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

Given that $a > b > 0$, and $a + b = 2$, find the minimum value of $$\frac {3a-b}{a^{2}+2ab-3b^{2}}$$.

Answer: $\frac {3+ \sqrt {5}}{4}$


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

A line passing through the focus of the parabola $y^2=4x$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. If $|AB|=7$, find the distance from the midpoint $M$ of $AB$ to the directrix of the parabola.

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


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

The graph of the function y=sin(2x+φ) is shifted to the left by π/6 units along the x-axis, resulting in an even function graph. Determine the value of φ such that the equation 2(x + π/6) + φ = -x + 2πk is satisfied for some integer k.

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>.

Let \( a, b, c \) be real numbers satisfying \( 9a^2 + 4b^2 + 36c^2 = 4 \). Find the minimum value of \( 3a + 6b + 12c \).

Answer: $-2\sqrt{14}$


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

Find the remainder when $2^{2^{2^2}}$ is divided by $500$.

Answer: $536$


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

In triangle \( \triangle ABC \), given that \( \frac{\cos A}{\sin B} + \frac{\cos B}{\sin A} = 2 \) and the perimeter of \( \triangle ABC \) is 12, find the maximum possible area of the triangle.

Answer: $36(3 - 2\sqrt{2})$


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

Given $f(x) = 3x^2 + 2x + 1$, if $\int_{-1}^{1} f(x)\,dx = 2f(a)$, then $a = \_\_\_\_\_\_$.

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 the function $f(x)=\cos (2x+\varphi)$, where $|\varphi| \leqslant \frac{\pi}{2}$, if $f\left( \frac{8\pi}{3}-x\right)=-f(x)$, determine the horizontal shift required to obtain the graph of $y=\sin 2x$.

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>.

Let \( p, q, r, \) and \( s \) be the roots of the polynomial
\[ x^4 + 10x^3 + 20x^2 + 15x + 6 = 0. \]
Find the value of
\[ \frac{1}{pq} + \frac{1}{pr} + \frac{1}{ps} + \frac{1}{qr} + \frac{1}{qs} + \frac{1}{rs}. \]

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


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 cone with a base radius of $5$ and a lateral area of $65π$, let the angle between the slant height and the height of the cone be $θ$. Find the value of $\sinθ$.

Answer: $\frac{5}{13}$


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>.

An ellipse C is defined by the equation $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ (a > b > 0), and its four vertices form a rhombus with side length $$\sqrt {3}$$ and area $$2 \sqrt {2}$$.
(1) Find the equation of ellipse C.
(2) A line l passes through point Q(0, -2) and intersects ellipse C at points A and B. If the product of the slopes of OA and OB (where O is the origin) is -1, find the length of segment AB.

Answer: $\frac {4 \sqrt {21}}{11}$


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

Given that $\alpha$ and $\beta$ are two interior angles of an oblique triangle, if $\frac{{\cos \alpha - \sin \alpha}}{{\cos \alpha + \sin \alpha}} = \cos 2\beta$, then the minimum value of $\tan \alpha + \tan \beta$ is ______.

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>.

Find the product of all constants $t$ such that the quadratic $x^2 + tx - 12$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers.

Answer: $1936$


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

The school assigns three people, A, B, and C, to participate in social practice activities in 7 different communities, with at most 2 people assigned to each community. There are $\boxed{\text{336}}$ different distribution schemes (answer in numbers).

Answer: $336$


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

Given an ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$, there is a point $P$ on the ellipse such that the distance to the two foci of the ellipse, $F_{1}$ and $F_{2}$, satisfies $|PF_{1}|+|PF_{2}|=10$, and the eccentricity $e= \dfrac {4}{5}$.
$(1)$ Find the standard equation of the ellipse.
$(2)$ If $\angle F_{1}PF_{2}=60^{\circ}$, find the area of $\triangle F_{1}PF_{2}$.

Answer: $3 \sqrt {3}$


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

For what value of $x$ does $3^{3x^{2} - 8x + 5} = 3^{3x^{2} + 5x - 6}$?

Answer: $\frac{11}{13}$


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>.

On an island, there are only knights, who always tell the truth, and liars, who always lie. One fine day, 30 islanders sat around a round table. Each of them can see everyone except himself and his neighbors. Each person in turn said the phrase: "Everyone I see is a liar." How many liars were sitting at the table?

Answer: $28$


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

The cells of a  $100 \times 100$  table are colored white. In one move, it is allowed to select some  $99$  cells from the same row or column and recolor each of them with the opposite color. What is the smallest number of moves needed to get a table with a chessboard coloring?

*S. Berlov*

Answer: $100$


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

Given the sequence $\{a_n\}$, $a_1=1$, $a_2=2$, and $a_{n+2}-a_{n}=1+(-1)^{n}$ $(n\in\mathbb{N}_{+})$, calculate the value of $S_{100}$.

