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

Assume that  $f(a+b) = f(a) + f(b) + ab$ , and that  $f(75) - f(51) = 1230$ .  Find  $f(100)$ .

Answer: $3825$


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

The binomial expansion of $((x-1)^{n}(n∈N_{+}))$ has only the coefficient of the fourth term the largest, find the constant term in the binomial expansion of $(2 \sqrt {x}- \frac{1}{ \sqrt {x}})^{n}$.

Answer: $-160$


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

Find the product of all positive integral values of $n$ such that $n^2 - 21n + 110 = p$ for some prime number $p$. Assume there is at least one such $n$.

Answer: $108$


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

To protect the ecological environment, a mountainous area in our city has started to implement the policy of returning farmland to forest since 2005. It is known that at the end of 2004, the forest coverage area of this mountainous area was $a$ acres.
(1) Assuming the annual natural growth rate of forest coverage area after returning farmland to forest is 2%, write the function relationship between the forest coverage area $y$ (in acres) and the number of years $x$ (in years) since the implementation of returning farmland to forest, and calculate the forest coverage area of this mountainous area at the end of 2009.
(2) If by the end of 2014, the forest coverage area of this mountainous area needs to be at least twice that at the end of 2004, additional artificial greening projects must be implemented. What is the minimum annual average growth rate of the forest coverage area required to meet this goal by the end of 2014?
(Reference data: $1.02^{4}=1.082$, $1.02^{5}=1.104$, $1.02^{6}=1.126$, $\lg2=0.301$, $\lg1.072=0.0301$)

Answer: $7.2\%$


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

In the set of numbers 1, 2, 3, 4, 5, select an even number a and an odd number b to form a vector $\overrightarrow{a} = (a, b)$ with the origin as the starting point. From all the vectors obtained with the origin as the starting point, select any two vectors as adjacent sides to form a parallelogram. Let the total number of parallelograms formed be n, and among them, let the number of parallelograms with an area not exceeding 4 be m. Calculate the value of $\frac{m}{n}$.

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


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

Given that D is a point on the hypotenuse BC of right triangle ABC, and $AC= \sqrt {3}DC$, $BD=2DC$. If $AD=2 \sqrt {3}$, then $DC=\_\_\_\_\_\_$.

Answer: $\sqrt {6}$


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

In acute triangle $\text{ABC}$, the lengths of the sides opposite to angles $\text{A}$, $\text{B}$, and $\text{C}$ are $a$, $b$, and $c$ respectively, and $a = 2b\sin{\text{A}}$.

(II) Find the measure of angle $\text{B}$;

(III) If $a = 3\sqrt{3}$ and $c = 5$, find $b$.

Answer: $\sqrt{7}$


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

Let \( a, b, c \) be non-negative real numbers satisfying
\[ a + b + c = 8 \]
and
\[ ab + bc + ca = 16.\]
Let \( m = \min \{ ab, bc, ca \} \). Find the maximum possible value of \( m \).

Answer: $16/9$


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>.

In right triangle $DEF$, where $DE=15$, $DF=9$, and $EF=12$ units. What is the distance from $F$ to the midpoint of segment $DE$?

Answer: $7.5$


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>.

Kiana has two older twin siblings, and their ages, together with her own, multiply together to equal 162. Calculate the sum of their ages.

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>.

To implement the "double reduction" policy and promote intangible cultural heritage, a certain school plans to organize extracurricular interest classes for students to visit the following projects: $A$ (Xiu Shui pottery art), $B$ (Xiu Shui tea-picking opera), $C$ (Jiujiang mountain song), $D$ (Dean Pan opera). Xiao Ming and Xiao Han randomly sign up to visit one of them.
$(1)$ The event "Xiao Ming visits Jiujiang mountain song" is ________; (Please fill in the number of the correct answer on the line)
① a certain event;
② an impossible event;
③ a random event.
$(2)$ Please calculate the probability that both Xiao Ming and Xiao Han will visit intangible cultural heritage projects in Xiu Shui using a list or tree diagram method.

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>.

The Ebbinghaus forgetting curve describes the law of human brain forgetting new things. Based on this, a research team found that after learning course $A$, 20% of the memorized content is forgotten every week. In order to ensure that the memorized content does not fall below $\frac{1}{12}$, the content needs to be reviewed after $n$ ($n\in N$) weeks. Find the value of $n$. ($\lg 3\approx 0.477$, $\lg 2\approx 0.3$)

Answer: $10$


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

In quadrilateral $ABCD$ with diagonals $AC$ and $BD$, intersecting at $O$, suppose $BO=5$, $OD = 7$, $AO=9$, $OC=4$, and $AB=7$. The length of $AD$ is:
**A)** $15$
**B)** $\sqrt{210}$
**C)** $14$
**D)** $\sqrt{220}$
**E)** $13$

Answer: $\sqrt{210}$


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

Given $ \frac {\pi}{2}\leqslant \beta\leqslant \alpha\leqslant \frac {3\pi}{4} $, $ \cos (\alpha-\beta) = \frac {12}{13} $, $ \sin (\alpha+\beta) = -\frac {3}{5} $, find the values of $ \sin 2\alpha $ and $ \cos 2\beta $.

