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

Jacqueline has 200 liters of a chemical solution. Liliane has 30% more of this chemical solution than Jacqueline, and Alice has 15% more than Jacqueline. Determine the percentage difference in the amount of chemical solution between Liliane and Alice.

Answer: $13.04\%$


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

In triangle $PQR$, $\angle Q=90^\circ$, $PQ=9$ and $QR=12$. Points $S$ and $T$ are on $\overline{PR}$ and $\overline{QR}$, respectively, and $\angle PTS=90^\circ$. If $ST=6$, then what is the length of $PS$?

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>.

Jo climbs a flight of 8 stairs every day but is never allowed to take a 3-step when on any even-numbered step. Jo can take the stairs 1, 2, or 3 steps at a time, if permissible, under the new restriction. Find the number of ways Jo can climb these eight stairs.

Answer: $54$


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

The Weston Junior Football Club has 24 players on its roster, including 4 goalies. During a training session, a drill is conducted wherein each goalie takes turns defending the goal while the remaining players (including the other goalies) attempt to score against them with penalty kicks.

How many penalty kicks are needed to ensure that every player has had a chance to shoot against each of the goalies?

Answer: $92$


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

Given the function $f(x)=(a+ \frac {1}{a})\ln x-x+ \frac {1}{x}$, where $a > 0$.
(I) If $f(x)$ has an extreme value point in $(0,+\infty)$, find the range of values for $a$;
(II) Let $a\in(1,e]$, when $x_{1}\in(0,1)$, $x_{2}\in(1,+\infty)$, denote the maximum value of $f(x_{2})-f(x_{1})$ as $M(a)$, does $M(a)$ have a maximum value? If it exists, find its maximum value; if not, explain why.

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


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

Jack and Jill run a 12 km circuit. First, they run 7 km to a certain point and then the remaining 5 km back to the starting point by different, uneven routes. Jack has a 12-minute head start and runs at the rate of 12 km/hr uphill and 15 km/hr downhill. Jill runs 14 km/hr uphill and 18 km/hr downhill. How far from the turning point are they when they pass each other, assuming their downhill paths are the same but differ in uphill routes (in km)?
A) $\frac{226}{145}$
B) $\frac{371}{145}$
C) $\frac{772}{145}$
D) $\frac{249}{145}$
E) $\frac{524}{145}$

Answer: $\frac{772}{145}$


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

In $\triangle ABC$, let the sides opposite to angles $A$, $B$, and $C$ be $a$, $b$, and $c$ respectively. If $\sin A = \sin B = -\cos C$.

$(1)$ Find the sizes of angles $A$, $B$, and $C$;

$(2)$ If the length of the median $AM$ on side $BC$ is $\sqrt{7}$, find the area of $\triangle ABC$.

Answer: $\sqrt{3}$


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

The points $(2, 9), (12, 14)$, and $(4, m)$, where $m$ is an integer, are vertices of a triangle. What is the sum of the values of $m$ for which the area of the triangle is a minimum?

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>.

A standard die is rolled eight times. What is the probability that the product of all eight rolls is divisible by 4?

Answer: $\frac{247}{256}$


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>.

Given the set $M=\{a, b, -(a+b)\}$, where $a\in \mathbb{R}$ and $b\in \mathbb{R}$, and set $P=\{1, 0, -1\}$. If there is a mapping $f:x \to x$ that maps element $x$ in set $M$ to element $x$ in set $P$ (the image of $x$ under $f$ is still $x$), then the set $S$ formed by the points with coordinates $(a, b)$ has \_\_\_\_\_\_\_\_\_\_\_ subsets.

Answer: $64$


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

Let $a \star b = \frac{\sqrt{a^2+b}}{\sqrt{a^2 - b}}$. If $y \star 15 = 5$, find $y$.

Answer: $\frac{\sqrt{65}}{2}$


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

Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy: $|\overrightarrow{a}| = \sqrt{2}$, $|\overrightarrow{b}| = 2$, and $(\overrightarrow{a} - \overrightarrow{b}) \perp \overrightarrow{a}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

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


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

Given that Chelsea is leading by 60 points halfway through a 120-shot archery competition, each shot can score 10, 7, 3, or 0 points, and Chelsea always scores at least 3 points. If Chelsea's next \(n\) shots are all for 10 points, she will secure her victory regardless of her opponent's scoring in the remaining shots. Find the minimum value for \(n\).

