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

Let $S_{n}$ be the sum of the first $n$ terms of a geometric sequence $\{a_{n}\}$. If $S_{4}=-5$ and $S_{6}=21S_{2}$, calculate the value of $S_{8}$.

Answer: $-85$


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

Find the smallest natural number that starts with the digit five, which becomes four times smaller if this five is removed from the beginning of its decimal representation and appended to the end.

Answer: $512820$


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

(1) Use the Horner's method to calculate the polynomial $f(x) = 3x^6 + 5x^5 + 6x^4 + 79x^3 - 8x^2 + 35x + 12$ when $x = -4$, find the value of $v_3$.
(2) Convert the hexadecimal number $210_{(6)}$ into a decimal number.

Answer: $78$


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

Cátia leaves school every day at the same time and rides her bicycle home. When she pedals at $20 \mathrm{~km} / \mathrm{h}$, she arrives home at $16 \mathrm{~h} 30 \mathrm{~m}$. If she pedals at $10 \mathrm{~km} / \mathrm{h}$, she arrives home at $17 \mathrm{~h} 15 \mathrm{~m}$. At what speed should she pedal to arrive home at $17 \mathrm{~h}$?

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>.

Sergey, while being a student, worked part-time at a student cafe throughout the year. Sergey's salary was 9000 rubles per month. In the same year, Sergey paid 100000 rubles for his treatment at a healthcare institution and purchased medication prescribed by a doctor for 20000 rubles (eligible for deduction). The following year, Sergey decided to claim a social tax deduction. What amount of personal income tax paid is subject to refund to Sergey from the budget? (Answer as a whole number, without spaces or units of measurement.)

Answer: $14040$


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

Cameron writes down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. How many integers are in Cameron's list?

Answer: $871$


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

Let \( a, b, c, d \) be real numbers defined by

$$
a=\sqrt{4-\sqrt{5-a}}, \quad b=\sqrt{4+\sqrt{5-b}}, \quad c=\sqrt{4-\sqrt{5+c}}, \quad d=\sqrt{4+\sqrt{5+d}}
$$

Calculate their product.

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>.

In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is given by
$$
\begin{cases}
x = 1 + t\cos\alpha \\
y = 2 + t\sin\alpha
\end{cases}
$$
where $t$ is the parameter. In the polar coordinate system, which uses the same unit length as $xOy$ and has the origin $O$ as the pole and the positive $x$-axis as the polar axis, the equation of circle $C$ is $\rho = 6\sin\theta$.
(I) Find the Cartesian coordinate equation of circle $C$.
(II) Suppose circle $C$ intersects line $l$ at points A and B. If point P has coordinates (1, 2), 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>.

An element is randomly chosen from among the first $20$ rows of Pascal's Triangle. What is the probability that the value of the element chosen is $1$?

Note: The 1 at the top is often labelled the "zeroth" row of Pascal's Triangle, by convention. So to count a total of 20 rows, use rows 0 through 19.

Answer: $\frac{39}{210}$


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

Calculate the limit of the function:

$$\lim _{x \rightarrow 0} \frac{3^{5 x}-2^{x}}{x-\sin 9 x}$$

Answer: $\frac{1}{8} \ln \frac{2}{243}$


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

A "pass level game" has the following rules: On the \( n \)-th level, a die is thrown \( n \) times. If the sum of the points that appear in these \( n \) throws is greater than \( 2^{n} \), then the player passes the level.

1. What is the maximum number of levels a person can pass in this game?
2. What is the probability of passing the first three levels in a row?

(Note: The die is a fair six-faced cube with numbers \( 1, 2, 3, 4, 5, \) and \( 6 \). After the die is thrown and comes to rest, the number on the top face is the outcome.)

Answer: $\frac{100}{243}$


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

Evaluate $\lfloor-5.77\rfloor+\lceil-3.26\rceil+\lfloor15.93\rfloor+\lceil32.10\rceil$.

Answer: $39$


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

In the tetrahedron \( P-ABC \), edges \( PA \), \( AB \), and \( AC \) are mutually perpendicular, and \( PA = AB = AC \). Let \( E \) and \( F \) be the midpoints of \( AB \) and \( PC \) respectively. Find the sine of the angle \(\theta\) between \( EF \) and the plane \( PBC \).

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>.

There are 7 volunteers to be arranged to participate in the promotional activities for the Shanghai World Expo on Saturday and Sunday, with 6 people participating each day and 3 people arranged for each day. There are __________ different arrangements possible (answer with a number).

Answer: $140$


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

Point \( M \) is the midpoint of side \( BC \) of triangle \( ABC \), where \( AB = 17 \), \( AC = 30 \), and \( BC = 19 \). A circle is constructed with diameter \( AB \). A point \( X \) is chosen arbitrarily on this circle. What is the minimum possible value of the length of segment \( MX \)?