Answer: $-\frac{63}{65}$


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

Given that a bin contains 10 kg of peanuts, 2 kg of peanuts are removed and 2 kg of raisins are added and thoroughly mixed in, and then 2 kg of this mixture are removed and 2 kg of raisins are added and thoroughly mixed in again, determine the ratio of the mass of peanuts to the mass of raisins in the final mixture.

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


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>.

A square sheet of paper  $ABCD$  is folded straight in such a way that point  $B$  hits to the midpoint of side  $CD$ . In what ratio does the fold line divide side  $BC$ ?

Answer: $5/3$


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 Maplewood Youth Soccer Team consists of 25 players, including four who are goalies. During a special drill, each goalie takes a turn in the net while the other 24 players (including the remaining goalies) attempt penalty kicks.
Calculate the total number of penalty kicks that must be taken so each player has the opportunity to take a penalty kick against each goalie.

Answer: $96$


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

Calculate the value of $\displaystyle\frac{245^2-225^2}{20}$.

Answer: $470$


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

Given $f(x) = (m^2+2m)x^{m^2+m-1}$, for what value(s) of $m$ does:
(Ⅰ) $f(x)$ become a power function;
(Ⅱ) $f(x)$ become a direct proportionality function;
(Ⅲ) $f(x)$ become an inverse proportionality function.

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>.

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

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


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

Thomas has constant speeds for both running and walking. When a down-escalator is moving, Thomas can run down it in 15 seconds or walk down it in 30 seconds. One day, when the escalator was broken (and stationary), it took Thomas 20 seconds to run down it. How long, in seconds, would it take Thomas to walk down the broken escalator?

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(x + $\frac{π}{4}$) = $\frac{1}{3}$, find the value of sin4x - 2cos3xsinx = ___.

Answer: $-\frac{7}{9}$


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>.

In the ancient Chinese mathematical work "Nine Chapters on the Mathematical Art," there is a problem as follows: "There is a golden rod in China, five feet long. When one foot is cut from the base, it weighs four catties. When one foot is cut from the end, it weighs two catties. How much does each foot weigh in succession?" Based on the given conditions of the previous question, if the golden rod changes uniformly from thick to thin, estimate the total weight of this golden rod to be approximately ____ catties.

Answer: $15$


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

Given triangle PQR with PQ = 60 and PR = 20, the area is 240. Let M be the midpoint of PQ and N be the midpoint of PR. An altitude from P to side QR intersects MN and QR at X and Y, respectively. Find the area of quadrilateral XYMR.

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>.

Three fair coins are tossed once. For each head that results, one fair die is rolled. Determine the probability that the sum of the results of the die rolls is odd.

Answer: $\frac{7}{16}$


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

A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside the cylinder?

Answer: $2\sqrt{61}$


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

Given $m>0$ and $|x+1|+|2x-1|\geq m$ always holds, $a, b, c \in \mathbb{R}$ satisfy $a^2+2b^2+3c^2=m$. Find the minimum value of $a+2b+3c$.

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>.

A board of size \(2022 \times 2022\) is given. Lisa and Varya take turns painting \(2 \times 2\) squares on the board with red and blue colors. They agreed that each cell can be painted no more than once in blue and no more than once in red. Cells that are painted blue and then red (or vice versa) become purple. Once all cells are painted, the girls count how many of them are purple. What counts could they have gotten?

Options:
- \(2022 \times 2022\)
- \(2022 \times 2020\)
- \(2021 \times 2022\)
- \(2021 \times 2020\)

Answer: $2021 \cdot 2020$


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

In $\triangle ABC$, $\tan A = \frac{1}{4}$ and $\tan B = \frac{3}{5}$.
(1) Find the measure of angle $C$;
(2) If the shortest side length of $\triangle ABC$ is $\sqrt{2}$, find the area of $\triangle ABC$.

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>.

In the Cartesian coordinate system $(xOy)$, the equation of circle $C$ is $((x-4)^{2}+y^{2}=1)$. If there exists at least one point on the line $y=kx-3$ such that a circle with this point as the center and $2$ as the radius intersects with circle $C$, then the maximum value of $k$ is _____.

