders arrived at \( B \) at the same time. Find the distance from \( C \) to \( B \), given that the rider from \( C \) traveled each kilometer 1 minute and 15 seconds faster than the rider from \( A \), and the rider from \( A \) reached \( B \) in 5 hours.

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>.

Given that $α \in (0, \frac{π}{2})$ and $\sin α - \cos α = \frac{1}{2}$, find the value of $\frac{\cos 2α}{\sin (α + \frac{π}{4})}$.

A) $-\frac{\sqrt{2}}{2}$
B) $\frac{\sqrt{2}}{2}$
C) $-1$
D) $1$

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>.

A newly designed car travels 4.2 kilometers further per liter of gasoline than an older model. The fuel consumption for the new car is 2 liters less per 100 kilometers. How many liters of gasoline does the new car consume per 100 kilometers? If necessary, round your answer to two decimal places.

Answer: $5.97$


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

Suppose  $n \ge 0$  is an integer and all the roots of  $x^3 +
\alpha x + 4 - ( 2 \times 2016^n) = 0$  are integers. Find all possible values of  $\alpha$ .

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>.

The government decided to privatize civil aviation. For each of the 127 cities in the country, the connecting airline between them is sold to one of the private airlines. Each airline must make all acquired airlines one-way but in such a way as to ensure the possibility of travel from any city to any other city (possibly with several transfers). What is the maximum number of companies that can buy the airlines?

Answer: $63$


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

An apartment and an office are sold for $15,000 each. The apartment was sold at a loss of 25% and the office at a gain of 25%. Determine the net effect of the transactions.

Answer: $2000$


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

Given the set
$$
M=\{1,2, \cdots, 2020\},
$$
for any non-empty subset $A$ of $M$, let $\lambda_{A}$ be the sum of the maximum and minimum numbers in the subset $A$. What is the arithmetic mean of all such $\lambda_{A}$?

Answer: $2021$


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

A parallelogram $ABCD$ has a base $\overline{BC}$ of 6 units and a height from $A$ to line $BC$ of 3 units. Extend diagonal $AC$ beyond $C$ to point $E$ such that $\overline{CE}$ is 2 units, making $ACE$ a right triangle at $C$. Find the area of the combined shape formed by parallelogram $ABCD$ and triangle $ACE$.

Answer: $18 + 3\sqrt{5}$


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>.

John has 15 marbles of different colors, including one red, one green, one blue, and one yellow marble. In how many ways can he choose 5 marbles, if at least one of the chosen marbles is red, green, or blue, but not yellow?

Answer: $1540$


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

Consider a dark rectangle created by merging two adjacent unit squares in an array of unit squares; part as shown below. If the first ring of squares around this center rectangle contains 10 unit squares, how many unit squares would be in the $100^{th}$ ring?

Answer: $802$


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

Given that both $\alpha$ and $\beta$ are acute angles, and $\sin \alpha = \frac{3}{5}$, $\tan (\alpha - \beta) = -\frac{1}{3}$.

(1) Find the value of $\sin (\alpha - \beta)$;
(2) Find the value of $\cos \beta$.

Answer: $\frac{9\sqrt{10}}{50}$


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>.

A gold coin is worth $x\%$ more than a silver coin. The silver coin is worth $y\%$ less than the gold coin. Both $x$ and $y$ are positive integers. How many possible values for $x$ are there?

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>.

A baker adds 0.45 kilograms of flour to a mixing bowl that already contains 2 3/4 kilograms of flour. How many kilograms of flour are in the bowl now?

Answer: $3.20$


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>.

If the binomial coefficient of only the fourth term in the expansion of $(x^{2} - \frac {1}{2x})^{n}$ is the largest, then the sum of all the coefficients in the expansion is $\boxed{\text{answer}}$.

Answer: $\frac {1}{64}$


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

A certain senior high school has a total of 3200 students, with 1000 students each in the second and third grades. A stratified sampling method is used to draw a sample of size 160. The number of first-grade students that should be drawn is ______ .