Answer: $2600$


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

For how many integer values of $b$ does the equation $$x^2 + bx + 12b = 0$$ have integer solutions for $x$?

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>.

In Rivertown, car plates each contain three symbols: two letters followed by a digit. The first letter is chosen from the set ${A, B, G, H, T}$, the second letter from ${E, I, O, U}$, and the digit from $0$ to $9$.

To accommodate an increase in the number of cars, Rivertown decides to expand each set by adding new symbols. Rivertown adds two new letters and one new digit. These additional symbols can be added entirely to one set or distributed among the sets. Determine the largest number of ADDITIONAL car plates that can be made possible by the most advantageous distribution of these new symbols.

Answer: $130$


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

In how many ways can  $17$  identical red and  $10$  identical white balls be distributed into  $4$  distinct boxes such that the number of red balls is greater than the number of white balls in each box?

Answer: $5720$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 "The line $ax - by = 0$ intersects the circle $(x - 2\sqrt{2})^2 + y^2 = 6$".
(1) If $a$ and $b$ are the numbers obtained by rolling a dice twice, find the probability of event $A$.
(2) If the real numbers $a$ and $b$ satisfy $(a - \sqrt{3})^2 + (b - 1)^2 \leq 4$, find the probability of event $A$.

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>.

Chloe wants to purchase a jacket that costs $\$45.50$. She checks her purse and finds she has four $\$10$ bills, ten quarters, and some nickels and dimes. What is the minimum number of dimes that she must have if she has 15 nickels?

Answer: $23$


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

Evaluate $101 \times 101$ using a similar mental math technique.

Answer: $10201$


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

Calculate the limit

$$
\lim _{x \rightarrow 0}\left(\frac{1+x^{2} 2^{x}}{1+x^{2} 5^{x}}\right)^{1 / \sin ^{3} x}
$$

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>.

If one vertex and the two foci of an ellipse form an equilateral triangle, determine the eccentricity of this ellipse.

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>.

In the Cartesian coordinate system $xOy$, given the parametric equations of circle $C_{1}$ as $\left\{\begin{array}{l}x=2+2\cos\alpha\\ y=1+2\sin\alpha\end{array}\right.$ ($\alpha$ is the parameter), establish a polar coordinate system with the coordinate origin as the pole and the positive x-axis as the polar axis.
$(1)$ Find the polar equation of $C_{1}$;
$(2)$ If the polar equation of line $C_{2}$ is $θ=\frac{π}{4}$ ($ρ∈R$), and the intersection points of $C_{2}$ and $C_{1}$ are $P$ and $Q$, find the area of $\triangle C_{1}PQ$.

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


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

Kai has begun to list, in ascending order, the positive integers which are not factors of 240. What is the sixth number on Kai's list?

Answer: $17$


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

Given that the polar coordinate equation of curve C is $\rho = \sqrt{3}$, and the parametric equations of line l are $x = 1 + \frac{\sqrt{2}}{2}t$, $y = \frac{\sqrt{2}}{2}t$ (where t is the parameter), and the parametric equations of curve M are $x = \cos \theta$, $y = \sqrt{3} \sin \theta$ (where $\theta$ is the parameter).
1. Write the Cartesian coordinate equation for curve C and line l.
2. If line l intersects curve C at points A and B, and P is a moving point on curve M, find the maximum area of triangle ABP.

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


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

In the addition shown, each of \( P \), \( Q \), and \( R \) is a digit.

\[
\begin{array}{r}
P Q R \\
+\quad Q R \\
\hline 1012
\end{array}
\]

What is the value of \( P + Q + R \)?

Answer: $20$


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

The points  $A$ ,  $B$ ,  $C$ ,  $D$ , and  $E$  lie in one plane and have the following properties:  $AB = 12, BC = 50, CD = 38,  AD = 100, BE = 30, CE = 40$ .

Find the length of the segment  $ED$ .

Answer: $74$


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

Determine the share of gold in the currency structure of the National Welfare Fund (NWF) as of December 1, 2022, using one of the following methods:

First Method:

a) Find the total amount of NWF funds allocated in gold as of December 1, 2022:
\[GOLD_{22} = 1388.01 - 41.89 - 2.77 - 478.48 - 309.72 - 0.24 = 554.91 \, (\text{billion rubles})\]

b) Determine the share of gold in the currency structure of NWF funds as of December 1, 2022:
\[\alpha_{22}^{GOLD} = \frac{554.91}{1388.01} \approx 39.98\% \]

c) Calculate by how many percentage points and in which direction the share of gold in the currency structure of NWF funds changed over the review period:
\[\Delta \alpha^{GOLD} = \alpha_{22}^{GOLD} - \alpha_{21}^{GOLD} = 39.98 - 31.8 = 8.18 \approx 8.2 \, (\text{p.p.})\]