Answer: $\frac{24}{7}$


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

Determine the value of
\[
3003 + \frac{1}{3} \left( 3002 + \frac{1}{3} \left( 3001 + \dots + \frac{1}{3} \left( 4 + \frac{1}{3} \cdot 3 \right) \right) \dotsb \right).
\]

Answer: $4503.75$


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

Two circles have centers at (1,3) and (4,1) respectively. A line is tangent to the first circle at point (4,6) and to the second circle at point (7,4). Find the slope of the tangent line at these points.

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 a frustum of a cone with the areas of its two bases being $4\pi$ and $25\pi$ respectively, and the height of the frustum is 4, find the volume and the lateral surface area of the frustum.

Answer: $35\pi$


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

The value of  $ 21!$  is  $ 51{,}090{,}942{,}171{,}abc{,}440{,}000$ , where  $ a$ ,  $ b$ , and  $ c$  are digits.  What is the value of  $ 100a \plus{} 10b \plus{} c$ ?

Answer: $709$


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

In triangle ABC, the lengths of the sides opposite to angles A, B, and C are a, b, and c, respectively.
1. If c = 2, $C = \frac{\pi}{3}$, and the area of triangle ABC is equal to $\sqrt{3}$, find cos(A + B) and the values of a and b.
2. If B is an obtuse angle, and $cosA = \frac{3}{5}, sinB = \frac{12}{13}$, find the value of sinC.

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


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

A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common fraction.

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>.

The "One Helmet, One Belt" safety protection campaign is a safety protection campaign launched by the Ministry of Public Security nationwide. It is also an important standard for creating a civilized city and being a civilized citizen. "One helmet" refers to a safety helmet. Drivers and passengers of electric bicycles should wear safety helmets. A certain shopping mall intends to purchase a batch of helmets. It is known that purchasing 8 type A helmets and 6 type B helmets costs $630, and purchasing 6 type A helmets and 8 type B helmets costs $700.
$(1)$ How much does it cost to purchase 1 type A helmet and 1 type B helmet respectively?
$(2)$ If the shopping mall is prepared to purchase 200 helmets of these two types, with a total cost not exceeding $10200, and sell type A helmets for $58 each and type B helmets for $98 each. In order to ensure that the total profit is not less than $6180, how many purchasing plans are there? How many type A and type B helmets are in the plan with the maximum profit? What is the maximum profit?

Answer: $6200$


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

$ABCD$ is a trapezoid with $AB \parallel CD$, $AB=6$, and $CD=15$. If the area of $\triangle AED=30$, what is the area of $\triangle AEB?$

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 the diagram, the number line between 0 and 2 is divided into 8 equal parts. The numbers 1 and \(S\) are marked on the line. What is the value of \(S\)?

Answer: $1.25$


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

An isosceles triangle $DEF$ has $DE = DF = 5\sqrt{3}$, and a circle with radius $3\sqrt{3}$ is tangent to line $DE$ at $E$ and to line $DF$ at $F$. What is the area of the circle that passes through vertices $D$, $E$, and $F$?
A) $63\pi$
B) $48\pi$
C) $72\pi$
D) $36\pi$
E) $54\pi$

Answer: $48\pi$


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

Given a geometric sequence $\{a_n\}$ where the sum of the first $n$ terms $S_n = a \cdot 3^n - 2$, find the value of $a_2$.

Answer: $12$


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

Given that the sum of the first three terms of a geometric sequence $\{a_n\}$ is $3$ and the sum of the first nine terms is $39$, calculate the value of the sum of the first six terms.

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>.

Determine the time the copy machine will finish all the paperwork if it starts at 9:00 AM and completes half the paperwork by 12:30 PM.

Answer: $4:00$


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

Consider a large square where each side is divided into four equal parts. At each division, a point is placed. An inscribed square is constructed such that its corners are at these division points nearest to the center of each side of the large square. Calculate the ratio of the area of the inscribed square to the area of the large square.
A) $\frac{1}{4}$
B) $\frac{1}{2}$
C) $\sqrt{2}$
D) $\frac{3}{4}$
E) $1$

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


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

Given that there are 10 light bulbs in a box, with 8 of good quality and 2 defective, calculate P(ξ=4), where ξ is the number of draws to draw 2 good quality bulbs, drawing one at a time without replacement.

Answer: $\frac{1}{15}$


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

Given the functions $f(x)= \begin{cases} 2^{x-2}-1,x\geqslant 0 \\ x+2,x < 0 \end{cases}$ and $g(x)= \begin{cases} x^{2}-2x,x\geqslant 0 \\ \frac {1}{x},x < 0. \end{cases}$, find the sum of all the zeros of the function $f[g(x)]$.