Answer: $60$


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

Let $n$ be a positive integer, and let $b_0, b_1, \dots, b_n$ be a sequence of real numbers such that $b_0 = 25$, $b_1 = 56$, $b_n = 0$, and
$$ b_{k+1} = b_{k-1} - \frac{7}{b_k} $$
for $k = 1, 2, \dots, n-1$. Find $n$.

Answer: $201$


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

If line $l_1: ax+2y+6=0$ is parallel to line $l_2: x+(a-1)y+(a^2-1)=0$, then the real number $a=$　.

Answer: $-1$


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

A man asks: "How old are you, my son?"

To this question, I answered:

"If my father were seven times as old as I was eight years ago, then one quarter of my father's current age would surely be fourteen years now. Please calculate from this how many years weigh upon my shoulders!"

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>.

Find the measure of the angle

$$
\delta=\arccos \left(\left(\sin 2907^{\circ}+\sin 2908^{\circ}+\cdots+\sin 6507^{\circ}\right)^{\cos 2880^{\circ}+\cos 2881^{\circ}+\cdots+\cos 6480^{\circ}}\right)
$$

Answer: $63$


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

Two teachers are taking a photo with 3 male students and 3 female students lined up in a row. The teachers can only stand at the ends, and the male students cannot be adjacent. How many different ways are there to arrange the photo?

Answer: $288$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 $1110011_2$ to its decimal equivalent.

Answer: $115$


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

A rectangular table of size \( x \) cm by 80 cm is covered with identical sheets of paper of size 5 cm by 8 cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet is placed in the top-right corner. What is the length \( x \) in centimeters?

Answer: $77$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 four terms of an arithmetic progression, as well as the sum of the first seven terms, are natural numbers. Furthermore, its first term \(a_1\) satisfies the inequality \(a_1 \leq \frac{2}{3}\). What is the greatest value that \(a_1\) can take?

Answer: $9/14$


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

In the isosceles trapezoid $ABCD$, $AD \parallel BC$, $\angle B = 45^\circ$. Point $P$ is on the side $BC$. The area of $\triangle PAD$ is $\frac{1}{2}$, and $\angle APD = 90^\circ$. Find the minimum value of $AD$.

Answer: $\sqrt{2}$


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

Given circle $C$: $x^{2}+y^{2}+8x+ay-5=0$ passes through the focus of parabola $E$: $x^{2}=4y$. The length of the chord formed by the intersection of the directrix of parabola $E$ and circle $C$ is $\_\_\_\_\_\_$.

Answer: $4 \sqrt{6}$


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

Given that \(0 \leq a_{k} \leq 1 \) for \(k=1,2, \ldots, 2020\), and defining \(a_{2021} = a_{1}\), \(a_{2022} = a_{2}\), find the maximum value of \(\sum_{k=1}^{2020}\left(a_{k} - a_{k+1}a_{k+2}\right)\).

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>.

In how many ways can a barrel with a capacity of 10 liters be emptied using two containers with capacities of 1 liter and 2 liters?

Answer: $89$


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

Calculate the limit of the function:

$$
\lim_{x \rightarrow 1} (2-x)^{\sin \left(\frac{\pi x}{2}\right) / \ln (2-x)}
$$

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 the geometric sequence $\{a_n\}$, $a_2a_3=5$ and $a_5a_6=10$. Calculate the value of $a_8a_9$.

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>.

How many five-digit natural numbers are divisible by 9, where the last digit is greater than the second last digit by 2?

Answer: $800$


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

In triangle $ABC$, $D$ is on segment $BC$ such that $BD:DC = 3:2$. Point $E$ bisects $\angle BAC$. Given $BD = 45$ units and $DC = 30$ units, find the length of $BE$.

Answer: $45$


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

To assess the shooting level of a university shooting club, an analysis group used stratified sampling to select the shooting scores of $6$ senior members and $2$ new members for analysis. After calculation, the sample mean of the shooting scores of the $6$ senior members is $8$ (unit: rings), with a variance of $\frac{5}{3}$ (unit: rings$^{2}$). The shooting scores of the $2$ new members are $3$ rings and $5$ rings, respectively. What is the variance of the shooting scores of these $8$ members?

