You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 $C_1$ is
$$
\begin{cases}
x=2\sqrt{2}-\frac{\sqrt{2}}{2}t \\
y=\sqrt{2}+\frac{\sqrt{2}}{2}t
\end{cases}
(t \text{ is the parameter}).
$$
In the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the equation of curve $C_2$ is $\rho=4\sqrt{2}\sin \theta$.

(Ⅰ) Convert the equation of $C_2$ into a Cartesian coordinate equation;

(Ⅱ) Suppose $C_1$ and $C_2$ intersect at points $A$ and $B$, and the coordinates of point $P$ are $(\sqrt{2},2\sqrt{2})$, find $|PA|+|PB|$.

Answer: $2\sqrt{7}$


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

At Stanford in 1988, human calculator Shakuntala Devi was asked to compute  $m = \sqrt[3]{61{,}629{,}875}$  and  $n = \sqrt[7]{170{,}859{,}375}$ .  Given that  $m$  and  $n$  are both integers, compute  $100m+n$ .

*Proposed by Evan Chen*

Answer: $39515$


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

In an isosceles triangle \( \triangle ABC \), the length of the altitude to one of the equal sides is \( \sqrt{3} \) and the angle between this altitude and the base is \( 60^\circ \). Calculate the area of \( \triangle ABC \).

Answer: $\sqrt{3}$


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

Let  $ a,\ b$  be real constants. Find the minimum value of the definite integral:

 $ I(a,\ b)\equal{}\int_0^{\pi} (1\minus{}a\sin x \minus{}b\sin 2x)^2 dx.$

Answer: $\pi - \frac{8}{\pi}$


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>.

In city "N", there are 10 horizontal and 12 vertical streets. A pair of horizontal and a pair of vertical streets form the rectangular boundary of the city, while the rest divide it into blocks shaped like squares with a side length of 100 meters. Each block has an address consisting of two integers \((i, j)\), \(i = 1, 2, \ldots, 9\), \(j = 1, 2, \ldots, 11\), representing the numbers of the streets that bound it from below and the left. A taxi transports passengers from one block to another, observing the following rules:

1. Pickup and drop-off are carried out at any point on the block's boundary at the passenger's request.
2. It is prohibited to enter inside the block.
3. Transportation is carried out by the shortest path.
4. For every 100 meters of travel, a fare of 1 coin is charged (with distance rounded up to the nearest multiple of 100 meters in favor of the driver).

How many blocks are there in the city? What is the maximum and minimum fare the driver can request for a ride from block \((7,1)\) to block \((2,10)\) without violating the rules?

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>.

The lengths of the six edges of a tetrahedron $ABCD$ are $7, 13, 18, 27, 36, 41$, and $AB = 41$. What is the length of $CD$?

Answer: $13$


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

Given the function $f(x) = \cos^4x + 2\sin x\cos x - \sin^4x$
(1) Determine the parity, the smallest positive period, and the intervals of monotonic increase for the function $f(x)$.
(2) When $x \in [0, \frac{\pi}{2}]$, find the maximum and minimum values of the function $f(x)$.

Answer: $-1$


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

Given the function $f(x) = \sin(x - \varphi)$ and $|\varphi| < \frac{\pi}{2}$, and $\int_{0}^{\frac{2\pi}{3}} f(x) \, dx = 0$, find the equation of one of the axes of symmetry of the graph of function $f(x)$.

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


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>.

Brand Z juice claims, "We offer 30% more juice than Brand W at a price that is 15% less." What is the ratio of the unit price of Brand Z juice to the unit price of Brand W juice? Express your answer as a common fraction.

Answer: $\frac{17}{26}$


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

A rectangle  $ABEF$  is drawn on the leg  $AB$  of a right triangle  $ABC$ , whose apex  $F$  is on the leg  $AC$ . Let  $X$  be the intersection of the diagonal of the rectangle  $AE$  and the hypotenuse  $BC$  of the triangle. In what ratio does point  $X$  divide the hypotenuse  $BC$  if it is known that  $| AC | = 3 | AB |$  and  $| AF | = 2 | AB |$ ?

Answer: $2:3$


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

Among the numbers $210_{(6)}$, $1000_{(4)}$, and $111111_{(2)}$, the smallest number is \_\_\_\_\_\_.

Answer: $111111_{(2)}$


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 \( b \) is an even number between 1 and 11 (inclusive), and \( c \) is any natural number, determine the number of quadratic equations \( x^{2} + b x + c = 0 \) that have two distinct real roots.

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>.

In the parallelogram  $ABCD$ , a line through  $C$  intersects the diagonal  $BD$  at  $E$  and  $AB$  at  $F$ . If  $F$  is the midpoint of  $AB$  and the area of  $\vartriangle BEC$  is  $100$ , find the area of the quadrilateral  $AFED$ .