Answer: $52$


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

In triangle $ABC$, $AB = 5$, $BC = 12$, and $AC = 13$. Let $BM$ be the median from vertex $B$ to side $AC$. If $BM = m \sqrt{2}$, then find $m$.

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


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

In $\triangle ABC$, $A=60^{\circ}$, $AC=4$, $BC=2\sqrt{3}$, the area of $\triangle ABC$ equals \_\_\_\_\_\_.

Answer: $2\sqrt{3}$


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

Let  $X=\{1,2,3,...,10\}$ . Find the number of pairs of  $\{A,B\}$  such that  $A\subseteq X, B\subseteq X, A\ne B$  and  $A\cap B=\{5,7,8\}$ .

Answer: $2186$


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

Let $r(x)$ be a monic quartic polynomial such that $r(1) = 5,$ $r(2) = 8,$ $r(3) = 13,$ and $r(4) = 20.$ Find $r(5).$

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>.

A rectangle can be divided into  $n$  equal squares. The same rectangle can also be divided into  $n+76$  equal squares. Find  $n$ .

Answer: $324$


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

A square with side length $x$ is inscribed in a right triangle with sides of length $6$, $8$, and $10$ so that one vertex of the square coincides with the right-angle vertex of the triangle. Another square with side length $y$ is inscribed in the same triangle, but one side of this square lies along the hypotenuse of the triangle. What is $\frac{x}{y}$?
A) $\frac{109}{175}$
B) $\frac{120}{190}$
C) $\frac{111}{175}$
D) $\frac{100}{160}$

Answer: $\frac{111}{175}$


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

Consider the following multiplicative magic square, where each row, column, and diagonal product equals the same value:

$\begin{tabular}{|c|c|c|} \hline 75 & \textit{b} & \textit{c} \\ \hline \textit{d} & \textit{e} & \textit{f} \\ \hline \textit{g} & \textit{h} & 3 \\ \hline \end{tabular}$

All entries are positive integers. Determine the possible values for $h$ and their sum.
A) 50
B) 75
C) 100
D) 125
E) 150

Answer: $150$


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

A math competition problem has probabilities of being solved independently by person \( A \), \( B \), and \( C \) as \( \frac{1}{a} \), \( \frac{1}{b} \), and \( \frac{1}{c} \) respectively, where \( a \), \( b \), and \( c \) are positive integers less than 10. When \( A \), \( B \), and \( C \) work on the problem simultaneously and independently, the probability that exactly one of them solves the problem is \( \frac{7}{15} \). Determine the probability that none of the three persons solve the problem.

Answer: $4/15$


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  $a,b$  be positive reals such that  $\frac{1}{a}+\frac{1}{b}\leq2\sqrt2$  and  $(a-b)^2=4(ab)^3$ . Find  $\log_a b$ .

Answer: $-1$


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

Given a deck consisting of three red cards labeled $A$, $B$, $C$, three green cards labeled $A$, $B$, $C$, and three blue cards labeled $A$, $B$, $C$, calculate the probability of drawing a winning set.

Answer: $\frac{1}{14}$


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

On a straight stretch of one-way, two-lane highway, vehicles obey a safety rule: the distance from the back of one vehicle to the front of another is exactly one vehicle length for each 20 kilometers per hour of speed or fraction thereof. Suppose a sensor on the roadside counts the number of vehicles that pass in one hour. Each vehicle is 5 meters long and they can travel at any speed. Let \( N \) be the maximum whole number of vehicles that can pass the sensor in one hour. Find the quotient when \( N \) is divided by 10.

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>.

When manufacturing a steel cable, it was found that the cable has the same length as the curve given by the system of equations:

$$
\left\{\begin{array}{l}
x+y+z=10 \\
x y+y z+x z=-22
\end{array}\right.
$$

Find the length of the cable.

Answer: $4 \pi \sqrt{\frac{83}{3}}$


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

Let the sides opposite to the internal angles $A$, $B$, and $C$ of triangle $\triangle ABC$ be $a$, $b$, and $c$ respectively. Given that $\sin B - \sin C = \sin (A-C)$.
$(1)$ Find the value of $A$.
$(2)$ If $AB=2$, $AC=5$, and the medians $AM$ and $BN$ on sides $BC$ and $AC$ intersect at point $P$, find the cosine value of $\angle MPN$.