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


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

Point \( D \) lies on side \( CB \) of right triangle \( ABC \left(\angle C = 90^{\circ} \right) \), such that \( AB = 5 \), \(\angle ADC = \arccos \frac{1}{\sqrt{10}}, DB = \frac{4 \sqrt{10}}{3} \). Find the area of triangle \( ABC \).

Answer: $15/4$


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>.

Let  $N$  denote the number of all natural numbers  $n$  such that  $n$  is divisible by a prime  $p> \sqrt{n}$  and  $p<20$ . What is the value of  $N$  ?

Answer: $69$


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

The only prime factors of an integer  $n$  are 2 and 3. If the sum of the divisors of  $n$  (including itself) is  $1815$ , find  $n$ .

Answer: $648$


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

In a $5 \times 18$ rectangle, the numbers from 1 to 90 are placed. This results in five rows and eighteen columns. In each column, the median value is chosen, and among the medians, the largest one is selected. What is the minimum possible value that this largest median can take?

Recall that among 99 numbers, the median is such a number that is greater than 49 others and less than 49 others.

Answer: $54$


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

A truncated pyramid has a square base with a side length of 4 units, and every lateral edge is also 4 units. The side length of the top face is 2 units. What is the greatest possible distance between any two vertices of the truncated pyramid?

Answer: $\sqrt{32}$


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

For a positive integer \( n \), let \( s(n) \) denote the sum of its digits, and let \( p(n) \) denote the product of its digits. If the equation \( s(n) + p(n) = n \) holds true, then \( n \) is called a coincidence number. What is the sum of all coincidence numbers?

Answer: $531$


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

Let $[x]$ represent the greatest integer less than or equal to the real number $x$. Define the sets
$$
\begin{array}{l}
A=\{y \mid y=[x]+[2x]+[4x], x \in \mathbf{R}\}, \\
B=\{1,2, \cdots, 2019\}.
\end{array}
$$

Find the number of elements in the intersection $A \cap B$.

Answer: $1154$


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

The sides of a triangle are 5, 6, and 7. Find the area of the orthogonal projection of the triangle onto a plane that forms an angle equal to the smallest angle of the triangle with the plane of the triangle.

Answer: $\frac{30 \sqrt{6}}{7}$


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

In how many ways can you select two letters from the word "УЧЕБНИК" such that one of the letters is a consonant and the other is a vowel?

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>.

Find the largest value of the parameter \(a\) for which the equation:

\[ (|x-2| + 2a)^2 - 3(|x-2| + 2a) + 4a(3-4a) = 0 \]

has exactly three solutions. Indicate the largest value in the answer.

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>.

Choose one digit from 0, 2, 4, and two digits from 1, 3, 5 to form a three-digit number without repeating digits. The total number of different three-digit numbers that can be formed is (    )

A 36      B 48       C 52       D 54

Answer: $48$


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

An electric company installed a total of 402 poles along both sides of the road, with a distance of 20 meters between each adjacent pole. Later, all poles were replaced, and only 202 poles were installed. The distance between each adjacent pole after the replacement is $\qquad$ meters.

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>.

Construct a square \(A B C D\) with side length \(6 \text{ cm}\). Construct a line \(p\) parallel to the diagonal \(A C\) passing through point \(D\). Construct a rectangle \(A C E F\) such that vertices \(E\) and \(F\) lie on the line \(p\).

Using the given information, calculate the area of rectangle \(A C E F\).

Answer: $36$


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

If $a$, $b$, $c$, $d$, $e$, and $f$ are integers for which $512x^3 + 125 = (ax^2 + bx + c)(dx^2 + ex + f)$ for all $x$, then what is $a^2+b^2+c^2+d^2+e^2+f^2$?

Answer: $6410$


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

The set $\{[x] + [2x] + [3x] \mid x \in \mathbb{R}\} \mid \{x \mid 1 \leq x \leq 100, x \in \mathbb{Z}\}$ has how many elements, where $[x]$ denotes the greatest integer less than or equal to $x$.