Answer: $250$


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

Given the data from a 2×2 contingency table calculates $k=4.073$, there is a \_\_\_\_\_\_ confidence that the two variables are related, knowing that $P(k^2 \geq 3.841) \approx 0.05$, $P(k^2 \geq 5.024) \approx 0.025$.

Answer: $95\%$


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

In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and $\overrightarrow{m}=(\sqrt{3}b-c,\cos C)$, $\overrightarrow{n}=(a,\cos A)$. Given that $\overrightarrow{m} \parallel \overrightarrow{n}$, determine the value of $\cos A$.

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


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

In the tetrahedron P-ABC, $PC \perpendicular$ plane ABC, $\angle CAB=90^\circ$, $PC=3$, $AC=4$, $AB=5$, then the surface area of the circumscribed sphere of this tetrahedron is \_\_\_\_\_\_.

Answer: $50\pi$


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>.

Consider a polynomial  $P(x) \in \mathbb{R}[x]$ , with degree  $2023$ , such that  $P(\sin^2(x))+P(\cos^2(x)) =1$  for all  $x \in \mathbb{R}$ . If the sum of all roots of  $P$  is equal to  $\dfrac{p}{q}$  with  $p, q$  coprime, then what is the product  $pq$ ?

Answer: $4046$


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

Given bed A has 600 plants, bed B has 500 plants, bed C has 400 plants, beds A and B share 60 plants, beds A and C share 80 plants, beds B and C share 40 plants, and beds A, B, and C share 20 plants collectively, calculate the total number of unique plants when considering just beds A, B, and C.

Answer: $1340$


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

Given that Adam has a triangular field ABC with AB = 5, BC = 8, and CA = 11, and he intends to separate the field into two parts by building a straight fence from A to a point D on side BC such that AD bisects ∠BAC, find the area of the part of the field ABD.

Answer: $\frac{5 \sqrt{21}}{4}$


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 trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 30 units, respectively, and the altitude is 15 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively. Calculate the area of trapezoid $EFCD$ if the height from $E$ to line $CD$ is 10 units.

Answer: $250$


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

The Cookie Monster now encounters a different cookie, which is bounded by the equation $(x-2)^2 + (y+1)^2 = 5$. He wonders if this cookie is big enough to share. Calculate the radius of this cookie and determine the area it covers.

Answer: $5\pi$


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>.

Football tickets are normally priced at $15 each. After buying 5 tickets, any additional tickets are sold at a discounted price of $12 each. If Jane has $150, what is the maximum number of tickets she can buy?

Answer: $11$


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

Given $f(x)=x^{3}+3ax^{2}+bx+a^{2}$ has an extremum of $0$ at $x=-1$, find $a-b=\_\_\_\_\_\_\_\_$.

Answer: $-7$


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

Let $\alpha \in \left(0, \frac{\pi}{3}\right)$, satisfying $\sqrt{3}\sin\alpha + \cos\alpha = \frac{\sqrt{6}}{2}$.

$(1)$ Find the value of $\cos\left(\alpha + \frac{\pi}{6}\right)$;
$(2)$ Find the value of $\cos\left(2\alpha + \frac{7\pi}{12}\right)$.

Answer: $\frac{\sqrt{2} - \sqrt{30}}{8}$


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>.

There are four points that are $7$ units from the line $y=20$ and $15$ units from the point $(10,20)$. What is the sum of the $x$- and $y$-coordinates of all four of these points?

Answer: $120$


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

The number of terms in the expansion of $(x+y+z)^4$ is 15.

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>.

There is a school that offers 10 courses for students to elect. Among them, Courses A, B, and C have conflicting schedules and thus at most one of these can be chosen. The school mandates that each student must elect three courses. How many different elective schemes are available for a student?

Answer: $98$


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

Given the function $f(x)=\frac{ax+b}{{x}^{2}+4}$ attains a maximum value of $1$ at $x=-1$, find the minimum value of $f(x)$.

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


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

Given an arithmetic sequence $\{a_n\}$, the sum of its first $n$ terms is $S_n$. It is known that $S_8 \leq 6$ and $S_{11} \geq 27$. Determine the minimum value of $S_{19}$.

Answer: $133$


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

How many different 8-digit positive integers exist if the digits from the second to the eighth can only be 0, 1, 2, 3, or 4?

Answer: $703125$


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

The repeating decimal for $\frac{5}{13}$ is $0.cdc\ldots$ What is the value of the sum $c+d$?

Answer: $11$


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

Find the value of $\dfrac{2\cos 10^\circ - \sin 20^\circ }{\sin 70^\circ }$.

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 $x$ and $y$ be positive integers such that $7x^5 = 11y^{13}.$ Find the prime factorization of the minimum possible value of $x$ and determine the sum of the exponents and the prime factors.