Answer: $\frac{4\sqrt{91}}{91}$


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

A four-digit number has the following properties:

(a) It is a perfect square;

(b) Its first two digits are equal

(c) Its last two digits are equal.

Find all such four-digit numbers.

Answer: $7744$


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

Write the digits from 0 to 9 in a line, in any order you choose. On the line below, combine the neighboring digits to form nine new numbers, and sum these numbers as in the example below:

| 2 |  | 1 |  | 3 |  | 7 |  | 4 |  | 9 |  | 5 |  | 8 |  | 0 |  | 6 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
|  | 21 |  | 13 |  | 37 |  | 74 |  | 49 |  | 95 |  | 58 |  | 80 |  | 06 |  |
| 1 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |

What is the maximum sum that can be obtained in this way?
A) 506
B) 494
C) 469
D) 447
E) 432

Answer: $494$


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

In the arithmetic sequence $\{a_n\}$, it is given that $a_1 > 0$, $a_{2012} + a_{2013} > 0$, and $a_{2012} \cdot a_{2013} < 0$, calculate the largest natural number n that satisfies S_n > 0.

Answer: $4024$


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

Let $x,$ $y,$ and $z$ be real numbers such that $x + y + z = 7$ and $x, y, z \geq 2.$ Find the maximum value of
\[\sqrt{2x + 3} + \sqrt{2y + 3} + \sqrt{2z + 3}.\]

Answer: $\sqrt{69}$


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>.

Three people, A, B, and C, visit three tourist spots, with each person visiting only one spot. Let event $A$ be "the three people visit different spots," and event $B$ be "person A visits a spot alone." Then, the probability $P(A|B)=$ ______.

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>.

Suppose a sequence starts with 1254, 2547, 5478, and ends with 4781. Let $T$ be the sum of all terms in this sequence. Find the largest prime factor that always divides $T$.

Answer: $101$


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

Given the equation $x^2 + y^2 = |x| + 2|y|$, calculate the area enclosed by the graph of this equation.

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


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

Given the function $f(x) = ax^3 + (a-1)x^2 + 27(a-2)x + b$, its graph is symmetric about the origin. Determine the monotonicity of $f(x)$ on the interval $[-4, 5]$ and find the maximum and minimum values of $f(x)$ on this interval.

Answer: $-54$


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

A company needs to renovate its new office building. If the renovation is done solely by Team A, it would take 18 weeks, and if done solely by Team B, it would take 12 weeks. The result of the bidding is that Team A will work alone for the first 3 weeks, and then both Team A and Team B will work together. The total renovation cost is 4000 yuan. If the payment for the renovation is based on the amount of work completed by each team, how should the payment be distributed?

Answer: $2000$


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

Hou Yi shot three arrows at each of three targets. On the first target, he scored 29 points, and on the second target, he scored 43 points. How many points did he score on the third target?

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>.

Mrs. Delta's language class has 52 students, each with unique initials, and no two students have initials that are alphabetically consecutive (e.g., AB cannot follow AC directly). Assuming Y is considered a consonant, what is the probability of randomly picking a student whose initials (each first and last name starts with the same letter, like AA, BB) are both vowels? Express your answer as a common fraction.

Answer: $\frac{5}{52}$


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

Let \( [x] \) denote the greatest integer less than or equal to the real number \( x \). Consider a sequence \( \{a_n\} \) defined by \( a_1 = 1 \) and \( a_n = \left[\sqrt{n a_{n-1}}\right] \). Find the value of \( a_{2017} \).

Answer: $2015$


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

Marius is entering a wildlife photo contest, and wishes to arrange his nine snow leopards in a row. Each leopard has a unique collar color. The two shortest leopards have inferiority complexes and demand to be placed at the ends of the row. Moreover, the two tallest leopards, which have a superiority complex, insist on being placed next to each other somewhere in the row. How many ways can Marius arrange the leopards under these conditions?

Answer: $2880$


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

In an addition problem where the digits were written on cards, two cards were swapped, resulting in an incorrect expression: $37541 + 43839 = 80280$. Find the error and write the correct value of the sum.

Answer: $81380$


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

Let \( x = 19.\overline{87} \). If \( 19.\overline{87} = \frac{a}{99} \), find \( a \).