Answer: $\frac{1}{2} + \sqrt{3}$


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

In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $$\begin{cases} x= \sqrt {3}- \frac { \sqrt {3}}{2}t \\ y= \frac {1}{2}t \end{cases}$$ (where $t$ is the parameter). In the polar coordinate system with the origin as the pole and the positive x-axis as the polar axis, the polar equation of curve $C$ is $$\rho=2 \sqrt {3}\sin\theta$$.
(1) Write the Cartesian coordinate equation of curve $C$;
(2) Given that the intersection of line $l$ with the x-axis is point $P$, and its intersections with curve $C$ are points $A$ and $B$. If the midpoint of $AB$ is $D$, find the length of $|PD|$.

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


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>.

Calculate the value of $\left(\sum_{k=1}^{10} \log_{4^k} 2^{k^2}\right)\cdot\left(\sum_{k=1}^{50} \log_{16^k} 64^k\right)$.

Answer: $2062.5$


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>.

In $\Delta ABC$, $\angle A$ satisfies the condition $\sqrt{3}\sin A+\cos A=1$, and $AB=2$, $BC=2\sqrt{3}$. Determine the area $S=$_______ of $\Delta ABC$.

Answer: $\sqrt{3}$


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

Given the hyperbola $\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1$ ($a>0$, $b>0$), a line passing through the origin with an inclination angle of $\frac {\pi}{3}$ intersects the left and right branches of the hyperbola at points P and Q respectively. A circle with the segment PQ as its diameter passes through the right focus F. Find the eccentricity of the hyperbola.

Answer: $\sqrt{(\sqrt{3}+1)^2}$


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

Determine the number of pairs of positive integers  $x,y$  such that  $x\le y$ ,  $\gcd (x,y)=5!$  and  $\text{lcm}(x,y)=50!$ .

Answer: $16384$


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

A person can commute by train or car. If he takes the train to work in the morning, he takes the car in the afternoon; if he takes the train home in the afternoon, he takes the car in the morning. Over $x$ days, this person took the train 9 times, took the car in the morning 8 times, and took the car in the afternoon 15 times. What is the value of $x$?
(26th American High School Mathematics Exam, 1975)

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>.

There are  $64$  booths around a circular table and on each one there is a chip. The chips and the corresponding booths are numbered  $1$  to  $64$  in this order. At the center of the table there are  $1996$  light bulbs which are all turned off. Every minute the chips move simultaneously in a circular way (following the numbering sense) as follows: chip  $1$  moves one booth, chip  $2$  moves two booths, etc., so that more than one chip can be in the same booth. At any minute, for each chip sharing a booth with chip  $1$  a bulb is lit. Where is chip  $1$  on the first minute in which all bulbs are lit?

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>.

A school arranges for five people, \( A \), \( B \), \( C \), \( D \), and \( E \), to enter into three classes, with each class having at least one person, and \( A \) and \( B \) cannot be in the same class. Calculate the total number of different arrangements.

Answer: $114$


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

Given the polar equation of curve $C$ is $\rho-4\sin \theta=0$. With the pole as the origin and the positive half-axis of the $x$-axis as the polar axis, a Cartesian coordinate system is established. Line $l$ passes through point $M(1,0)$ with an inclination angle of $\dfrac{3\pi}{4}$.
$(1)$ Find the Cartesian equation of curve $C$ and the parametric equation of line $l$;
$(2)$ Suppose line $l$ intersects curve $C$ at points $A$ and $B$, calculate $|MA|+|MB|$.

Answer: $3\sqrt{2}$


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

**Q13.** Determine the greatest value of the sum  $M=11xy+3x+2012yz$ , where  $x,y,z$  are non negative integers satisfying condition  $x+y+z=1000.$

Answer: $503000000$


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

How many four-digit positive integers are divisible by both 12 and 20, but are not divisible by 16?

Answer: $113$


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

The Greater Eighteen Hockey League has three divisions, with six teams in each division. Each team plays each of the other teams in its own division three times and every team in the other divisions twice. How many league games are scheduled?

Answer: $351$


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

Given points $S$, $A$, $B$, $C$ on the surface of a sphere $O$, $SA \perp$ plane $ABC$, $AB \perp BC$, $SA = AB = 1$, $BC = \sqrt{2}$, calculate the surface area of sphere $O$.

Answer: $4\pi$


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

Calculate $7 \cdot 9\frac{2}{5}$.

Answer: $65\frac{4}{5}$


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>.