Answer: $31$


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

According to the classification standard of the Air Pollution Index (API) for city air quality, when the air pollution index is not greater than 100, the air quality is good. The environmental monitoring department of a city randomly selected the air pollution index for 5 days from last month's air quality data, and the data obtained were 90, 110, x, y, and 150. It is known that the average of the air pollution index for these 5 days is 110.
$(1)$ If x < y, from these 5 days, select 2 days, and find the probability that the air quality is good for both of these 2 days.
$(2)$ If 90 < x < 150, find the minimum value of the variance of the air pollution index for these 5 days.

Answer: $440$


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

Given vectors $\overrightarrow{a}=(\cos α,\sin α)$, $\overrightarrow{b}=(\cos β,\sin β)$, and $|\overrightarrow{a}- \overrightarrow{b}|= \frac {4 \sqrt {13}}{13}$.
(1) Find the value of $\cos (α-β)$;
(2) If $0 < α < \frac {π}{2}$, $- \frac {π}{2} < β < 0$, and $\sin β=- \frac {4}{5}$, find the value of $\sin α$.

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


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

In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is given that $\sqrt{3}\sin C - c\cos A = c$.
$(1)$ Find the value of angle $A$.
$(2)$ If $b = 2c$, point $D$ is the midpoint of side $BC$, and $AD = \sqrt{7}$, find the area of triangle $\triangle ABC$.

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>.

Consider the graph of $y=f(x)$, which consists of five line segments as described below:
- From $(-5, -4)$ to $(-3, 0)$
- From $(-3, 0)$ to $(-1, -1)$
- From $(-1, -1)$ to $(1, 3)$
- From $(1, 3)$ to $(3, 2)$
- From $(3, 2)$ to $(5, 6)$

What is the sum of the $x$-coordinates of all points where $f(x) = 2.3$?

Answer: $4.35$


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>.

Olympus Corporation has released a new version of its popular Vowel Soup. In this version, each vowel (A, E, I, O, U) appears 7 times in each bowl. How many six-letter "words" can be formed from this Olympus Vowel Soup? Note: Words do not need to be actual words in the English language.

Answer: $15625$


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

Given the equations
$$
z^{2}=4+4 \sqrt{15} i \text { and } z^{2}=2+2 \sqrt{3} i,
$$
the roots are the coordinates of the vertices of a parallelogram in the complex plane. If the area $S$ of the parallelogram can be expressed as $p \sqrt{q} - r \sqrt{s}$ (where $p, q, r, s \in \mathbf{Z}_{+}$, and $r$ and $s$ are not perfect squares), find the value of $p+q+r+s$.

Answer: $20$


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

Given the definition: $min\{a,b\} = \begin{cases} a,\quad a\leqslant b \\ b,\quad a > b \end{cases}$. If a point $P(x,y)$ is randomly selected within the region defined by $\begin{cases} 0\leqslant x\leqslant 2 \\ 0\leqslant y\leqslant 6 \end{cases}$, find the probability that $x$ and $y$ satisfy $min\{3x-2y+6,x-y+4\}=x-y+4$.

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


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

Given the function $f(x) = f'(1)e^{x-1} - f(0)x + \frac{1}{2}x^2$ (where $f'(x)$ is the derivative of $f(x)$, and $e$ is the base of the natural logarithm), and $g(x) = \frac{1}{2}x^2 + ax + b$ ($a \in \mathbb{R}, b \in \mathbb{R}$):

(Ⅰ) Find the explicit formula for $f(x)$ and its extremum;

(Ⅱ) If $f(x) \geq g(x)$, find the maximum value of $\frac{b(a+1)}{2}$.

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


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 $\dfrac {x^{2}}{4}+ \dfrac {y^{2}}{3}=1$ with its left and right foci denoted as $F_{1}$ and $F_{2}$ respectively, and a point $P$ on the ellipse. If $\overrightarrow{PF_{1}}\cdot \overrightarrow{PF_{2}}= \dfrac {5}{2}$, calculate $| \overrightarrow{PF_{1}}|\cdot| \overrightarrow{PF_{2}}|$.

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


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

Given $f(n) = n^2 \cos(n\pi)$ and $a_n = f(n) + f(n+1)$, find the sum of $a_1 + a_2 + a_3 + \cdots + a_{100}$.

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>.

Compute the following expressions:

(1) $2 \sqrt{12} -6 \sqrt{ \frac{1}{3}} + \sqrt{48}$

(2) $(\sqrt{3}-\pi)^{0}-\frac{\sqrt{20}-\sqrt{15}}{\sqrt{5}}+(-1)^{2017}$

Answer: $\sqrt{3} - 2$


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

In a triangle \( \triangle ABC \), \(a\), \(b\), and \(c\) are the sides opposite to angles \(A\), \(B\), and \(C\) respectively, with \(B= \dfrac {2\pi}{3}\). If \(a^{2}+c^{2}=4ac\), then find the value of \( \dfrac {\sin (A+C)}{\sin A\sin C} \).