Answer: $6.5$


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

A point  $ M$  is taken on the perpendicular bisector of the side  $ AC$  of an acute-angled triangle  $ ABC$  so that  $ M$  and  $ B$  are on the same side of  $ AC$ . If  $ \angle BAC\equal{}\angle MCB$  and  $ \angle ABC\plus{}\angle MBC\equal{}180^{\circ}$ , find  $ \angle BAC.$

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>.

In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $b\sin 2C=c\sin B$.
$(1)$ Find the value of angle $C$;
$(2)$ If $\sin \left(B- \frac {\pi}{3}\right)= \frac {3}{5}$, find the value of $\sin A$.

Answer: $\frac {4 \sqrt {3}-3}{10}$


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>.

Coins are arranged in a row from left to right. It is known that two of them are fake, they lie next to each other, the left one weighs 9 grams, the right one weighs 11 grams, and all the remaining are genuine and each weighs 10 grams. The coins are weighed on a balance scale, which either shows which of the two sides is heavier, or is in balance indicating the weights on both sides are equal. What is the maximum number \( n \) of coins for which it is possible to find the 11-gram coin in three weighings?

Answer: $28$


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

If in the expansion of the binomial \((x- \frac {2}{ \sqrt {x}})^{n}\), only the coefficient of the fifth term is the largest, then the coefficient of the term containing \(x^{2}\) is ______.

Answer: $1120$


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

Yasmine makes her own chocolate beverage by mixing volumes of milk and syrup in the ratio \(5: 2\). Milk comes in \(2 \text{ L}\) bottles and syrup comes in \(1.4 \text{ L}\) bottles. Yasmine has a limitless supply of full bottles of milk and of syrup. Determine the smallest volume of chocolate beverage that Yasmine can make that uses only whole bottles of both milk and syrup.

Answer: $19.6$


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

Given that $x$ and $y$ are positive numbers, $\theta \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right)$, and it satisfies $\frac{\sin\theta}{x} = \frac{\cos\theta}{y}$ and $\frac{\cos^2\theta}{x^2} + \frac{\sin^2\theta}{y^2} = \frac{10}{3(x^2+y^2)}$, determine the value of $\frac{x}{y}$.

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>.

Evaluate the definite integral:

$$
\int_{0}^{\frac{\pi}{2}} \frac{\cos x \, dx}{(1+\cos x+\sin x)^{2}}
$$

Answer: $\ln(2) - \frac{1}{2}$


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>.

A container is already filled with water. There are three lead balls: large, medium, and small. The first time, the small ball is submerged in the water; the second time, the small ball is removed, and the medium ball is submerged in the water; the third time, the medium ball is removed, and the large ball is submerged in the water. It is known that the water spilled the first time is 3 times the water spilled the second time, and the water spilled the third time is 3 times the water spilled the first time. Find the ratio of the volumes of the three balls.

Answer: $3 : 4 : 13$


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>.

How many two-digit numbers have digits whose sum is a perfect square less than or equal to 25?

Answer: $17$


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

There are five gifts priced at 2 yuan, 5 yuan, 8 yuan, 11 yuan, and 14 yuan, and five boxes priced at 1 yuan, 3 yuan, 5 yuan, 7 yuan, and 9 yuan. Each gift is paired with one box. How many different total prices are possible?

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>.

Let $p, q, r, s, t, u, v, w$ be distinct elements in the set $\{-8, -6, -4, -1, 1, 3, 5, 14\}$. What is the minimum possible value of $(p+q+r+s)^2 + (t+u+v+w)^2$?

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 centers of faces $ABCD$ and $ADD'A'$ of the cube $ABCD-A'B'C'D'$ are $M$ and $N$ respectively. Find the sine of the angle between the skew lines $MN$ and $BD'$.

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


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>.

The average of a set of data 1, 3, 2, 5, $x$ is 3. What is the standard deviation of this set of data?

Answer: $\sqrt{2}$


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

Find the smallest integer  $n$  such that an  $n\times n$  square can be partitioned into  $40\times 40$  and  $49\times 49$  squares, with both types of squares present in the partition, if
a)  $40|n$ ; b)  $49|n$ ; c)  $n\in \mathbb N$ .

Answer: $1960$


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

Triangle $\triangle DEF$ has a right angle at $F$, $\angle D = 60^\circ$, and $DF = 6$. Find the radius of the incircle and the circumference of the circumscribed circle around $\triangle DEF$.

Answer: $12\pi$


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

Let's call any natural number "very prime" if any number of consecutive digits (in particular, a digit or number itself) is a prime number. For example,  $23$  and  $37$  are "very prime" numbers, but  $237$  and  $357$  are not. Find the largest "prime" number (with justification!).