Second Method:

a) Determine the share of euro in the currency structure of NWF funds as of December 1, 2022:
\[\alpha_{22}^{EUR} = \frac{41.89}{1388.01} \approx 3.02\% \]

b) Determine the share of gold in the currency structure of NWF funds as of December 1, 2022:
\[\alpha_{22}^{GOLD} = 100 - 3.02 - 0.2 - 34.47 - 22.31 - 0.02 = 39.98\%\]

c) Calculate by how many percentage points and in which direction the share of gold in the currency structure of NWF funds changed over the review period:
\[\Delta \alpha^{GOLD} = \alpha_{22}^{GOLD} - \alpha_{21}^{GOLD} = 39.98 - 31.8 = 8.18 \approx 8.2 \, (\text{p.p.})\]

Answer: $8.2$


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

In a factory, a total of $100$ parts were produced. Among them, A produced $0$ parts, with $35$ being qualified. B produced $60$ parts, with $50$ being qualified. Let event $A$ be "Selecting a part from the $100$ parts at random, and the part is qualified", and event $B$ be "Selecting a part from the $100$ parts at random, and the part is produced by A". Then, the probability of $A$ given $B$ is \_\_\_\_\_\_.

Answer: $\dfrac{7}{8}$


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

Let $f(x) = x - 3$ and $g(x) = x/2$. Compute \[f(g^{-1}(f^{-1}(g(f^{-1}(g(f(23))))))).\]

Answer: $16$


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

Given $\alpha $, $\beta \in (0, \frac{π}{2})$, $\sin \alpha = \frac{{\sqrt{5}}}{5}$, $\cos \beta = \frac{1}{{\sqrt{10}}}$, find the value of $\alpha - \beta$.

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


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

Given an ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ whose right focus is $F$, and a circle with center $F$: $x^2 + y^2 - 4x - 32 = 0$ that is tangent to the right directrix of the ellipse, find the eccentricity of the ellipse.

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 an ellipse $C$: $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$ with its left and right foci being $F_{1}$ and $F_{2}$ respectively, and its eccentricity is $\dfrac {1}{2}$. Let $M$ be any point on the ellipse and the perimeter of $\triangle MF_{1}F_{2}$ equals $6$.

(Ⅰ) Find the equation of the ellipse $C$;
(Ⅱ) With $M$ as the center and $MF_{1}$ as the radius, draw a circle $M$. When circle $M$ and the line $l$: $x=4$ have common points, find the maximum area of $\triangle MF_{1}F_{2}$.

Answer: $\dfrac { \sqrt {15}}{3}$


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 square has been divided into  $2022$  rectangles with no two of them having a common interior point. What is the maximal number of distinct lines that can be determined by the sides of these rectangles?

Answer: $2025$


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

Let $h(x) = \sqrt[3]{\frac{2x+5}{5}}$. For what value of $x$ will $h(3x) = 3(h(x))$? Express your answer in simplest form.

Answer: $-\frac{65}{24}$


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>.

Construct a new shape by adding an eighth unit cube to the previously described configuration of seven cubes. Place this new cube adjacent to one of the six outlying cubes from the central cube. What is the ratio of the volume in cubic units to the surface area in square units for this new configuration?
A) $\frac{8}{31}$
B) $\frac{8}{32}$
C) $\frac{8}{33}$
D) $\frac{8}{34}$

Answer: $\frac{8}{33}$


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

Given $(2x-1)^{2015} = a_{0} + a_{1}x + a_{2}x^{2} + \ldots + a_{2015}x^{2015}$ ($x \in \mathbb{R}$), evaluate the expression $\frac {1}{2}+ \frac {a_{2}}{2^{2}a_{1}}+ \frac {a_{3}}{2^{3}a_{1}}+\ldots+ \frac {a_{2015}}{2^{2015}a_{1}}$.

Answer: $\frac {1}{4030}$


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

Given the line $y=kx+1$ intersects the parabola $C: x^2=4y$ at points $A$ and $B$, and a line $l$ is parallel to $AB$ and is tangent to the parabola $C$ at point $P$, find the minimum value of the area of triangle $PAB$ minus the length of $AB$.

Answer: $-\frac{64}{27}$


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

Point $P$ is a moving point on the parabola $y^{2}=4x$. The minimum value of the sum of the distances from point $P$ to point $A(0,-1)$ and from point $P$ to the line $x=-1$ is ______.

Answer: $\sqrt{2}$


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

Point $(x,y)$ is chosen randomly from the rectangular region with vertices at $(0,0)$, $(3036,0)$, $(3036,3037)$, and $(0,3037)$. What is the probability that $x > 3y$? Express your answer as a common fraction.