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


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

There are 552 weights with masses of 1g, 2g, 3g, ..., 552g. Divide them into three equal weight piles.

Answer: $50876$


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

A point $Q$ is chosen in the interior of $\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\triangle DEF$, the resulting smaller triangles $u_{1}$, $u_{2}$, and $u_{3}$ have areas $16$, $25$, and $36$, respectively. Furthermore, a circle centered at $Q$ inside $\triangle DEF$ cuts off a segment from $u_3$ with area $9$. Find the area of $\triangle DEF$.

Answer: $225$


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

Given that the terminal side of angle $\alpha$ is in the second quadrant and intersects the unit circle at point $P(m, \frac{\sqrt{15}}{4})$.
$(1)$ Find the value of the real number $m$;
$(2)$ Let $f(\alpha) = \frac{\cos(2\pi - \alpha) + \tan(3\pi + \alpha)}{\sin(\pi - \alpha) \cdot \cos(\alpha + \frac{3\pi}{2})}$. Find the value of $f(\alpha)$.

Answer: $-\frac{4 + 16\sqrt{15}}{15}$


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>.

Square $XYZW$ has area $144$. Point $P$ lies on side $\overline{XW}$, such that $XP = 2WP$. Points $Q$ and $R$ are the midpoints of $\overline{ZP}$ and $\overline{YP}$, respectively. Quadrilateral $XQRW$ has an area of $20$. Calculate the area of triangle $RWP$.

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>.

The energy stored by a pair of positive charges is inversely proportional to the distance between them, and directly proportional to their charges. Four identical point charges are initially placed at the corners of a square with each side length $d$. This configuration stores a total of $20$ Joules of energy. How much energy, in Joules, would be stored if two of these charges are moved such that they form a new square with each side doubled (i.e., side length $2d$)?

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>.

The diagram shows a rhombus and two sizes of regular hexagon. What is the ratio of the area of the smaller hexagon to the area of the larger hexagon?

Answer: $1:4$


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

Given that five boys, A, B, C, D, and E, are randomly assigned to stay in 3 standard rooms (with at most two people per room), calculate the probability that A and B stay in the same standard room.

Answer: $\frac{1}{5}$


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>.

Use the Horner's method to calculate the value of the polynomial $f(x) = 7x^7 + 6x^6 + 5x^5 + 4x^4 + 3x^3 + 2x^2 + x$ when $x = 3$, and find the value of $v_3$.

Answer: $262$


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

The eccentricity of the ellipse $\frac {x^{2}}{9}+ \frac {y^{2}}{4+k}=1$ is $\frac {4}{5}$. Find the value of $k$.

Answer: $21$


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

$A_{2n}^{n+3} + A_{4}^{n+1} = \boxed{\text{\_\_\_\_\_\_\_\_}}$.

Answer: $744$


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

How many positive integers less than $800$ are either a perfect cube or a perfect square?

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>.

How many three-digit numbers are increased by 99 when their digits are reversed?

Answer: $80$


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

In the sequence $\{a_{n}\}$, the sum of the first $n$ terms is $S_{n}=2^{n}+2$. Find the sum $T_{20}$ of the first 20 terms in the sequence $\{\log _{2}a_{n}\}$.

Answer: $192$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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, the distance from the center of the circle $\rho=4\cos\theta$ ($\rho\in\mathbb{R}$) to the line $\theta= \frac {\pi}{3}$ can be found using the formula for the distance between a point and a line in polar coordinates.

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>.

The sum of the digits in the product of $\overline{A A A A A A A A A} \times \overline{B B B B B B B B B}$.

Answer: $81$


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

Given that Sofia has a $5 \times 7$ index card, if she shortens the length of one side by $2$ inches and the card has an area of $21$ square inches, find the area of the card in square inches if instead she shortens the length of the other side by $1$ inch.

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>.

Solve the equations:
$(1)x^{2}-5x+1=0$;
$(2) 2\left(x-5\right)^{2}+x\left(x-5\right)=0$.

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>.

What is the probability that when two numbers are randomly selected from the set {1, 2, 3, 4}, one number is twice the other?

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 positive integers $a$ and $b$ are members of a set where $a \in \{2, 3, 5, 7\}$ and $b \in \{2, 4, 6, 8\}$, and the sum of $a$ and $b$ must be even, determine the smallest possible value for the expression $2 \cdot a - a \cdot b$.

Answer: $-12$


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

Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=\sqrt{2}$, and $\overrightarrow{a}\perp(\overrightarrow{a}+2\overrightarrow{b})$, calculate the projection of $\overrightarrow{b}$ in the direction of $\overrightarrow{a}$.

Answer: $-1$


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

The parabolas $y = (x - 2)^2$ and $x + 6 = (y - 2)^2$ intersect at four points $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4)$. Find
\[
x_1 + x_2 + x_3 + x_4 + y_1 + y_2 + y_3 + y_4.
\]

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>.