Answer: $373$


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

Let  $l$  some line, that is not parallel to the coordinate axes. Find minimal  $d$  that always exists point  $A$  with integer coordinates, and distance from  $A$  to  $l$  is   $\leq d$

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


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

Each day 289 students are divided into 17 groups of 17. No two students are ever in the same group more than once. What is the largest number of days that this can be done?

Answer: $18$


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

Points \( K \) and \( N \) are located on the sides \( AB \) and \( AC \) of triangle \( ABC \) respectively, such that \( AK = BK \) and \( AN = 2NC \).

In what ratio does the segment \( KN \) divide the median \( AM \) of triangle \( ABC \)?

Answer: $4:3$


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

Given the function
$$
f(x)=
\begin{cases}
1 & (1 \leq x \leq 2) \\
\frac{1}{2}x^2 - 1 & (2 < x \leq 3)
\end{cases}
$$
define $h(a) = \max\{f(x) - ax \mid x \in [1, 3]\} - \min\{f(x) - ax \mid x \in [1, 3]\}$ for any real number $a$.
1. Find the value of $h(0)$.
2. Find the expression for $h(a)$ and its minimum value.

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


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

Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $\frac{x^{2}}{16}{+}\frac{y^{2}}{b^{2}}{=}1$, and the line $l$ passing through $F_1$ intersects the ellipse at points $A$ and $B$. If the maximum value of $|AF_2|+|BF_2|$ is $10$, find the eccentricity of the ellipse.

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


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

Let triangle $ABC$ be an equilateral triangle. A point $M$ in the plane of the triangle satisfies $\overrightarrow{AM}=\frac{1}{3}\overrightarrow{AB}+\frac{2}{3}\overrightarrow{AC}$. The cosine value of the angle between the vectors $\overrightarrow{AM}$ and $\overrightarrow{BC}$ is ______.

Answer: $\frac{\sqrt{7}}{14}$


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>.

Scatterbrained Scientist had a sore knee. The doctor prescribed 10 pills for the knee, to be taken one pill daily. These pills help in $90 \%$ of cases, but in $2 \%$ of cases, there is a side effect—it eliminates scatterbrainedness, if present.

Another doctor prescribed the Scientist pills for scatterbrainedness, also to be taken one per day for 10 consecutive days. These pills cure scatterbrainedness in $80 \%$ of cases, but in $5 \%$ of cases, there is a side effect—the knee pain stops.

The two bottles of pills look similar, and when the Scientist went on a ten-day business trip, he took one bottle with him but paid no attention to which one. He took one pill daily for ten days and returned completely healthy: the scatterbrainedness was gone and the knee pain was no more. Find the probability that the Scientist took the pills for scatterbrainedness.

Answer: $0.69$


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

Determine how many integers are in the list containing the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them.

Answer: $871$


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

Xiaoming and Xiaojun start simultaneously from locations A and B, heading towards each other. If both proceed at their original speeds, they meet after 5 hours. If both increase their speeds by 2 km/h, they meet after 3 hours. The distance between locations A and B is 30 km.

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>.

Compute the definite integral:

$$
\int_{0}^{\frac{\pi}{4}} \left( x^{2} + 17.5 \right) \sin 2x \, dx
$$

Answer: $\frac{68 + \pi}{8}$


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>.

On an island, there are knights, liars, and followers; each one knows who is who among them. All 2018 island inhabitants were lined up and each was asked to answer "Yes" or "No" to the question: "Are there more knights than liars on the island?" The inhabitants answered one by one in such a way that the others could hear. Knights told the truth, liars lied. Each follower answered the same as the majority of those who had answered before them, and if the number of "Yes" and "No" answers was equal, they could give either answer. It turned out that there were exactly 1009 "Yes" answers. What is the maximum number of followers that could be among the island inhabitants?

Answer: $1009$


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

Find the largest integer \( a \) such that the expression

\[
a^2 - 15a - (\tan x - 1)(\tan x + 2)(\tan x + 5)(\tan x + 8)
\]

is less than 35 for all values of \( x \in (-\pi/2, \pi/2) \).

Answer: $10$


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

Given a parallelogram \(ABCD\) with sides \(AB=2\) and \(BC=3\), find the area of this parallelogram, given that the diagonal \(AC\) is perpendicular to the segment \(BE\), where \(E\) is the midpoint of side \(AD\).

Answer: $\sqrt{35}$


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>.

Let **v** be a vector such that
\[
\left\| \mathbf{v} + \begin{pmatrix} 4 \\ 2 \end{pmatrix} \right\| = 10.
\]
Find the smallest possible value of $\|\mathbf{v}\|$.

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


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>.