Answer: $\frac{506}{3037}$


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

In the diagram, \( AB \) and \( CD \) intersect at \( E \). If \(\triangle BCE\) is equilateral and \(\triangle ADE\) is a right-angled triangle, what is the value of \( x \)?

Answer: $30$


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

\begin{align*}
4a + 2b + 5c + 8d &= 67 \\
4(d+c) &= b \\
2b + 3c &= a \\
c + 1 &= d \\
\end{align*}
Given the above system of equations, find \(a \cdot b \cdot c \cdot d\).

Answer: $\frac{1201 \times 572 \times 19 \times 124}{105^4}$


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

In triangle $ABC$, if median $\overline{AD}$ makes an angle of $30^\circ$ with side $\overline{BC},$ then find the value of $|\cot B - \cot C|.$

Answer: $2\sqrt{3}$


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

Given a sequence $\{a_{n}\}$ such that $a_{2}=2a_{1}=4$, and $a_{n+1}-b_{n}=2a_{n}$, where $\{b_{n}\}$ is an arithmetic sequence with a common difference of $-1$.
$(1)$ Investigation: Determine whether the sequence $\{a_{n}-n\}$ is an arithmetic sequence or a geometric sequence, and provide a justification.
$(2)$ Find the smallest positive integer value of $n$ that satisfies $a_{1}+a_{2}+\ldots +a_{n} \gt 2200$.

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  $a$  and  $b$  be the roots of the polynomial  $x^2+2020x+c$ . Given that  $\frac{a}{b}+\frac{b}{a}=98$ , compute  $\sqrt c$ .

Answer: $202$


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

Given that point $O$ is the origin of coordinates, point $A$ in the first quadrant lies on the graph of the inverse proportional function $y=\frac{1}{x}$ for $x>0$, and point $B$ in the second quadrant lies on the graph of the inverse proportional function $y=-\frac{4}{x}$ for $x<0$, and $O A$ is perpendicular to $O B$, find the value of $\tan \angle A B O$.

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 $\sin 2α = - \frac{24}{25}$, and $α ∈ (\frac{3π}{4},π)$, find the value of $\sin α$.

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


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

Given \\(\{x_{1},x_{2},x_{3},x_{4}\} \subseteq \{x | (x-3) \cdot \sin \pi x = 1, x > 0\}\\), find the minimum value of \\(x_{1}+x_{2}+x_{3}+x_{4}\\).

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>.

How many whole numbers between 1 and 500 do not contain the digit 2?

Answer: $323$


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

Joe's quiz scores were 88, 92, 95, 81, and 90, and then he took one more quiz and scored 87. What was his mean score after all six quizzes?

Answer: $88.83$


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

The graph of \(y^2 + 2xy + 60|x| = 900\) partitions the plane into several regions. What is the area of the bounded region?

Answer: $1800$


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

In a right triangle $PQR$, where $\angle P = 90^\circ$, suppose $\cos Q = \frac{4}{5}$. The length of side $PQ$ (adjacent to $\angle Q$) is $12$. What is the length of $PR$ (hypotenuse)?

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>.

The diagram shows a triangle \(ABC\) and two lines \(AD\) and \(BE\), where \(D\) is the midpoint of \(BC\) and \(E\) lies on \(CA\). The lines \(AD\) and \(BE\) meet at \(Z\), the midpoint of \(AD\). What is the ratio of the length \(CE\) to the length \(EA\)?

Answer: $2:1$


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

Two arithmetic sequences, $C$ and $D$, begin with the terms 25 and 40 respectively, both with common differences of absolute value 15, with sequence $C$ increasing and sequence $D$ decreasing. What is the absolute value of the difference between the 60th terms of sequences $C$ and $D$?

Answer: $1755$


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

In a 6 by 5 grid, how many 10-step paths are there from $W$ to $X$ that must pass through a point $H$? Assume $W$ is located at the top-left corner, $X$ at the bottom-right corner, and $H$ is three squares to the right and two squares down from $W$.

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 the ellipse $E$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, its eccentricity is $\frac{\sqrt{2}}{2}$, point $F$ is the left focus of the ellipse, point $A$ is the right vertex, and point $B$ is the upper vertex. Additionally, $S_{\triangle ABF} = \frac{\sqrt{2}+1}{2}$.
(I) Find the equation of the ellipse $E$;
(II) If the line $l$: $x - 2y - 1 = 0$ intersects the ellipse $E$ at points $P$ and $Q$, find the perimeter and area of $\triangle FPQ$.