If \( \frac{\sqrt{3}}{b \sqrt{7} - \sqrt{3}} = \frac{2 \sqrt{21} + 3}{c} \), find \( c \).

If \( f(y) = 4 \sin y^{\circ} \) and \( f(a - 18) = b \), find \( b \).

Answer: $25$


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

A point  $P$  is chosen uniformly at random in the interior of triangle  $ABC$  with side  lengths  $AB = 5$ ,  $BC = 12$ ,  $CA = 13$ . The probability that a circle with radius  $\frac13$  centered at  $P$  does not intersect the perimeter of  $ABC$  can be written as  $\frac{m}{n}$  where  $m, n$  are relatively prime positive integers. Find  $m + n$ .

Answer: $61$


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

The third quartile of the data $13$, $11$, $12$, $15$, $16$, $18$, $21$, $17$ is ______.

Answer: $17.5$


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>.

Determine the value of $x^2 + y^2$ if $x - y = 25$ and $xy = 36$. Additionally, find what $x+y$ equals.

Answer: $\sqrt{769}$


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

How many integer triples  $(x,y,z)$ are there such that $\begin{array}{rcl} x - yz^2&\equiv & 1 \pmod {13} \\ xz+y&\equiv& 4 \pmod {13} \end{array}$ where $0\leq x < 13$ , $0\leq y <13$ , and $0\leq z< 13$ .

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>.

Find the number of rationals  $\frac{m}{n}$  such that

(i)  $0 < \frac{m}{n} < 1$ ;

(ii)  $m$  and  $n$  are relatively prime;

(iii)  $mn = 25!$ .

Answer: $256$


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>.

Given the sets \( A = \{(x, y) \mid |x| + |y| = a, a > 0\} \) and \( B = \{(x, y) \mid |xy| + 1 = |x| + |y| \} \), if the intersection \( A \cap B \) is the set of vertices of a regular octagon in the plane, determine the value of \( a \).

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>.

Let A and B be fixed points in the plane with distance AB = 1. An ant walks on a straight
line from point A to some point C in the plane and notices that the distance from itself to B
always decreases at any time during this walk. Compute the area of the region in the plane
containing all points where point C could possibly be located.

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


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

In an isosceles right triangle \( \triangle ABC \), \( \angle A = 90^\circ \), \( AB = 1 \). \( D \) is the midpoint of \( BC \), \( E \) and \( F \) are two other points on \( BC \). \( M \) is the other intersection point of the circumcircles of \( \triangle ADE \) and \( \triangle ABF \); \( N \) is the other intersection point of line \( AF \) with the circumcircle of \( \triangle ACE \); \( P \) is the other intersection point of line \( AD \) with the circumcircle of \( \triangle AMN \). Find the length of \( AP \).

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>.

It is given that  $x = -2272$ ,  $y = 10^3+10^2c+10b+a$ , and  $z = 1$  satisfy the equation  $ax + by + cz = 1$ , where  $a, b, c$  are positive integers with  $a < b < c$ . Find  $y.$

Answer: $1987$


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

Estimate the range of the submissions for this problem. Your answer must be between  $[0, 1000]$ . An estimate  $E$  earns  $\frac{2}{1+0.05|A-E|}$  points, where  $A$  is the actual answer.

*2022 CCA Math Bonanza Lightning Round 5.2*

Answer: $500$


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

A random simulation method is used to estimate the probability of a shooter hitting the target at least 3 times out of 4 shots. A calculator generates random integers between 0 and 9, where 0 and 1 represent missing the target, and 2 through 9 represent hitting the target. Groups of 4 random numbers represent the results of 4 shots. After randomly simulating, 20 groups of random numbers were generated:
7527   0293   7140   9857   0347   4373   8636   6947   1417   4698
0371   6233   2616   8045   6011   3661   9597   7424   7610   4281
Estimate the probability that the shooter hits the target at least 3 times out of 4 shots based on the data above.

Answer: $0.75$


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

In the polygon shown, each side is perpendicular to its adjacent sides, and all 24 of the sides are congruent. The perimeter of the polygon is 48. Find the area of the polygon.

Answer: $48$


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

In $\triangle ABC$, $AB = 6$, $BC = 10$, $CA = 8$, and side $BC$ is extended to a point $P$ such that $\triangle PAB$ is similar to $\triangle PCA$. Calculate the length of $PC$.

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>.