For all real numbers $x$ and $y$, define the mathematical operation $\diamond$ such that $x \diamond 0 = 2x, x \diamond y = y \diamond x$, and $(x + 1) \diamond y = (x \diamond y) \cdot (y + 2)$. What is the value of $6 \diamond 3$?

Answer: $93750$


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

Let  $\mathcal S$  be a set of  $16$  points in the plane, no three collinear. Let  $\chi(S)$  denote the number of ways to draw  $8$  lines with endpoints in  $\mathcal S$ , such that no two drawn segments intersect, even at endpoints. Find the smallest possible value of  $\chi(\mathcal S)$  across all such  $\mathcal S$ .

*Ankan Bhattacharya*

Answer: $1430$


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

Let  $ y_0$  be chosen randomly from  $ \{0, 50\}$ , let  $ y_1$  be chosen randomly from  $ \{40, 60, 80\}$ , let  $ y_2$  be chosen randomly from  $ \{10, 40, 70, 80\}$ , and let  $ y_3$  be chosen randomly from  $ \{10, 30, 40, 70, 90\}$ . (In each choice, the possible outcomes are equally likely to occur.) Let  $ P$  be the unique polynomial of degree less than or equal to  $ 3$  such that  $ P(0) \equal{} y_0$ ,  $ P(1) \equal{} y_1$ ,  $ P(2) \equal{} y_2$ , and  $ P(3) \equal{} y_3$ . What is the expected value of  $ P(4)$ ?

Answer: $107$


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

In $\triangle ABC$, $\cos A= \frac{\sqrt{3}}{3}$, $c=\sqrt{3}$, and $a=3\sqrt{2}$. Find the value of $\sin C$ and the area of $\triangle ABC$.

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


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

Madam Mim has a deck of  $52$  cards, stacked in a pile with their backs facing up. Mim separates the small pile consisting of the seven cards on the top of the deck, turns it upside down, and places it at the bottom of the deck. All cards are again in one pile, but not all of them face down; the seven cards at the bottom do, in fact, face up. Mim repeats this move until all cards have their backs facing up again. In total, how many moves did Mim make  $?$

Answer: $52$


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

Two circles of radius \( r \) are externally tangent to each other and internally tangent to the ellipse \( x^2 + 4y^2 = 8 \). Find \( r \).

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


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

Find the product of all positive integral values of $x$ such that $x^2 - 40x + 399 = q$ for some prime number $q$. Note that there must be at least one such $x$.

Answer: $396$


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

A particle with charge  $8.0 \, \mu\text{C}$  and mass  $17 \, \text{g}$  enters a magnetic field of magnitude  $\text{7.8 mT}$  perpendicular to its non-zero velocity. After 30 seconds, let the absolute value of the angle between its initial velocity and its current velocity, in radians, be  $\theta$ . Find  $100\theta$ .

*(B. Dejean, 5 points)*

Answer: $1.101$


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

Given the line $y=ax$ intersects the circle $C:x^2+y^2-2ax-2y+2=0$ at points $A$ and $B$, and $\Delta ABC$ is an equilateral triangle, then the area of circle $C$ is __________.

Answer: $6\pi$


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>.

In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, with $b=1$, and $2\cos C - 2a - c = 0$.

(Ⅰ) Find the magnitude of angle $B$;
(Ⅱ) Find the distance from the circumcenter of $\triangle ABC$ to side $AC$.

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


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

In Mr. Johnson's class, 12 out of 20 students received an 'A' grade and the rest received a 'B' grade. Mrs. Smith, teaching a different class, observed that the proportion of students getting 'A' was the same. If Mrs. Smith has 30 students total, how many students received an 'A' grade? Moreover, if the same proportion received 'B' as in Mr. Johnson’s class, how many students in Mrs. Smith’s class received 'B'?

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>.

The sizes of circular pizzas are determined by their diameter. If Lana's initial pizza was 14 inches in diameter and she decides to order a larger pizza with a diameter of 18 inches instead, what is the percent increase in the area of her pizza?

Answer: $65.31\%$


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

Given the function $f(x) = x^3 - ax^2 + 3x$, and $x=3$ is an extremum of $f(x)$.
(Ⅰ) Determine the value of the real number $a$;
(Ⅱ) Find the equation of the tangent line $l$ to the graph of $y=f(x)$ at point $P(1, f(1))$;
(Ⅲ) Find the minimum and maximum values of $f(x)$ on the interval $[1, 5]$.

Answer: $15$


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

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

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


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

In a right triangle $\triangle PQR$, we know that $\tan Q = 0.5$ and the length of $QP = 16$. What is the length of $QR$?