Find a ten-digit number where the first digit indicates how many times the digit 0 appears in the number, the second digit indicates how many times the digit 1 appears, and so forth, with the tenth digit indicating how many times the digit 9 appears in the number.

Generalize and solve the problem for a number system with base $n$.

Answer: $6210001000$


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

A number contains only two kinds of digits: 3 or 4, and both 3 and 4 appear at least once. The number is a multiple of both 3 and 4. What is the smallest such number?

Answer: $3444$


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

In the complex plane, the points \( 0, z, \frac{1}{z}, z+\frac{1}{z} \) form a parallelogram with an area of \( \frac{35}{37} \). If the real part of \( z \) is greater than 0, find the minimum value of \( \left| z + \frac{1}{z} \right| \).

Answer: $\frac{5 \sqrt{74}}{37}$


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

In triangle $ABC$, if $A = \frac{\pi}{3}$, $\tan B = \frac{1}{2}$, and $AB = 2\sqrt{3} + 1$, then find the length of $BC$.

Answer: $\sqrt{15}$


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 \(A B C D\) be a square of side length 13. Let \(E\) and \(F\) be points on rays \(A B\) and \(A D\), respectively, so that the area of square \(A B C D\) equals the area of triangle \(A E F\). If \(E F\) intersects \(B C\) at \(X\) and \(B X = 6\), determine \(D F\).

Answer: $\sqrt{13}$


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>.

Mathematician Wiener, the founder of cybernetics, was asked about his age during his Ph.D. awarding ceremony at Harvard University because he looked very young. Wiener's interesting response was: "The cube of my age is a four-digit number, and the fourth power of my age is a six-digit number. These two numbers together use all the digits from 0 to 9 exactly once, with no repetition or omission." What is Wiener's age that year? (Note: The cube of a number \(a\) is equal to \(a \times a \times a\), and the fourth power of a number \(a\) is equal to \(a \times a \times a \times a\)).

Answer: $18$


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

In trapezoid $ABCD$, sides $AB$ and $CD$ are parallel with lengths of 10 and 24 units respectively, and the altitude is 15 units. Points $G$ and $H$ are the midpoints of sides $AD$ and $BC$, respectively. Determine the area of quadrilateral $GHCD$.

Answer: $153.75$


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

Let $b_1, b_2, \ldots$ be a sequence determined by the rule $b_n= \frac{b_{n-1}}{2}$ if $b_{n-1}$ is even and $b_n=3b_{n-1}+1$ if $b_{n-1}$ is odd. For how many positive integers $b_1 \le 1000$ is it true that $b_1$ is less than each of $b_2$, $b_3$, and $b_4$?

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 expression  $\circ \ 1\  \circ \ 2 \ \circ 3 \ \circ \dots \circ \ 2012$  is written on a blackboard. Catherine places a  $+$  sign or a  $-$  sign into each blank. She then evaluates the expression, and finds the remainder when it is divided by 2012. How many possible values are there for this remainder?

*Proposed by Aaron Lin*

Answer: $1006$


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

What is the ratio of the sides of a triangle in which the sum of the lengths of the altitudes taken two at a time corresponds to the ratio 5:7:8?

Answer: $10:15:6$


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>.

The number $7.21\times 10^{11}$ has how many digits in the original number.

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>.

Using the seven digits $1, 2, 3, 4, 5, 6, 7$ to appropriately arrange them into a 7-digit number so that it is a multiple of 11, how many such numbers can be formed?

Answer: $576$


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

Which are more: three-digit numbers where all digits have the same parity (all even or all odd), or three-digit numbers where adjacent digits have different parity?

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>.

Simplify first, then evaluate: $\dfrac{x^{2}-4x+4}{2x}\div \dfrac{x^{2}-2x}{x^{2}}+1$. Choose a suitable number from $0$, $1$, $2$, substitute it in and evaluate.

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


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

Given that a 1-step requires 4 toothpicks, a 2-step requires 10 toothpicks, a 3-step requires 18 toothpicks, and a 4-step requires 28 toothpicks, determine the number of additional toothpicks needed to build a 6-step staircase.

Answer: $26$


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>.

In the number \(2016 * * * * 02 *\), each of the 5 asterisks needs to be replaced by any of the digits \(0, 2, 4, 7, 8, 9\) (digits can repeat) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done?

Answer: $1728$


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

Lily is riding her bicycle at a constant rate of 15 miles per hour and Leo jogs at a constant rate of 9 miles per hour. If Lily initially sees Leo 0.75 miles in front of her and later sees him 0.75 miles behind her, determine the duration of time, in minutes, that she can see Leo.

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 the function \( f(x) \):
\[ f(x) = \begin{cases}
\ln x & \text{if } x > 1, \\
\frac{1}{2} x + \frac{1}{2} & \text{if } x \leq 1
\end{cases} \]
If \( m < n \) and \( f(m) = f(n) \), what is the minimum value of \( n - m \)?