If $cos2α=-\frac{{\sqrt{10}}}{{10}}$, $sin({α-β})=\frac{{\sqrt{5}}}{5}$, and $α∈({\frac{π}{4},\frac{π}{2}})$, $β∈({-π,-\frac{π}{2}})$, then $\alpha +\beta =$____.

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


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

In triangle \( \triangle ABC \), given that \( \sin A = 10 \sin B \sin C \) and \( \cos A = 10 \cos B \cos C \), find the value of \( \tan A \).

Answer: $-9$


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

Given the following four conclusions:
\\((1)\\) The center of symmetry of the function \\(f(x)= \dfrac {x-1}{2x+1}\\) is \\((- \dfrac {1}{2},- \dfrac {1}{2})\\);
\\((2)\\) If the equation in \\(x\\), \\(x- \dfrac {1}{x}+k=0\\), has no real roots for \\(x \in (0,1)\\), then the range of \\(k\\) is \\(k \geqslant 2\\);
\\((3)\\) Given that point \\(P(a,b)\\) and point \\(Q(1,0)\\) are on opposite sides of the line \\(2x-3y+1=0\\), then \\(3b-2a > 1\\);
\\((4)\\) If the graph of the function \\(f(x)=\sin (2x- \dfrac {\pi}{3})\\) is shifted to the right by \\(\varphi (\varphi > 0)\\) units and becomes an even function, then the minimum value of \\(\varphi\\) is \\( \dfrac {\pi}{12}\\),
among these conclusions, the correct ones are: \_\_\_\_\_\_ .

Answer: $(3)(4)$


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

Given the function \( f(x) = \sin^4 x \),
1. Let \( g(x) = f(x) + f\left(\frac{\pi}{2} - x\right) \). Find the maximum and minimum values of \( g(x) \) in the interval \(\left[\frac{\pi}{6}, \frac{3\pi}{8}\right]\).
2. Find the value of \(\sum_{k=1}^{89} f\left(\frac{k\pi}{180}\right)\).

Answer: $\frac{133}{4}$


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>.

What is the maximum number of rooks that can be placed in an \(8 \times 8 \times 8\) cube so that they do not attack each other?

Answer: $64$


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

In triangle $XYZ$, which is equilateral with a side length $s$, lines $\overline{LM}$, $\overline{NO}$, and $\overline{PQ}$ are parallel to $\overline{YZ}$, and $XL = LN = NP = QY$. Determine the ratio of the area of trapezoid $PQYZ$ to the area of triangle $XYZ$.

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


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

How many paths are there from the starting point $C$ to the end point $D$, if every step must be up or to the right in a grid of 8 columns and 7 rows?

Answer: $6435$


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

Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Define $A = \left[\frac{7}{8}\right] + \left[\frac{7^2}{8}\right] + \cdots + \left[\frac{7^{2016}}{8}\right]$. Find the remainder when $A$ is divided by 50.

Answer: $42$


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

Given vectors $\overrightarrow{a}=(\sin \theta+\cos \theta,1)$ and $\overrightarrow{b}=(5,1)$, which are orthogonal, and $\theta \in (0,\pi)$, find the value of $\tan \theta$.

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


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

Simplify: \\( \dfrac {\sin 7 ^{\circ} + \cos 15 ^{\circ} \sin 8 ^{\circ} }{\cos 7 ^{\circ} - \sin 15 ^{\circ} \sin 8 ^{\circ} }= \) \_\_\_\_\_\_ .

Answer: $2- \sqrt {3}$


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

The graph of the function $y=\sin 2x-\sqrt{3}\cos 2x$ can be obtained by shifting the graph of the function $y=\sin 2x+\sqrt{3}\cos 2x$ to the right by $\frac{\pi}{3}$ units.

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


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

Let  $n$  be largest number such that \[ \frac{2014^{100!}-2011^{100!}}{3^n} \] is still an integer.  Compute the remainder when  $3^n$  is divided by  $1000$ .

Answer: $83$


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

How many nonnegative integers can be expressed as \[b_6\cdot4^6+b_5\cdot4^5+b_4\cdot4^4+b_3\cdot4^3+b_2\cdot4^2+b_1\cdot4^1+b_0\cdot4^0,\] where $b_i \in \{-1,0,1\}$ for $0 \leq i \leq 6$?