Distribute 7 students into two dormitories, A and B, with each dormitory having at least 2 students. How many different distribution plans are there?

Answer: $112$


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

Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 8.\]

Answer: $\frac{89}{9}$


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>.

A line passing through point $P(-2,2)$ intersects the hyperbola $x^2-2y^2=8$ such that the midpoint of the chord $MN$ is exactly at $P$. Find the length of $|MN|$.

Answer: $2 \sqrt{30}$


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

Given the definition that an arithmetic sequence is a sequence in which the difference between any term and its preceding term is a constant. For example, the sequence $2, 4, 6, 8, \ldots$ is an arithmetic sequence. A second-order arithmetic sequence is defined as a sequence in which the differences between successive terms form a new sequence that is an arithmetic sequence itself. For instance, the sequence $1, 3, 7, 13, 21, \ldots$ is a second-order arithmetic sequence since the differences between successive terms form the sequence $2, 4, 6, 8, \ldots$, which is an arithmetic sequence.
1. Find the 100th term and the sum of the first 100 terms of the arithmetic sequence $2, 4, 6, 8, \ldots$.
2. Find the 6th term of the second-order arithmetic sequence $1, 3, 7, 13, 21, \ldots$.
3. Find the 2013th term of the second-order arithmetic sequence $1, 3, 7, 13, 21, \ldots$.

Answer: $4050157$


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

Triangle \( ABC \) has \( AB=24 \), \( AC=26 \), and \( BC=22 \). Points \( D \) and \( E \) are located on \( \overline{AB} \) and \( \overline{AC} \), respectively, so that \( \overline{DE} \) is parallel to \( \overline{BC} \) and contains the center of the inscribed circle of triangle \( ABC \). Calculate \( DE \) and express it in the simplest form.

Answer: $\frac{275}{18}$


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>.

How does the area of the quadrilateral formed by connecting the midpoints of the sides of a concave quadrilateral compare to the area of the original quadrilateral?

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 probability that Claudia gets at least 2 more heads than tails if she flips 12 coins?

Answer: $\frac{793}{2048}$


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

How many four-digit integers are divisible by both 7 and 5?

Answer: $257$


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

Given that a floor is tiled in a similar pattern with a $4 \times 4$ unit repeated pattern and each of the four corners looks like the scaled down version of the original, determine the fraction of the tiled floor made up of darker tiles, assuming symmetry and pattern are preserved.

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>.

Calculate:
$(1)-4\times 9$;
$(2)10-14-\left(-5\right)$;
$(3)-3×(-\frac{1}{3})^3$;
$(4)-56+(-8)×(\frac{1}{8})$.

Answer: $-57$


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

Points  $ A$  and  $ B$  lie on a circle centered at  $ O$ , and  $ \angle AOB=60^\circ$ . A second circle is internally tangent to the first and tangent to both  $ \overline{OA}$  and  $ \overline{OB}$ . What is the ratio of the area of the smaller circle to that of the larger circle?

Answer: $\frac{1}{9}$


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

Given a cone whose vertex and the circumference of its base are both on the same sphere, if the radius of the sphere is $1$, then when the volume of the cone is maximized, the height of the cone is ______.

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


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

Assign five volunteers randomly to three different positions A, B, and C for service, with each position having at least one volunteer.
(1) Calculate the probability that exactly two volunteers are assigned to position A.
(2) Let the random variable $\xi$ represent the number of these five volunteers serving at position A, find the probability distribution of $\xi$.

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


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

Find the smallest number, written using only ones and zeros, that would be divisible by 225.

Answer: $11111111100$


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

A point $Q$ is chosen at random in the interior of equilateral triangle $DEF$. What is the probability that $\triangle DEQ$ has a greater area than each of $\triangle DFQ$ and $\triangle EFQ$?

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


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

Given a geometric sequence $\{a_n\}$ composed of positive numbers, with a common ratio $q=2$, and $a_1 \cdot a_2 \cdot a_3 \cdot \ldots \cdot a_{30} = 2^{45}$, calculate the product of the terms $a_1 \cdot a_4 \cdot a_7 \cdot \ldots \cdot a_{28}$.

Answer: $2^5$


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

What is the value of $\dfrac{\sqrt[5]{11}}{\sqrt[7]{11}}$ expressed as 11 raised to what power?

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


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

The minimum possible sum of the three dimensions of a rectangular box with a volume of $1729$ in$^3$.

Answer: $39$


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

In the Cartesian coordinate system xOy, the parametric equations of line l are given by: $$\begin{cases}x=1+t\cos α \\ y=2+t\sin α\end{cases}$$ (t is a parameter, 0≤a<π). Establish a polar coordinate system with O as the pole and the positive semi-axis of the x-axis as the polar axis. The polar equation of curve C is ρ=6sinθ.