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


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

Let $x=1$ be an extremum point of the function $f(x)=a_{n+1}x^{3}-a_{n}x^{2}-a_{n+2}x+1(n\in N_{+})$, where the sequence $\{a_{n}\}$ is given by $a_{1}=1$, $a_{2}=2$, and $b_{n}=\log _{2}a_{2n}$. Determine the value of $\left\lfloor \dfrac {2018}{b_{1}b_{2}}+ \dfrac {2018}{b_{2}b_{3}}+\ldots+ \dfrac {2018}{b_{2018}b_{2019}}\right\rfloor$.

Answer: $1008$


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

A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered.

Answer: $12\%$


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

If the moving point $P$ is on the line $y=x+1$, and the moving point $Q$ is on the curve $x^{2}=-2y$, calculate the minimum value of $|PQ|$.

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


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

A solid is formed by rotating a triangle with sides of lengths 3, 4, and 5 around the line containing its shortest side. Find the surface area of this solid.

Answer: $36\pi$


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

Let $P(x) = b_0 + b_1x + b_2x^2 + \dots + b_mx^m$ be a polynomial with integer coefficients, where $0 \le b_i < 5$ for all $0 \le i \le m$.

Given that $P(\sqrt{5})=23+19\sqrt{5}$, compute $P(3)$.

Answer: $132$


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

Calculate \(14 \cdot 31\) and \(\left\lfloor\frac{2+\sqrt{2}}{2}\right\rfloor + \left\lfloor\frac{3+\sqrt{3}}{3}\right\rfloor + \left\lfloor\frac{4+\sqrt{4}}{4}\right\rfloor + \cdots + \left\lfloor\frac{1989+\sqrt{1989}}{1989}\right\rfloor + \left\lfloor\frac{1990+\sqrt{1990}}{1990}\right\rfloor\).

Answer: $1989$


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

How many ways are there to put 7 balls in 2 boxes if the balls are distinguishable but the boxes are not?

Answer: $64$


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

In $\triangle ABC$, $b^{2}=ac$, and $a+c=3$, $\cos B= \frac{3}{4}$, then $\overset{→}{AB} \cdot \overset{→}{BC} =$______.

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>.

Tim plans a weeklong prank to repeatedly steal Nathan's fork during lunch. He involves different people each day:
- On Monday, he convinces Joe to do it.
- On Tuesday, either Betty or John could undertake the prank.
- On Wednesday, there are only three friends from whom he can seek help, as Joe, Betty, and John are not available.
- On Thursday, neither those involved earlier in the week nor Wednesday's helpers are willing to participate, but four new individuals are ready to help.
- On Friday, Tim decides he could either do it himself or get help from one previous assistant who has volunteered again.
How many different combinations of people could be involved in the prank over the week?

Answer: $48$


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

What is the smallest positive integer $n$ such that $\frac{n}{n+103}$ is equal to a terminating decimal?

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>.

Let \(a\), \(b\), \(c\), and \(d\) be positive integers with \(a < 3b\), \(b < 4c\), and \(c < 5d\). If \(d < 80\), find the largest possible value for \(a\).

Answer: $4724$


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

If for any \( x \in \mathbf{R} \), the function \( f(x) \) satisfies the equation \( f(x+2009) = -f(x+2008) \), and \( f(2009) = -2009 \), determine the value of \( f(-1) \).

Answer: $-2009$


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

A road of 1500 meters is being repaired. In the first week, $\frac{5}{17}$ of the total work was completed, and in the second week, $\frac{4}{17}$ was completed. What fraction of the total work was completed in these two weeks? And what fraction remains to complete the entire task?

Answer: $\frac{8}{17}$


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

Real numbers between 0 and 1, inclusive, are chosen based on the outcome of flipping two fair coins. If two heads are flipped, then the chosen number is 0; if a head and a tail are flipped (in any order), the number is 0.5; if two tails are flipped, the number is 1. Another number is chosen independently in the same manner. Calculate the probability that the absolute difference between these two numbers, x and y, is greater than $\frac{1}{2}$.

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


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

Given functions $f(x)=xe^x$ and $g(x)=-\frac{lnx}{x}$, if $f(x_{1})=g(x_{2})=t\left( \gt 0\right)$, find the maximum value of $\frac{{x}_{1}}{{x}_{2}{e}^{t}}$.

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


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

Given a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a>0, b>0$) with a point C on it, a line passing through the center of the hyperbola intersects the hyperbola at points A and B. Let the slopes of the lines AC and BC be $k_1$ and $k_2$ respectively. Find the eccentricity of the hyperbola when $\frac{2}{k_1 k_2} + \ln{k_1} + \ln{k_2}$ is minimized.

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 a sequence of positive terms $\{a\_n\}$, with $a\_1=2$, $(a\_n+1)a_{n+2}=1$, and $a\_2=a\_6$, find the value of $a_{11}+a_{12}$.