Answer: $3 - 2\ln 2$


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

There are three types of snacks for the kitten. It eats a stick of cat food every 1 day, an egg yolk every 2 days, and nutritional cream every 3 days. The kitten ate cat stick and nutritional cream on March 23, and ate cat stick and egg yolk on March 25. Which day in March does the kitten eat all three types of snacks for the first time?

Answer: $29$


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

Find the number of four-digit numbers in which all digits are different, the first digit is divisible by 2, and the sum of the first and last digits is divisible by 3.

Answer: $672$


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

Given a triangle \( ABC \) with the condition:

\[ \cos (\angle A - \angle B) + \sin (\angle A + \angle B) = 2 \]

Find the side \( BC \) if \( AB = 4 \).

Answer: $2\sqrt{2}$


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

In triangle ABC, the lengths of the three sides are three consecutive natural numbers, and the largest angle is twice the smallest angle. Calculate the area of this triangle.

Answer: $\frac {15 \sqrt {7}}{4}$


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 real numbers $x$ and $y$ that satisfy the system of inequalities $\begin{cases} x - 2y - 2 \leqslant 0 \\ x + y - 2 \leqslant 0 \\ 2x - y + 2 \geqslant 0 \end{cases}$, if the minimum value of the objective function $z = ax + by + 5 (a > 0, b > 0)$ is $2$, determine the minimum value of $\frac{2}{a} + \frac{3}{b}$.

Answer: $\frac{10 + 4\sqrt{6}}{3}$


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

Reading material: In general, the equation $\frac{1}{x}+\frac{1}{y}=1$ is not valid, but some special real numbers can make it valid. For example, when $x=2$ and $y=2$, $\frac{1}{2}+\frac{1}{2}=1$ is valid. We call $(2,2)$ a "magical number pair" that makes $\frac{1}{x}+\frac{1}{y}=1$ valid. Please complete the following questions:
$(1)$ Among the pairs $(\frac{4}{3},4), (1,1)$, the "magical number pair" that makes $\frac{1}{x}+\frac{1}{y}=1$ valid is ______;
$(2)$ If $(5-t,5+t)$ is a "magical number pair" that makes $\frac{1}{x}+\frac{1}{y}=1$ valid, find the value of $t$;
$(3)$ If $(m,n)$ is a "magical number pair" that makes $\frac{1}{x}+\frac{1}{y}=1$ valid, and $a=b+m$, $b=c+n$, find the minimum value of the algebraic expression $\left(a-c\right)^{2}-12\left(a-b\right)\left(b-c\right)$.

Answer: $-36$


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

The polynomial \( x^8 - 4x^7 + 7x^6 + \cdots + a_0 \) has all its roots positive and real numbers. Find the possible values for \( a_0 \).

Answer: $\frac{1}{256}$


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

The sequence is defined recursively:

\[ x_{0} = 0, \quad x_{n+1} = \frac{(n^2 + n + 1) x_{n} + 1}{n^2 + n + 1 - x_{n}}. \]

Find \( x_{8453} \).

Answer: $8453$


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

The population size (in number of animals) of a certain animal species is given by the equation $y=a\log_{2}(x+1)$. Suppose that the population size of this animal species in the first year was 100 animals. What will be the population size in the 15th year.

Answer: $400$


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

Find the angle $D A C$ given that $A B = B C$ and $A C = C D$, and the lines on which points $A, B, C, D$ lie are parallel with equal distances between adjacent lines. Point $A$ is to the left of $B$, $C$ is to the left of $B$, and $D$ is to the right of $C$.

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>.

A high school is holding a speech contest with 10 participants. There are 3 students from Class 1, 2 students from Class 2, and 5 students from other classes. Using a draw to determine the speaking order, what is the probability that the 3 students from Class 1 are placed consecutively (in consecutive speaking slots) and the 2 students from Class 2 are not placed consecutively?

Answer: $\frac{1}{20}$


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 $ \tan \left( \alpha + \frac{\pi}{4} \right) = -\frac{1}{2} $ and $\frac{\pi}{2} < \alpha < \pi$, calculate the value of $ \frac{\sin 2\alpha - 2\cos^2 \alpha}{\sin \left(\alpha - \frac{\pi}{4} \right)} $.

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


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>.

Express $\frac{214_8}{32_5} + \frac{343_9}{133_4}$ in base 10.

Answer: $\frac{9134}{527}$


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

For a positive integer  $n$ , let  $f(n)$  be the sum of the positive integers that divide at least one of the nonzero base  $10$  digits of  $n$ . For example,  $f(96)=1+2+3+6+9=21$ . Find the largest positive integer  $n$  such that for all positive integers  $k$ , there is some positive integer  $a$  such that  $f^k(a)=n$ , where  $f^k(a)$  denotes  $f$  applied  $k$  times to  $a$ .