1. Find the Cartesian equation of curve C.
2. If point P(1, 2), and curve C intersects with line l at points A and B, find the minimum value of |PA| + |PB|.

Answer: $2\sqrt{7}$


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

Let event $A$ be "Point $M(x,y)$ satisfies $x^{2}+y^{2}\leqslant a(a > 0)$", and event $B$ be "Point $M(x,y)$ satisfies $\begin{cases} & x-y+1\geqslant 0 \\ & 5x-2y-4\leqslant 0 \\ & 2x+y+2\geqslant 0 \end{cases}$. If $P(B|A)=1$, then find the maximum value of the real number $a$.

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


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>.

Jacqueline has 2 liters of soda. Liliane has 60% more soda than Jacqueline, and Alice has 40% more soda than Jacqueline. Calculate the percentage difference between the amount of soda Liliane has compared to Alice.

Answer: $14.29\%$


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

Find the minimum distance from a point on the curve y=e^{x} to the line y=x.

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>.

Find the number of positive integers \( n \) that satisfy
\[
(n - 1)(n - 3)(n - 5) \dotsm (n - 99) < 0.
\]

Answer: $25$


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

Determine the binomial coefficient and the coefficient of the 4th term in the expansion of $\left( \left. x^{2}- \frac{1}{2x} \right. \right)^{9}$.

Answer: $- \frac{21}{2}$


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

Given the function $f(x)=4\sin({8x-\frac{π}{9}})$, $x\in \left[0,+\infty \right)$, determine the initial phase of this harmonic motion.

Answer: $-\frac{\pi}{9}$


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>.

In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a\cos B - b\cos A = c$, and $C = \frac{π}{5}$, calculate the value of $\angle B$.

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


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

Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to the real number $x$, such as $\lfloor 3.2 \rfloor = 3$, $\lfloor -4.5 \rfloor = -5$. The area of the shape formed by points $(x, y)$ on the plane that satisfy $\lfloor x \rfloor^2 + \lfloor y \rfloor^2 = 50$ is.

Answer: $12$


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

Given that the point \( P(x, y) \) satisfies the equation \( (x-4 \cos \theta)^{2}+(y-4 \sin \theta)^{2}=4(\theta \in \mathbf{R}) \), find the area of the region where the point \( P(x, y) \) can be located.

Answer: $32\pi$


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

What are the rightmost three digits of $7^{1984}$?

Answer: $401$


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

Given that $θ$ is an angle in the second quadrant, if $\tan \left(θ+ \frac {π}{4}\right)= \frac {1}{2}$, calculate the value of $\sin θ-\cos θ$.

Answer: $\frac {2 \sqrt {10}}{5}$


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

Let $1$; $5$; $9$; $\ldots$ and $8$; $15$; $22$; $\ldots$ be two arithmetic progressions. The set $S$ is the union of the first $2100$ terms of each sequence. How many distinct numbers are in $S$?
A) 3800
B) 3900
C) 4000
D) 4100
E) 4200

Answer: $3900$


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

Given three points $A$, $B$, and $C$ in the Cartesian coordinate system that lie on the same straight line, where $\overrightarrow{OA}=(-2, m)$, $\overrightarrow{OB}=(n, 1)$, $\overrightarrow{OC}=(5, -1)$, and $OA \perp OB$, with $O$ being the origin of the coordinate system.

(I) Find the values of the real numbers $m$ and $n$.
(II) Let $G$ be the centroid of $\triangle OAC$, and assume $\overrightarrow{OB}=\frac{3}{2}\overrightarrow{OG}$. Find the area of $\triangle AOC$.

Answer: $\frac{13}{2}$


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>.

$A$ and $B$ are two distinct points on the parabola $y=3-x^{2}$ that are symmetric with respect to the line $x+y=0$. Find $|AB|$.

Answer: $3 \sqrt{2}$


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

Given $a \gt 0$, $b \gt 0$, if ${a}^{2}+{b}^{2}-\sqrt{3}ab=1$, determine the maximum value of $\sqrt{3}{a}^{2}-ab$.

Answer: $2 + \sqrt{3}$


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

There are 7 line segments with integer lengths in centimeters: $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$, satisfying $a_1 < a_2 < a_3 < a_4 < a_5 < a_6 < a_7$, and any three of these line segments cannot form a triangle. If $a_1 = 1$ cm and $a_7 = 21$ cm, then $a_6 = \ $.