Answer: $\frac{1}{9}+\frac{\sqrt{5}}{2}$


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

Subtract $123.45$ from $567.89.$ Express the result as a decimal to the nearest hundredth.

Answer: $444.44$


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

The value of \((4 + 44 + 444) \div 4\) is:

Answer: $123$


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

In the sequence $\{a_n\}$, if $a_1=-2$ and for any $n\in\mathbb{N}^*$, $a_{n+1}=1+2a_n$, then the sum of the first $10$ terms of the sequence $\{a_n\}$ is ______.

Answer: $-1033$


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

A class has prepared 6 programs to participate in the Xiamen No.1 Middle School Music Square event. The order of the programs has the following requirements: Programs A and B must be adjacent, and Programs C and D cannot be adjacent. How many possible arrangements of the program order are there for this event?

Answer: $144$


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

5 students stand in a row for a photo, where students A and B must stand next to each other, and A cannot stand at either end. Calculate the total number of possible arrangements.

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>.

Lead used to make twelve solid lead balls, each with a radius of 2 cm, is reused to form a single larger solid lead sphere. What is the radius of this larger sphere?

Answer: $\sqrt[3]{96}$


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>.

Given $\sin\alpha= \frac {2 \sqrt {2}}{3}$, $\cos(\alpha+\beta)=- \frac {1}{3}$, and $\alpha, \beta\in(0, \frac {\pi}{2})$, determine the value of $\sin(\alpha-\beta)$.

Answer: $\frac {10 \sqrt {2}}{27}$


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

Cylinder $C$'s height is equal to the diameter of cylinder $D$ and cylinder $C$'s diameter is equal to the height $h$ of cylinder $D$. If the volume of cylinder $D$ is three times the volume of cylinder $C$, the volume of cylinder $D$ can be written as $M \pi h^3$ cubic units. Find the value of $M$.

Answer: $\frac{9}{4}$


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

Find
\[
\cos \left( 4 \arccos \frac{2}{5} \right).
\]

Answer: $-\frac{47}{625}$


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

Triangle $\triangle ABC$ has a right angle at $C$, $\angle A = 45^\circ$, and $AC=12$. Find the radius of the incircle of $\triangle ABC$.

Answer: $6 - 3\sqrt{2}$


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

In an opaque bag, there are four identical balls labeled with numbers $3$, $4$, $5$, and $6$ respectively. Outside the bag, there are two balls labeled with numbers $3$ and $6$. Determine the probability that a triangle with the drawn ball and the numbers on the two balls outside the bag forms an isosceles triangle.

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>.

Simplify and write the result as a common fraction: $$\sqrt[4]{\sqrt[3]{\sqrt{\frac{1}{65536}}}}$$

Answer: $\frac{1}{\sqrt[3]{4}}$


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

If the vector $\overrightarrow{a} = (x, y-1)$ is collinear with the vector $\overrightarrow{b} = (3, -2)$, then the minimum value of $z = \log_{2}(4^x + 8^y)$ is ______.

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


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

Given two fixed points on the plane, \\(A(-2,0)\\) and \\(B(2,0)\\), and a moving point \\(T\\) satisfying \\(|TA|+|TB|=2 \sqrt {6}\\).
\\((\\)I\\()\\) Find the equation of the trajectory \\(E\\) of point \\(T\\);
\\((\\)II\\()\\) A line passing through point \\(B\\) and having the equation \\(y=k(x-2)\\) intersects the trajectory \\(E\\) at points \\(P\\) and \\(Q\\) \\((k\neq 0)\\). If \\(PQ\\)'s midpoint is \\(N\\) and \\(O\\) is the origin, the line \\(ON\\) intersects the line \\(x=3\\) at point \\(M\\). Find the maximum value of \\( \dfrac {|PQ|}{|MB|}\\).

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>.

Let the probability of germination for each seed be 0.9, and 1000 seeds have been planted. For each seed that does not germinate, 2 more seeds need to be replanted. Let X be the number of replanted seeds. Calculate the expected value of X.

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 the arithmetic sequence $\left\{ a_n \right\}$ where each term is positive, the sum of the first $n$ terms is $S_n$. When $n \in N^*, n \geqslant 2$, it holds that $S_n = \frac{n}{n-1}\left( a_n^2 - a_1^2 \right)$. Find the value of $S_{20} - 2S_{10}$.

Answer: $50$


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

If the price of a stamp is 45 cents, what is the maximum number of stamps that could be purchased with $50?