Answer: $8 \sqrt{5}$


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

In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is
$$
\begin{cases}
x=2+t\cos \alpha \\
y=1+t\sin \alpha
\end{cases}
(t \text{ is the parameter}),
$$
In the polar coordinate system (which uses the same unit length as the Cartesian coordinate system $xOy$, with the origin as the pole and the positive $x$-axis as the polar axis), the equation of circle $C$ is $\rho = 6\cos \theta$.

(1) Find the Cartesian coordinate equation of circle $C$.

(2) Suppose circle $C$ intersects line $l$ at points $A$ and $B$. If point $P$ has coordinates $(2,1)$, find the minimum value of $|PA|+|PB|$.

Answer: $2\sqrt{7}$


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

There was a bonus fund in a certain institution. It was planned to distribute the fund such that each employee of the institution would receive $50. However, it turned out that the last employee on the list would receive only $45. Then, in order to ensure fairness, it was decided to give each employee $45, leaving $95 undistributed, which would be carried over to the fund for the next year. What was the amount of the initial fund?

Answer: $950$


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

One interior angle in a triangle measures $50^{\circ}$. What is the angle between the bisectors of the remaining two interior angles?

Answer: $65$


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

Evaluate the argument $\theta$ of the complex number
\[
e^{11\pi i/60} + e^{31\pi i/60} + e^{51 \pi i/60} + e^{71\pi i /60} + e^{91 \pi i /60}
\]
expressed in the form $r e^{i \theta}$ with $0 \leq \theta < 2\pi$.

Answer: $\frac{17\pi}{20}$


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

Let set $\mathcal{A}$ be a 70-element subset of $\{1,2,3,\ldots,120\}$, and let $S$ be the sum of the elements of $\mathcal{A}$. Find the number of possible values of $S$.

Answer: $3501$


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

Let $d$ be a positive number such that when $144$ is divided by $d$, the remainder is $9$. Compute the sum of all possible two-digit values of $d$.

Answer: $87$


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

Let  $ABC$  be an equilateral triangle . Let  point  $D$  lie on side  $AB,E$   lie on side  $AC, D_1$  and  $E_1$  lie on side BC such that  $AB=DB+BD_1$  and  $AC=CE+CE_1$ . Calculate the smallest angle between the lines  $DE_1$  and  $ED_1$ .

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>.

The sides of a triangle have lengths \( 13, 17, \) and \( k, \) where \( k \) is a positive integer. For how many values of \( k \) is the triangle obtuse?

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>.

An integer $N$ is selected at random in the range $1 \leq N \leq 2030$. Calculate the probability that the remainder when $N^{12}$ is divided by $7$ is $1$.

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


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

A chessboard’s squares are labeled with numbers as follows:

[asy]
unitsize(0.8 cm);

int i, j;

for (i = 0; i <= 8; ++i) {
  draw((i,0)--(i,8));
  draw((0,i)--(8,i));
}

for (i = 0; i <= 7; ++i) {
for (j = 0; j <= 7; ++j) {
  label("$\frac{1}{" + string(9 - i + j) + "}$", (i + 0.5, j + 0.5));
}}
[/asy]

Eight of the squares are chosen such that each row and each column has exactly one selected square. Find the maximum sum of the labels of these eight chosen squares.

Answer: $\frac{8}{9}$


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

If $2\tan\alpha=3\tan \frac{\pi}{8}$, then $\tan\left(\alpha- \frac{\pi}{8}\right)=$ ______.

Answer: $\frac{5\sqrt{2}+1}{49}$


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

After reading the following solution, answer the question: Xiaofang found in the simplification of $\sqrt{7+4\sqrt{3}}$ that first, $\sqrt{7+4\sqrt{3}}$ can be simplified to $\sqrt{7+2\sqrt{12}}$. Since $4+3=7$ and $4\times 3=12$, that is, ${(\sqrt{4})^2}+{(\sqrt{3})^2}=7$, $\sqrt{4}×\sqrt{3}=\sqrt{12}$, so $\sqrt{7+4\sqrt{3}}=\sqrt{7+2\sqrt{12}}=\sqrt{{{(\sqrt{4})}^2}+2\sqrt{4×3}+{{(\sqrt{3})}^2}}=\sqrt{{{(\sqrt{4}+\sqrt{3})}^2}}=2+\sqrt{3}$. The question is:
$(1)$ Fill in the blanks: $\sqrt{4+2\sqrt{3}}=$______, $\sqrt{5-2\sqrt{6}}=$______;
$(2)$ Further research reveals that the simplification of expressions in the form of $\sqrt{m±2\sqrt{n}}$ can be done by finding two positive numbers $a$ and $b\left(a \gt b\right)$ such that $a+b=m$, $ab=n$, that is, ${(\sqrt{a})^2}+{(\sqrt{b})^2}=m$, $\sqrt{a}×\sqrt{b}=\sqrt{n}$, then we have: $\sqrt{m±2\sqrt{n}}=\_\_\_\_\_\_.$
$(3)$ Simplify: $\sqrt{4-\sqrt{15}$ (Please write down the simplification process).