Answer: $67$


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

In the trapezoid \(ABCD\) with bases \(AD \parallel BC\), the diagonals intersect at point \(E\). Given the areas \(S(\triangle ADE) = 12\) and \(S(\triangle BCE) = 3\), find the area of the trapezoid.

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>.

In a class at school, all students are the same age, except seven of them who are 1 year younger and two of them who are 2 years older. The sum of the ages of all the students in this class is 330. How many students are in this class?

Answer: $37$


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

If the functions \( f(x) \) and \( g(x) \) are defined on \( \mathbf{R} \), and \( f(x-y)=f(x)g(y)-g(x)f(y) \), with \( f(-2)=f(1) \neq 0 \), what is \( g(1) + g(-1) \)?

Answer: $-1$


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

Given that the cross-section of a cylinder is a square and the height of the cylinder is equal to the diameter of a sphere, calculate the ratio of the total surface area of the cylinder to the surface area of the sphere.

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


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

How many perfect squares less than 20,000 can be represented as the difference of squares of two integers that differ by 2?

Answer: $70$


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

An isosceles triangle and a rectangle both have perimeters of 60 inches. The rectangle's length is twice its width. What is the ratio of the length of one of the equal sides of the triangle to the width of the rectangle? Express your answer as a common fraction.

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


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

In triangle $\triangle ABC$, $\sin A = \frac{3}{5}$, $\tan B = 2$. Find the value of $\tan 2\left(A+B\right)$.

Answer: $\frac{44}{117}$


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

In a right cone with a base radius of \(15\) cm and a height of \(30\) cm, a sphere is inscribed. The radius of the sphere can be expressed as \(b\sqrt{d} - b\) cm. Find the value of \(b + d\).

Answer: $12.5$


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

Given that a ship travels in one direction and Emily walks parallel to the riverbank in the opposite direction, counting 210 steps from back to front and 42 steps from front to back, determine the length of the ship in terms of Emily's equal steps.

Answer: $70$


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

Find the last 3 digits of \(1 \times 3 \times 5 \times 7 \times \cdots \times 2005\).

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>.

Square $PQRS$ has sides of length 1. Points $M$ and $N$ are on $\overline{QR}$ and $\overline{RS},$ respectively, so that $\triangle PMN$ is equilateral. A square with vertex $Q$ has sides that are parallel to those of $PQRS$ and a vertex on $\overline{PM}.$ The length of a side of this smaller square is $\frac{d-\sqrt{e}}{f},$ where $d, e,$ and $f$ are positive integers and $e$ is not divisible by the square of any prime. Find $d+e+f.$

Answer: $12$


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

In triangle \( \triangle ABC \), \( |AB| = 13 \), \( |BC| = 14 \), \( |CA| = 15 \), an internal point \( P \) satisfies
\[
\overrightarrow{BP} \cdot \overrightarrow{CA} = 18 \text{ and } \overrightarrow{CP} \cdot \overrightarrow{BA} = 32.
\]
What is \( \overrightarrow{AP} \cdot \overrightarrow{BC} \)?

Answer: $14$


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

On the extensions of the medians \(A K\), \(B L\), and \(C M\) of triangle \(A B C\), points \(P\), \(Q\), and \(R\) are taken such that \(K P = \frac{1}{2} A K\), \(L Q = \frac{1}{2} B L\), and \(M R = \frac{1}{2} C M\). Find the area of triangle \(P Q R\) if the area of triangle \(A B C\) is 1.

Answer: $25/16$


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

Let function $G(n)$ denote the number of solutions to the equation $\cos x = \sin nx$ on the interval $[0, 2\pi]$. For each integer $n$ greater than 2, what is the sum $\sum_{n=3}^{100} G(n)$?

Answer: $10094$


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

If $k \in [-2, 2]$, find the probability that for the value of $k$, there can be two tangents drawn from the point A(1, 1) to the circle $x^2 + y^2 + kx - 2y - \frac{5}{4}k = 0$.

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>.

On a line, 5 points \( P, Q, R, S, T \) are marked in that order. It is known that the sum of the distances from \( P \) to the other 4 points is 67, and the sum of the distances from \( Q \) to the other 4 points is 34. Find the length of segment \( PQ \).