Answer: $111$


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

In the polar coordinate system, let curve \(C_1\) be defined by \(\rho\sin^2\theta = 4\cos\theta\). Establish a Cartesian coordinate system \(xOy\) with the pole as the origin and the polar axis as the positive \(x\)-axis. The curve \(C_2\) is described by the parametric equations:
\[ \begin{cases}
x = 2 + \frac{1}{2}t \\
y = \frac{\sqrt{3}}{2}t
\end{cases} \]
where \(t\) is the parameter.
(1) Find the Cartesian equations for \(C_1\) and \(C_2\).
(2) If \(C_1\) and \(C_2\) intersect at points \(A\) and \(B\), and there is a fixed point \(P\) with coordinates \((2,0)\), find the value of \(|PA| \cdot |PB|\).

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


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

Given that x > 0, y > 0, and x + 2y = 4, find the minimum value of $$\frac {(x+1)(2y+1)}{xy}$$.

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>.

Given that the sum of the first $n$ terms of an arithmetic sequence $\{a\_n\}$ is $S\_n$, with $a\_2 = 4$ and $S\_{10} = 110$, find the minimum value of $\frac{S\_n + 64}{a\_n}$.

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


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

Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=101$. What is $x$?

Answer: $50$


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

Let  $f : N \to N$  be a strictly increasing function such that  $f(f(n))= 3n$ , for all  $n \in N$ . Find  $f(2010)$ .

Note:  $N = \{0,1,2,...\}$

Answer: $3015$


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

It is currently 3:00:00 PM, as shown on a 12-hour digital clock. In 300 hours, 55 minutes, and 30 seconds, what will the time be and what is the sum of the hours, minutes, and seconds?

Answer: $88$


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

A pyramid has a square base $ABCD$ and a vertex $E$. The area of square $ABCD$ is $256$, and the areas of $\triangle ABE$ and $\triangle CDE$ are $120$ and $136$, respectively. The distance from vertex $E$ to the midpoint of side $AB$ is $17$. What is the volume of the pyramid?
- **A)** $1024$
- **B)** $1200$
- **C)** $1280$
- **D)** $1536$
- **E)** $1600$

Answer: $1280$


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

Given $\overrightarrow{m}=(\sin \omega x,-1)$, $\overrightarrow{n}=(1,- \sqrt {3}\cos \omega x)$ where $x\in\mathbb{R}$, $\omega > 0$, and $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$, and the distance between a certain highest point and its adjacent lowest point on the graph of function $f(x)$ is $5$,

$(1)$ Find the interval of monotonic increase for the function $f(x)$;

$(2)$ If $f\left( \dfrac {3\theta}{\pi}\right)= \dfrac {6}{5}$ where $\theta\in\left(- \dfrac {5\pi}{6}, \dfrac {\pi}{6}\right)$, then find the value of $f\left( \dfrac {6\theta}{\pi}+1\right)$.

Answer: $\dfrac {48}{25}$


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

Let  $ABC$  be triangle such that  $|AB| = 5$ ,  $|BC| = 9$  and  $|AC| = 8$ . The angle bisector of  $\widehat{BCA}$  meets  $BA$  at  $X$  and the angle bisector of  $\widehat{CAB}$  meets  $BC$  at  $Y$ . Let  $Z$  be the intersection of lines  $XY$  and  $AC$ . What is  $|AZ|$ ? $
\textbf{a)}\ \sqrt{104}
\qquad\textbf{b)}\ \sqrt{145}
\qquad\textbf{c)}\ \sqrt{89}
\qquad\textbf{d)}\ 9
\qquad\textbf{e)}\ 10
$

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>.

When \( q(x) = Dx^4 + Ex^2 + Fx + 6 \) is divided by \( x - 2 \), the remainder is 14. Find the remainder when \( q(x) \) is divided by \( x + 2 \).

Answer: $14$


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

Given the sequence ${a_n}$, $a_1=1$ and $a_n a_{n+1} + \sqrt{3}(a_n - a_{n+1}) + 1 = 0$. Determine the value of $a_{2016}$.

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>.

In the rectangular coordinate system $xOy$, the ordinary equation of curve $C_1$ is $x^2+y^2-2x=0$. Establish a polar coordinate system with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of curve $C_2$ is $\rho^{2}= \frac {3}{1+2\sin^{2}\theta }$.
(I) Find the parametric equation of $C_1$ and the rectangular equation of $C_2$;
(II) The ray $\theta= \frac {\pi}{3}(\rho\geq0)$ intersects $C_1$ at a point $A$ distinct from the pole, and $C_2$ at point $B$. Find $|AB|$.

Answer: $\frac { \sqrt {30}}{5}-1$


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

Two arithmetic sequences $\{a_{n}\}$ and $\{b_{n}\}$ have the sums of the first $n$ terms as $S_{n}$ and $T_{n}$, respectively. It is known that $\frac{{S}_{n}}{{T}_{n}}=\frac{7n+2}{n+3}$. Find $\frac{{a}_{7}}{{b}_{7}}$.