Answer: $\frac{\sqrt{10}}{2}-\frac{\sqrt{6}}{2}$


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

Calculate $3.6 \times 0.3$.

Answer: $1.08$


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

If $\tan \alpha= \sqrt {2}$, then $2\sin ^{2}\alpha-\sin \alpha\cos \alpha+\cos ^{2}\alpha=$ \_\_\_\_\_\_ .

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


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

There are two targets, A and B. A shooter shoots at target A once, with a probability of $\frac{3}{4}$ of hitting it and scoring $1$ point, or missing it and scoring $-1$ point. The shooter shoots at target B twice, with a probability of $\frac{2}{3}$ of hitting it and scoring $2$ points each time, or missing it and scoring $0$ points. The results of each shot are independent. Assuming the shooter completes all three shots, what is the probability of scoring $3$ points?

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


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

For an arithmetic sequence $b_1, b_2, b_3, \dots,$ let
\[S_n = b_1 + b_2 + b_3 + \dots + b_n,\]and let
\[T_n = S_1 + S_2 + S_3 + \dots + S_n.\]Given the value of $S_{2023},$ then you can uniquely determine the value of $T_n$ for some integer $n.$ What is this integer $n$?

Answer: $3034$


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

Compute the following sum:
\[
\frac{1}{2^{2024}} \sum_{n = 0}^{1011} (-3)^n \binom{2024}{2n}.
\]

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 decomposition rate $v$ of a certain type of garbage approximately satisfies the relationship $v=a\cdot b^{t}$, where $a$ and $b$ are non-zero constants, and $v=5\%$ after $6$ months and $v=10\%$ after $12$ months, determine the time needed for this type of garbage to completely decompose.

Answer: $32$


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

If the line $y=ax+b$ is a tangent line of the graph of the function $f(x)=\ln{x}-\frac{1}{x}$, then the minimum value of $a+b$ is $\_\_\_\_\_\_\_\_$.

Answer: $-1$


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

Given the function $y=\left[x\right]$, which is called the Gaussian function and represents the greatest integer not exceeding $x$, such as $\left[\pi \right]=3$, $\left[-2.5\right]=-3$. The solution set of the inequality $\frac{[x]}{[x]-4}＜0$ is ______; when $x \gt 0$, the maximum value of $\frac{[x]}{[x]^2+4}$ 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>.

Given the line $l: y = kx + m$ intersects the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ at points $A$ and $P$, and intersects the $x$-axis and $y$-axis at points $N$ and $M$ respectively, with $PM = MN$. Point $Q$ is the reflection of point $P$ across the $x$-axis, and the extension of line $QM$ intersects the ellipse at point $B$. Perpendicular lines to the $x$-axis are drawn from points $A$ and $B$, with the feet of the perpendiculars being $A_1$ and $B_1$ respectively.

(Ⅰ) If the left and right foci of the ellipse $C$ and one endpoint of its minor axis form the vertices of an equilateral triangle, and point $D(1,\frac{3}{2})$ lies on the ellipse $C$, find the equation of the ellipse $C$;

(Ⅱ) When $k = \frac{1}{2}$, if point $N$ bisects the line segment $A_1B_1$, find the eccentricity of the ellipse $C$.

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>.

Two concentric circles  $\omega, \Omega$  with radii  $8,13$  are given.  $AB$  is a diameter of  $\Omega$  and the tangent from  $B$  to  $\omega$  touches  $\omega$  at  $D$ . What is the length of  $AD$ .

Answer: $19$


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

In a race, all runners must start at point $A$, touch any part of a 1500-meter wall, and then stop at point $B$. Given that the distance from $A$ directly to the wall is 400 meters and from the wall directly to $B$ is 600 meters, calculate the minimum distance a participant must run to complete this. Express your answer to the nearest meter.

Answer: $1803$


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

Kelvin the Frog is playing the game of Survival. He starts with two fair coins. Every minute, he flips all his coins one by one, and throws a coin away if it shows tails. The game ends when he has no coins left, and Kelvin's score is the *square* of the number of minutes elapsed. What is the expected value of Kelvin's score? For example, if Kelvin flips two tails in the first minute, the game ends and his score is 1.

Answer: $\frac{64}{9}$


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

Simplify first, then evaluate: $(1-\frac{2}{{m+1}})\div \frac{{{m^2}-2m+1}}{{{m^2}-m}}$, where $m=\tan 60^{\circ}-1$.

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


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

In $\triangle ABC$, it is known that $\cos C + (\cos A - \sqrt{3} \sin A) \cos B = 0$.

(1) Find the measure of angle $B$.

(2) If $\sin (A - \frac{\pi}{3}) = \frac{3}{5}$, find $\sin 2C$.