*2021 CCA Math Bonanza Lightning Round #4.3*

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>.

If \( a^3 + b^3 + c^3 = 3abc = 6 \) and \( a^2 + b^2 + c^2 = 8 \), find the value of \( \frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a} \).

Answer: $-8$


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>.

Using one each of the coins and bills of 1 jiao, 2 jiao, 5 jiao, 1 yuan, 2 yuan, and 5 yuan, how many different monetary values can be formed?

Answer: $63$


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

Calculate: \((56 \times 0.57 \times 0.85) \div(2.8 \times 19 \times 1.7) =\)

Answer: $0.3$


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>.

Rectangle $PQRS$ is inscribed in a semicircle with diameter $\overline{GH}$, such that $PR=20$, and $PG=SH=12$. Determine the area of rectangle $PQRS$.
A) $120\sqrt{6}$
B) $150\sqrt{6}$
C) $160\sqrt{6}$
D) $180\sqrt{6}$
E) $200\sqrt{6}$

Answer: $160\sqrt{6}$


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

Find the infinite sum of \(\frac{1^{3}}{3^{1}}+\frac{2^{3}}{3^{2}}+\frac{3^{3}}{3^{3}}+\frac{4^{3}}{3^{4}}+\cdots\).

求 \(\frac{1^{3}}{3^{1}}+\frac{2^{3}}{3^{2}}+\frac{3^{3}}{3^{3}}+\frac{4^{3}}{3^{4}}+\cdots\) 無限項之和。

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


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>.

Find the smallest natural number that consists of identical digits and is divisible by 18.

Answer: $666$


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

The function $f(x) = (m^2 - m - 1)x^m$ is a power function, and it is a decreasing function on $x \in (0, +\infty)$. The value of the real number $m$ 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>.

Three of the four vertices of a rectangle are $(3, 7)$, $(12, 7)$, and $(12, -4)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 3)^2 + (y + 4)^2 = 16$?

Answer: $4\pi$


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

Five bricklayers working together finish a job in  $3$  hours. Working alone, each bricklayer takes at most  $36$  hours to finish the job. What is the smallest number of minutes it could take the fastest bricklayer to complete the job alone?

*Author: Ray Li*

Answer: $270$


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

$8[x]$ represents the greatest integer not exceeding the real number $x$. Then, $\left[\log _{2} 1\right]+\left[\log _{2} 2\right]+\left[\log _{2} 3\right]+\cdots+\left[\log _{2} 2012\right]=$ $\qquad$ .

Answer: $18084$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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) $x(x+4)=-5(x+4)$
(2) $(x+2)^2=(2x-1)^2$

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


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

In the rectangular coordinate system, a polar coordinate system is established with the origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. Given the curve $C:ρ\sin^2θ=2a\cos θ (a > 0)$, the line $l:\begin{cases}x=-2+\frac{\sqrt{2}}{2}t\\y=-4+\frac{\sqrt{2}}{2}t\end{cases} (t$ is the parameter$) $ passes through point $P(-2,-4)$ and intersects curve $C$ at points $M$ and $N$.

(I) Find the ordinary equations of curve $C$ and line $l$;

(II) If $|PM|, |MN|, |PN|$ form a geometric sequence, find the value of real number $a$.

Answer: $a=1$


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>.

All natural numbers from 1 to 1000 inclusive are divided into two groups: even and odd. In which group is the sum of all the digits used to write the numbers greater and by how much?

Answer: $499$


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

Rotate an equilateral triangle with side length $2$ around one of its sides to form a solid of revolution. The surface area of this solid is ______.

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


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>.

A cylindrical tank with radius 6 feet and height 7 feet is lying on its side. The tank is filled with water to a depth of 3 feet. Find the volume of water in the tank, in cubic feet.

Answer: $84\pi - 63\sqrt{3}$


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

Given that 2 students exercised 0 days, 4 students exercised 1 day, 5 students exercised 2 days, 3 students exercised 4 days, 7 students exercised 5 days, and 2 students exercised 6 days, calculate the average number of days exercised last week by the students in Ms. Brown's class.

Answer: $3.17$


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

At a meeting of cactus enthusiasts, 80 cactophiles presented their collections, each consisting of cacti of different species. It turned out that no single species of cactus is found in all collections simultaneously, but any 15 people have cacti of the same species. What is the minimum total number of cactus species that can be in all collections?

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>.

If a 31-day month is taken at random, find \( c \), the probability that there are 5 Sundays in the month.

Answer: $3/7$


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

In a circular arrangement of 101 natural numbers, it is known that among any 5 consecutive numbers, there are at least two even numbers. What is the minimum number of even numbers that can be among the listed numbers?