Answer: $\frac{93}{16}$


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>.

Four steel balls, each with a radius of 1, are to be completely packed into a container shaped as a regular tetrahedron. What is the minimum height of the tetrahedron?

Answer: $2 + \frac{2 \sqrt{6}}{3}$


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

a) Vanya flips a coin 3 times, and Tanya flips a coin 2 times. What is the probability that Vanya gets more heads than Tanya?

b) Vanya flips a coin $n+1$ times, and Tanya flips a coin $n$ times. What is the probability that Vanya gets more heads than Tanya?

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


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

Given the exponential function y=f(x) whose graph passes through the point $\left( \frac{1}{2}, \frac{\sqrt{2}}{2} \right)$, find the value of $\log_2 f(2)$.

Answer: $-2$


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

Simplify first, then evaluate: $(1-\frac{2}{x+1})÷\frac{x^2-1}{2x+2}$, where $x=\pi ^{0}+1$.

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


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

Mayuki walks once around a track shaped with straight sides and semicircular ends at a constant speed daily. The track has a width of \(4\) meters, and it takes her \(24\) seconds longer to walk around the outside edge than the inside edge. Determine Mayuki's speed in meters per second.

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>.

In writing the integers from 20 through 199 inclusive, how many times is the digit 7 written?

Answer: $38$


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

If 2023 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers?

Answer: $48$


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

Let the base areas of two cylinders be $S_1$ and $S_2$, and their volumes be $\upsilon_1$ and $\upsilon_2$, respectively. If their lateral areas are equal, and $$\frac {S_{1}}{S_{2}}= \frac {16}{9},$$ then the value of $$\frac {\upsilon_{1}}{\upsilon_{2}}$$ is \_\_\_\_\_\_.

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


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

Given that the random variable $\xi$ follows a normal distribution $N(1,4)$, if $p(\xi > 4)=0.1$, then $p(-2 \leqslant \xi \leqslant 4)=$ _____ .

Answer: $0.8$


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

Define a function $g(x),$ for positive integer values of $x,$ by
\[
g(x) = \left\{
\begin{aligned}
\log_3 x & \quad \text{if } \log_3 x \text{is an integer} \\
1 + g(x + 1) & \quad \text{otherwise}.
\end{aligned}
\right.
\]
Compute $g(50)$.

Answer: $35$


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

In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases} & x=2\sqrt{3}\cos a \\ & y=2\sin a \end{cases}$, where $a$ is a parameter and $a \in (0, \pi)$. In the polar coordinate system with the origin $O$ as the pole and the positive half axis of $x$ as the polar axis, the polar coordinates of point $P$ are $(4\sqrt{2}, \frac{\pi}{4})$, and the polar equation of line $l$ is $\rho \sin(\theta - \frac{\pi}{4}) + 5\sqrt{2} = 0$.

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

(2) Suppose $Q$ is a moving point on curve $C$, and $M$ is the midpoint of segment $PQ$. Find the maximum distance from point $M$ to the line $l$.

Answer: $6\sqrt{2}$


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

The lengths of the three sides of a triangle are \( 10 \), \( y+5 \), and \( 3y-2 \). The perimeter of the triangle is \( 50 \). What is the length of the longest side of the triangle?

Answer: $25.75$


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>.

In the expansion of $(\frac{3}{{x}^{2}}+x+2)^{5}$, the coefficient of the linear term in $x$ is ____.

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 that angle DEF is a right angle and the sides of triangle DEF are the diameters of semicircles, the area of the semicircle on segment DE equals $18\pi$, and the arc of the semicircle on segment DF has length $10\pi$. Determine the radius of the semicircle on segment EF.

Answer: $\sqrt{136}$


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

A regular hexahedron with an edge length of $1$ is cut by planes passing through the common vertex of three edges and their respective midpoints. After removing the $8$ triangular pyramids, the volume of the remaining convex polyhedron is $\_\_\_\_\_\_$.

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


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>.

A sphere intersects the $xy$-plane in a circle centered at $(3,5,0)$ with a radius of 2. The sphere also intersects the $yz$-plane in a circle centered at $(0,5,-8),$ with radius $r.$ Find $r.$

Answer: $\sqrt{59}$


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

Two people, A and B, participate in a general knowledge competition, with a total of 4 different questions, including 2 multiple-choice questions and 2 true/false questions. A and B each draw one question (without repetition).
$(1)$ What is the probability that A draws a multiple-choice question and B draws a true/false question?
$(2)$ What is the probability that at least one of A and B draws a multiple-choice question?

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


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>.

From the five points consisting of the four vertices and the center of a square, any two points are chosen. The probability that the distance between these two points is not less than the side length of the square is ______.

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