Answer: $\frac{24 + 7\sqrt{3}}{50}$


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>.

James has 6 ounces of tea in a ten-ounce mug and 6 ounces of milk in a separate ten-ounce mug. He first pours one-third of the tea from the first mug into the second mug and stirs well. Then he pours one-fourth of the mixture from the second mug back into the first. What fraction of the liquid in the first mug is now milk?
A) $\frac{1}{6}$
B) $\frac{1}{5}$
C) $\frac{1}{4}$
D) $\frac{1}{3}$
E) $\frac{1}{2}$

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>.

A number is composed of 6 millions, 3 tens of thousands, and 4 thousands. This number is written as ____, and when rewritten in terms of "ten thousands" as the unit, it becomes ____ ten thousands.

Answer: $603.4$


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

Given that in triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $2\sin A + \sin B = 2\sin C\cos B$ and the area of $\triangle ABC$ is $S = \frac{\sqrt{3}}{2}c$, find the minimum value of $ab$.

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>.

What is the total number of digits used when the first 3003 positive even integers are written?

Answer: $11460$


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

Calculate $52103_{8} - 1452_{9}$ in base 10.

Answer: $20471$


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

There are  $63$  houses at the distance of  $1, 2, 3, . . . , 63 \text{ km}$  from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants,  $63$  elfs named  $E_1, E_2, . . . , E_{63}$  , deliever the vaccine to each house; each elf will deliever vaccine to exactly one house and never return. Suppose that the elf  $E_n$   takes  $n$  minutes to travel  $1 \text{ km}$  for each  $n = 1,2,...,63$  , and that all elfs leave the north pole simultaneously. What is the minimum amount of time to complete the delivery?

Answer: $1024$


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

A class has $50$ students. The math scores $\xi$ of an exam follow a normal distribution $N(100, 10^{2})$. Given that $P(90 \leqslant \xi \leqslant 100)=0.3$, estimate the number of students with scores of $110$ or higher.

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>.

Evaluate the value of $\frac{1}{4}\cdot\frac{8}{1}\cdot\frac{1}{32}\cdot\frac{64}{1} \dotsm \frac{1}{1024}\cdot\frac{2048}{1}$.

Answer: $32$


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

Given an arithmetic sequence $\{a_n\}$, where $a_n \in \mathbb{N}^*$, and $S_n = \frac{1}{8}(a_n+2)^2$. If $b_n = \frac{1}{2}a_n - 30$, find the minimum value of the sum of the first \_\_\_\_\_\_ terms of the sequence $\{b_n\}$.

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>.

Let $g$ be a function taking the nonnegative integers to the nonnegative integers, such that
\[2g(a^2 + b^2) = [g(a)]^2 + [g(b)]^2\] for all nonnegative integers $a$ and $b.$

Let $n$ be the number of possible values of $g(16),$ and let $s$ be the sum of the possible values of $g(16).$  Find $n \times s.$

Answer: $99$


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

The diagonal lengths of a rhombus are 18 units and 26 units. Calculate both the area and the perimeter of the rhombus.

Answer: $20\sqrt{10}$


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

A parallelogram has side lengths of 10, $12x-2$, $5y+5$, and $4$. What is the value of $x+y$?

[asy]draw((0,0)--(24,0)--(30,20)--(6,20)--cycle);
label("$12x-2$",(15,0),S);
label("10",(3,10),W);
label("$5y+5$",(18,20),N);
label("4",(27,10),E);
[/asy]

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


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

The difference between two perfect squares is 221. What is the smallest possible sum of the two perfect squares?

Answer: $24421$


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

Given a line segment $\overline{AB}=10$ cm, a point $C$ is placed on $\overline{AB}$ such that $\overline{AC} = 6$ cm and $\overline{CB} = 4$ cm. Three semi-circles are drawn with diameters $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$, external to the segment. If a line $\overline{CD}$ is drawn perpendicular to $\overline{AB}$ from point $C$ to the boundary of the smallest semi-circle, find the ratio of the shaded area to the area of a circle taking $\overline{CD}$ as its radius.

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


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

In $\triangle ABC$, medians $\overline{AM}$ and $\overline{BN}$ are perpendicular. If $AM = 15$ and $BN = 20$, find the length of side $AB$.

Answer: $\frac{50}{3}$


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>.

Sarah subscribes to a virtual fitness class platform that charges a monthly membership fee plus a per-class fee. If Sarah paid a total of $30.72 in February for 4 classes, and $54.72 in March for 8 classes, with the monthly membership fee increasing by 10% from February to March, calculate the fixed monthly membership fee.

Answer: $7.47$


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

Calculate the volumes of solids formed by rotating the region bounded by the function graphs about the \( O y \) (y-axis).

$$
y = x^{2} + 1, \quad y = x, \quad x = 0, \quad x = 1
$$

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