Answer: $11$


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

Given the numbers \(5^{1971}\) and \(2^{1971}\) written consecutively, what is the number of digits in the resulting number?

Answer: $1972$


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

Starting with a pair of adult rabbits, the number of pairs of adult rabbits after one year can be calculated using the recurrence relation of the Fibonacci sequence, and the result is given by $F(13).

Answer: $233$


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

In the right parallelopiped  $ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime}$ , with  $AB=12\sqrt{3}$  cm and  $AA^{\prime}=18$  cm, we consider the points  $P\in AA^{\prime}$  and  $N\in A^{\prime}B^{\prime}$  such that  $A^{\prime}N=3B^{\prime}N$ . Determine the length of the line segment  $AP$  such that for any position of the point  $M\in BC$ , the triangle  $MNP$  is right angled at  $N$ .

Answer: $27/2$


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 $\sin (α- \frac{π}{3})= \frac{3}{5} $, where $a∈(\frac{π}{4}, \frac{π}{2})$, find $\tan α=\_\_\_\_\_\_\_\_\_$.

Answer: $- \frac{48+25 \sqrt{3}}{11}$


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>.

An infinite geometric series has a first term of \( 512 \) and its sum is \( 8000 \). What is its common ratio?

Answer: $0.936$


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

For the positive integer \( n \), if the expansion of \( (xy - 5x + 3y - 15)^n \) is combined and simplified, and \( x^i y^j \) (where \( i, j = 0, 1, \ldots, n \)) has at least 2021 terms, what is the minimum value of \( n \)?

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>.

Find the sum of all integers  $x$ ,  $x \ge 3$ , such that  $201020112012_x$  (that is,  $201020112012$  interpreted as a base  $x$  number) is divisible by  $x-1$

Answer: $32$


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

Compute
$$
\lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{3}+4 h\right)-4 \sin \left(\frac{\pi}{3}+3 h\right)+6 \sin \left(\frac{\pi}{3}+2 h\right)-4 \sin \left(\frac{\pi}{3}+h\right)+\sin \left(\frac{\pi}{3}\right)}{h^{4}}
$$

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


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

There are nine parts in a bag, including five different genuine parts and four different defective ones. These parts are being drawn and inspected one by one. If the last defective part is found exactly on the fifth draw, calculate the total number of different sequences of draws.

Answer: $480$


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

Given \(a, b, c, x, y, z \in \mathbf{R}_{+}\) satisfying the equations:
\[
cy + bz = a, \quad az + cx = b, \quad bx + ay = c,
\]
find the minimum value of the function
\[
f(x, y, z) = \frac{x^2}{1 + x} + \frac{y^2}{1 + y} + \frac{z^2}{1 + z}.
\]

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


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

What is the smallest positive integer $n$ such that $5n \equiv 2024 \pmod{26}$?

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>.

Allie and Betty play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $g(n)$ points, where \[g(n) = \left\{
\begin{array}{cl}
8 & \text{ if } n \text{ is a multiple of 3 and 4}, \\
3 & \text{ if } n \text{ is only a multiple of 3}, \\
1 & \text{ if } n \text{ is only a multiple of 4}, \\
0 & \text{ if } n \text{ is neither a multiple of 3 nor 4}.
\end{array}
\right.\]
Allie rolls the die four times and gets a 6, 3, 4, and 1. Betty rolls and gets 12, 9, 4, and 2. Compute the product of Allie's total points and Betty's total points.

Answer: $84$


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

A sports team's members have unique numbers taken from the integers 1 to 100. If no member's number is the sum of the numbers of any two other members, nor is it twice the number of any other member, what is the maximum number of people this team can have?

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>.

Petya and Vasya competed in a 60-meter race. When Petya finished, Vasya was 9 meters behind him. In the second race, Petya stood exactly 9 meters behind Vasya at the starting point. Who finished first in the second race and by how many meters did he lead his opponent? (Assume that each boy ran both times at his own constant speed).

Answer: $1.35$