Answer: $41$


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

A "clearance game" has the following rules: in the $n$-th round, a die is rolled $n$ times. If the sum of the points from these $n$ rolls is greater than $2^n$, then the player clears the round. Questions:
(1) What is the maximum number of rounds a player can clear in this game?
(2) What is the probability of clearing the first three rounds consecutively?
(Note: The die is a uniform cube with the numbers $1, 2, 3, 4, 5, 6$ on its faces. After rolling, the number on the top face is the result of the roll.)

Answer: $\frac{100}{243}$


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

Adam and Bettie are playing a game. They take turns generating a random number between  $0$  and  $127$  inclusive. The numbers they generate are scored as follows: $\bullet$  If the number is zero, it receives no points. $\bullet$  If the number is odd, it receives one more point than the number one less than it. $\bullet$  If the number is even, it receives the same score as the number with half its value.
if Adam and Bettie both generate one number, the probability that they receive the same score is  $\frac{p}{q}$  for relatively prime positive integers  $p$  and  $q$ . Find  $p$ .

Answer: $429$


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

Given that $\overrightarrow{AB} \perp \overrightarrow{AC}$, $|\overrightarrow{AB}|= \frac{1}{t}$, $|\overrightarrow{AC}|=t$, and point $P$ is a point on the plane of $\triangle ABC$ such that $\overrightarrow{AP}= \frac{\overrightarrow{AB}}{|\overrightarrow{AB}|} + \frac{4\overrightarrow{AC}}{|\overrightarrow{AC}|}$. Find the real value(s) of $t$ that satisfy $\overrightarrow{AP} \perp \overrightarrow{BC}$.

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


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

Calculate the limit of the function:

$\lim _{x \rightarrow \frac{1}{4}} \frac{\sqrt[3]{\frac{x}{16}}-\frac{1}{4}}{\sqrt{\frac{1}{4}+x}-\sqrt{2x}}$

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


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 the tetrahedron \( A B C D \),
$$
\begin{array}{l}
AB=1, BC=2\sqrt{6}, CD=5, \\
DA=7, AC=5, BD=7.
\end{array}
$$

Find its volume.

Answer: $\frac{\sqrt{66}}{2}$


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>.

If a class of 30 students is seated in a movie theater, then in any case at least two classmates will be in the same row. If the same is done with a class of 26 students, then at least three rows will be empty. How many rows are in the theater?

Answer: $29$


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

What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{30} \cdot 5^3}$ as a decimal?

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>.

For a four-digit natural number $M$, let the digit in the thousands place be $a$, in the hundreds place be $b$, in the tens place be $c$, and in the units place be $d$. The two-digit number formed by the thousands and units digits of $M$ is $A=10a+d$, and the two-digit number formed by the tens and hundreds digits of $M$ is $B=10c+b$. If the difference between $A$ and $B$ is equal to the negative of the sum of the thousands and hundreds digits of $M$, then $M$ is called an "open number." Determine whether $1029$ is an "open number" (fill in "yes" or "no"). If $M$ is an "open number," let $G(M)=\frac{b+13}{c-a-d}$. Find the maximum value of $M$ that satisfies the condition when $G(M)$ is divisible by $7$.

Answer: $8892$


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

In the triangular prism \(A-BCD\), the side edges \(AB, AC, AD\) are mutually perpendicular. The areas of triangles \(\triangle ABC\), \(\triangle ACD\), and \(\triangle ADB\) are \(\frac{\sqrt{2}}{2}\), \(\frac{\sqrt{3}}{2}\), and \(\frac{\sqrt{6}}{2}\) respectively. Find the volume of the circumscribed sphere of the triangular prism \(A-BCD\).

Answer: $\sqrt{6}\pi$


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>.

A woman wants freshly baked cookies delivered exactly at 18:00 for an event. Delivery trucks, upon finishing baking, travel with varying speeds due to potential traffic conditions:
- If there is moderate traffic, the trucks travel at an average speed of 60 km/h and would arrive at 17:45.
- If there are traffic jams, the trucks travel at an average speed of 20 km/h and would arrive at 18:15.

Determine the average speed the delivery truck must maintain to arrive exactly at 18:00.

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>.

In acute triangle $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and $4\sin ^{2} \frac {B+C}{2}-\cos 2A= \frac {7}{2}$.

1. Find the measure of angle $A$.
2. If the altitude on side $BC$ is $1$, find the minimum area of $\triangle ABC$.

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


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>.

Arrange for 7 staff members to be on duty from May 1st to May 7th. Each person is on duty for one day, with both members A and B not being scheduled on May 1st and 2nd. The total number of different scheduling methods is $\_\_\_\_\_\_\_$.