Answer: $13$


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

Given that $0 < α < \dfrac {π}{2}$, and $\cos (2π-α)-\sin (π-α)=- \dfrac { \sqrt {5}}{5}$,
(1) Find the value of $\sin α+\cos α$;
(2) Find the value of $\sin (2α- \dfrac {π}{4})$.

Answer: $\dfrac {7 \sqrt {2}}{10}$


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

Given the function $f(x)= \begin{cases} \left( \frac {1}{2}\right)^{x} & (x\geqslant 4) \\ f(x+1) & (x < 4)\end{cases}$, find the value of $f(\log _{2}3)$.

Answer: $\frac{1}{24}$


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>.

Eight students from Adams school worked for $4$ days, six students from Bentley school worked for $6$ days, and seven students from Carter school worked for $10$ days. If a total amount of $\ 1020$ was paid for the students' work, with each student receiving the same amount for a day's work, determine the total amount earned by the students from Carter school.

Answer: $517.39$


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

If the line \(x=\frac{\pi}{4}\) intersects the curve \(C: (x-\arcsin a)(x-\arccos a) + (y-\arcsin a)(y+\arccos a)=0\), determine the minimum value of the chord length as \(a\) varies.

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


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

Find the minimum positive period of the function \( f(x) = |\tan 2x| \).

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


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

Compute the sum \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\]

Answer: $75$


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

Given two points $A(-2,0)$ and $B(0,2)$, and point $C$ is any point on the circle $x^{2}+y^{2}-2x=0$, find the minimum area of $\triangle ABC$.

Answer: $3 - \sqrt{2}$


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

Given that $\theta$ is an angle in the second quadrant and $\tan(\theta + \frac{\pi}{4}) = \frac{1}{2}$, find the value of $\sin\theta + \cos\theta$.

Answer: $-\frac{\sqrt{10}}{5}$


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

The total price of 4 floppy disks and 5 CDs is not less than 20 yuan, and the total price of 6 floppy disks and 3 CDs is not more than 24 yuan. What is the minimum amount of money needed to buy 3 floppy disks and 9 CDs?

Answer: $22$


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

The first term of a geometric sequence is 250. If the sum of the first 50 terms is 625 and the sum of the first 100 terms is 1225, find the sum of the first 150 terms.

Answer: $1801$


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

Given that $W,X,Y$ and $Z$ are four different digits selected from the set ${1,2,3,4,5,6,7,8,9}$, find the value of $\frac{W}{X} + \frac{Y}{Z}$ when it is as small as possible.

Answer: $\frac{25}{72}$


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

In $\triangle ABC$, $AB=3$, $AC=2$, $\angle BAC=60^{\circ}$, $D$ is the midpoint of $BC$, $\cos \angle BAD=$ __________.

Answer: $\frac{4\sqrt{19}}{19}$


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>.

Given that $\triangle ABC$ is an acute triangle, vector $\overrightarrow{m}=(\cos (A+ \frac{\pi}{3}),\sin (A+ \frac{\pi}{3}))$, $\overrightarrow{n}=(\cos B,\sin B)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
(I) Find the value of $A-B$;
(II) If $\cos B= \frac{3}{5}$, and $AC=8$, find the length of $BC$.

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


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

Given unit vectors $a$ and $b$ satisfying $|a+3b|=\sqrt{13}$, find the angle between $a$ and $b$.

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


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

Create three-digit numbers without repeating digits using the numbers 0, 1, 2, 3, 4, 5:
(1) How many of them have a ones digit smaller than the tens digit?
(2) How many of them are divisible by 5?

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 an even function $f(x)$ defined on $\mathbb{R}$, for $x \geq 0$, $f(x) = x^2 - 4x$
(1) Find the value of $f(-2)$;
(2) For $x < 0$, find the expression for $f(x)$;
(3) Let the maximum value of the function $f(x)$ on the interval $[t-1, t+1]$ (where $t > 1$) be $g(t)$, find the minimum value of $g(t)$.

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>.

An equilateral triangle  $ABC$  is divided by nine lines parallel to  $BC$  into ten bands that are equally wide. We colour the bands alternately red and blue, with the smallest band coloured red. The difference between the total area in red and the total area in blue is  $20$   $\text{cm}^2$ .

What is the area of triangle  $ABC$ ?

Answer: $200$


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

Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$) with its left focus at F and the eccentricity $e = \frac{\sqrt{2}}{2}$, the line segment cut by the ellipse from the line passing through F and perpendicular to the x-axis has length $\sqrt{2}$.
(Ⅰ) Find the equation of the ellipse.
(Ⅱ) A line $l$ passing through the point P(0,2) intersects the ellipse at two distinct points A and B. Find the length of segment AB when the area of triangle OAB is at its maximum.