Answer: $2400$


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

Peter has three times as many sisters as brothers. His sister Louise has twice as many sisters as brothers. How many children are there in the family?

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>.

If \( N \) is a multiple of 84 and \( N \) contains only the digits 6 and 7, what is the smallest \( N \) that meets these conditions?

Answer: $76776$


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

In the Cartesian coordinate system $xOy$, the parametric equations of curve $C$ are $\left\{\begin{array}{l}{x=2+\sqrt{5}\cos\theta,}\\{y=\sqrt{5}\sin\theta}\end{array}\right.$ ($\theta$ is the parameter). Line $l$ passes through point $P(1,-1)$ with a slope of $60^{\circ}$ and intersects curve $C$ at points $A$ and $B$.
$(1)$ Find the general equation of curve $C$ and a parametric equation of line $l$;
$(2)$ Find the value of $\frac{1}{|PA|}+\frac{1}{|PB|}$.

Answer: $\frac{\sqrt{16+2\sqrt{3}}}{3}$


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

Calculate the area of one petal of the curve $\rho = \sin^2 \varphi$.

Answer: $\frac{3\pi}{16}$


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

The cost of 60 copies of the first volume and 75 copies of the second volume is 2700 rubles. In reality, the total payment for all these books was only 2370 rubles because a discount was applied: 15% off the first volume and 10% off the second volume. Find the original price of these books.

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>.

Write $\mathbf{2012}$ as the sum of $N$ distinct positive integers, where $N$ is at its maximum. What is the maximum value of $N$?

Answer: $62$


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

Given that $F_{1}$ and $F_{2}$ are the two foci of the ellipse $\frac{x^{2}}{20} + \frac{y^{2}}{4} = 1$, a line passing through $F_{2}$ intersects the ellipse at points $A$ and $B$. If $|F_{1}A| + |F_{1}B| = 5\sqrt{5}$, then $|AB| = $ ______.

Answer: $3\sqrt{5}$


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

Let  $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$  be positive reals satisfying \begin{align*} a + b &= c  a + b + c &= d  a + b + c + d &= e.\end{align*} If  $c=5$ , compute  $a+b+c+d+e$ .

*Proposed by Evan Chen*

Answer: $40$


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

When Xiaohong was 3 years old, her mother's age was the same as Xiaohong's age this year. When her mother is 78 years old, Xiaohong's age is the same as her mother’s age this year. How old is Xiaohong’s mother this year?

Answer: $53$


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

Two right triangles $\triangle ABC$ and $\triangle ABD$ share a side $AB$ in such way that $AB=8$, $BC=12$, and $BD=10$. Let $E$ be a point on $BC$ such that $BE=9$. Determine the area of $\triangle ABE$.

Answer: $36$


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

In the plane rectangular coordinate system $(xOy)$, establish a polar coordinate system $(.$ with $O$ as the pole and the positive semi-axis of $x$ as the polar axis. If the polar coordinate equation of the line $l$ is $\sqrt {2}ρ\cos (θ- \dfrac {π}{4})-2=0$, and the polar coordinate equation of the curve $C$ is: $ρ\sin ^{2}θ=\cos θ$, shrink the abscissa of all points on the curve $C$ to half of the original, and the ordinate remains unchanged. Then, translate it to the right by one unit to obtain the curve $C_{1}$.
(I) Find the rectangular coordinate equation of the curve $C_{1}$;
(II) It is known that the line $l$ intersects with the curve $C_{1}$ at points $A$ and $B$. Point $P(2,0)$, find the value of $(|PA|+|PB|)$.

Answer: $2 \sqrt {6}$


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

In the triangular pyramid $P-ABC$, $PA\bot $ plane $ABC$, $\triangle ABC$ is an isosceles triangle, where $AB=BC=2$, $\angle ABC=120{}^\circ $, and $PA=4$. The surface area of the circumscribed sphere of the triangular pyramid $P-ABC$ is __________.

Answer: $32\pi$


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

Given a circle $C: (x-3)^{2}+y^{2}=25$ and a line $l: (m+1)x+(m-1)y-2=0$ (where $m$ is a parameter), the minimum length of the chord intercepted by the circle $C$ and the line $l$ is ______.

Answer: $4\sqrt{5}$


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>.

\[
1.047. \left(\frac{\sqrt{561^{2} - 459^{2}}}{4 \frac{2}{7} \cdot 0.15 + 4 \frac{2}{7} : \frac{20}{3}} + 4 \sqrt{10}\right) : \frac{1}{3} \sqrt{40}
\]

Answer: $125$


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

Let \( A \) be a subset of \(\{1, 2, 3, \ldots, 2019\}\) having the property that the difference between any two of its elements is not a prime number. What is the largest possible number of elements in \( A \)?

Answer: $505$