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>.

What is the number of square units in the area of the octagon below?

[asy]
unitsize(0.5cm);
defaultpen(linewidth(0.7)+fontsize(10));
dotfactor = 4;
int i,j;
for(i=0;i<=5;++i)

{

for(j=-4;j<=4;++j)

{

dot((i,j));

}

}

for(i=1;i<=5;++i)

{

draw((i,-1/3)--(i,1/3));

}
for(j=1;j<=4;++j)

{

draw((-1/3,j)--(1/3,j));

draw((-1/3,-j)--(1/3,-j));

}

real eps = 0.2;

draw((4,4.5+eps)--(4,4.5-eps));
draw((5,4.5+eps)--(5,4.5-eps));
draw((4,4.5)--(5,4.5));

label("1 unit",(4.5,5));

draw((5.5-eps,3)--(5.5+eps,3));
draw((5.5-eps,4)--(5.5+eps,4));
draw((5.5,3)--(5.5,4));

label("1 unit",(6.2,3.5));

draw((-1,0)--(6,0));
draw((0,-5)--(0,5));
draw((0,0)--(1,4)--(4,4)--(5,0)--(4,-4)--(1,-4)--cycle,linewidth(2));
[/asy]

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>.

Explore the pattern, observe the following equations, and answer the questions.
$1=1^{2}$;
$1+3=4=2^{2}$;
$1+3+5=9=3^{2}$;
$1+3+5+7=16=4^{2}$;
$1+3+5+7+9=25=5^{2}$;
$\ldots $
$(1)$ Please guess $1+3+5+7+9+\ldots +21=$______;
$(2)$ Please guess $1+3+5+7+9+\ldots +(2n-1)=\_\_\_\_\_\_;(n$ is an integer and $n\geqslant 1)$
$(3)$ Calculate: $201+203+\ldots +297+299$.

Answer: $12500$


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

1. Calculate $\log _{2.5}6.25+ \lg 0.01+ \ln \sqrt {e}-2\,^{1+\log _{2}3}$
2. Given $\tan \alpha=-3$, and $\alpha$ is an angle in the second quadrant, find $\sin \alpha$ and $\cos \alpha$.

Answer: $- \frac { \sqrt {10}}{10}$


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

The sum of the digits of the positive integer \(N\) is three times the sum of the digits of \(N+1\). What is the smallest possible sum of the digits of \(N\)?

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 \\(\alpha\\) be an acute angle. If \\(\sin \left(\alpha+ \frac {\pi}{6}\right)= \frac {3}{5}\\), then \\(\cos \left(2\alpha- \frac {\pi}{6}\right)=\\) ______.

Answer: $\frac {24}{25}$


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

Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$ and satisfies $f(x+2)=f(x)$ for all $x \in \mathbb{R}$, and when $x \in (-1, 0)$, $f(x)=2^x$, find the value of $f(\log_2 5)$.

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


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>.

Let $\triangle PQR$ be a right triangle such that $Q$ is a right angle. A circle with diameter $QR$ intersects side $PR$ at $S$. If $PS = 2$ and $QS = 9$, find the length of $RS$.

Answer: $40.5$


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>.

In a small reserve, a biologist counted a total of 300 heads comprising of two-legged birds, four-legged mammals, and six-legged insects. The total number of legs counted was 980. Calculate the number of two-legged birds.

Answer: $110$


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

Given two points A and B on a number line, their distance is 2, and the distance between point A and the origin O is 3. Then, the sum of all possible distances between point B and the origin O equals to     .

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 fair six-sided die with faces numbered 1 to 6 is rolled twice. Let $a$ and $b$ denote the numbers obtained in the two rolls.

1. Find the probability that $a + b \geq 9$.
2. Find the probability that the line $ax + by + 5 = 0$ is tangent to the circle $x^2 + y^2 = 1$.
3. Find the probability that the lengths $a$, $b$, and $5$ form an isosceles triangle.

Answer: $\frac{7}{18}$


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>.

Four identical regular tetrahedrons are thrown simultaneously on a table. Calculate the probability that the product of the four numbers on the faces touching the table is divisible by 4.

Answer: $\frac{13}{16}$


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>.

Translate the function $y=\sqrt{3}\cos x+\sin x$ $(x\in\mathbb{R})$ to the left by $m$ $(m > 0)$ units, and the resulting graph is symmetric about the $y$-axis. Find the minimum value of $m$.

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


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

Determine the number of ordered pairs $(m, n)$ that satisfy $m$ and $n \in \{-1,0,1,2,3\}$, and the equation $mx^2 + 2x + n = 0$ has real solutions.

Answer: $17$