Answer: $5462$


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

In an acute-angled triangle, the sides $a$ and $b$ are the roots of the equation $x^{2}-2 \sqrt {3}x+2=0$. The angles $A$ and $B$ satisfy the equation $2\sin (A+B)- \sqrt {3}=0$. Find the value of the side $c$ and the area of $\triangle ABC$.

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


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

Answer: $4044\frac{2022}{2023}$


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

The graph of $xy = 4$ is a hyperbola. Find the distance between the foci of this hyperbola.

Answer: $4\sqrt{2}$


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

Given the hyperbola $C$: $\frac{x^{2}}{a^{2}}-y^{2}=1 (a > 0)$ and the line $l$: $x+y=1$ intersect at two distinct points $A$ and $B$.
(I) Find the range of the eccentricity $e$ of the hyperbola $C$.
(II) Let $P$ be the intersection point of line $l$ and the $y$-axis, and $\overrightarrow{PA} = \frac{5}{12}\overrightarrow{PB}$. Find the value of $a$.

Answer: $\frac{17}{13}$


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

Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $\overrightarrow{b}=(4\cos \alpha,-4\sin \alpha)$, and $\overrightarrow{a}\perp (\overrightarrow{a}- \overrightarrow{b})$, let the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ be $\theta$, then $\theta$ equals \_\_\_\_\_\_.

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


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

Given that $\{a_n\}$ is a geometric sequence with a common ratio of $q$, and $a_m$, $a_{m+2}$, $a_{m+1}$ form an arithmetic sequence.

(Ⅰ) Find the value of $q$;
(Ⅱ) Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$. Determine whether $S_m$, $S_{m+2}$, $S_{m+1}$ form an arithmetic sequence and explain why.

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>.

To ensure the safety of property during the Spring Festival holiday, an office needs to arrange for one person to be on duty each day for seven days. Given that there are 4 people in the office, and each person needs to work for either one or two days, the number of different duty arrangements is \_\_\_\_\_\_ . (Answer with a number)

Answer: $2520$


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

In triangle $ABC$, $AB = 8$, $BC = 8$, and $CA = 6$.

Point $P$ is randomly selected inside triangle $ABC$. What is the probability that $P$ is closer to vertex $C$ than it is to either vertex $A$ or $B$?

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


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

Given the ellipse $\Gamma$: $\dfrac {x^{2}}{a^{2}}+y^{2}=1(a > 1)$, its left focus is $F_{1}$, the right vertex is $A_{1}$, and the top vertex is $B_{1}$. The circle $P$ that passes through points $F_{1}$, $A_{1}$, and $B_{1}$ has its center coordinates at $\left( \dfrac { \sqrt {3}- \sqrt {2}}{2}, \dfrac {1- \sqrt {6}}{2}\right)$.
(Ⅰ) Find the equation of the ellipse;
(Ⅱ) If the line $l$: $y=kx+m$ ($k,m$ are constants, $k\neq 0$) intersects the ellipse $\Gamma$ at two distinct points $M$ and $N$.
(i) When the line $l$ passes through $E(1,0)$, and $\overrightarrow{EM}+2 \overrightarrow{EN}= \overrightarrow{0}$, find the equation of the line $l$;
(ii) When the distance from the origin $O$ to the line $l$ is $\dfrac { \sqrt {3}}{2}$, find the maximum area of $\triangle MON$.

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


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

Given the function $f(x) = \cos x \cdot \sin \left( x + \frac{\pi}{3} \right) - \sqrt{3}\cos^2 x + \frac{\sqrt{3}}{4}, x \in \mathbb{R}$,

$(1)$ Find the smallest positive period of $f(x)$;

$(2)$ Find the maximum and minimum values of $f(x)$ on the closed interval $\left[ -\frac{\pi}{4}, \frac{\pi}{4} \right]$.

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>.

The length of edge AB is 51 units, and the lengths of the other five edges of the tetrahedron ABCD measure 10, 20, 25, 34, and 45 units. Determine the length of edge CD.

Answer: $25$


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

Given a function $f(x)$ whose domain is $D$, if for any $x_1, x_2 \in D$, it holds that $f(x_1) \leq f(x_2)$ whenever $x_1 < x_2$, then the function $f(x)$ is called a non-decreasing function on $D$. Suppose $f(x)$ is a non-decreasing function on the interval $[0,1]$ and satisfies the following three conditions:
1. $f(0) = 0$;
2. $f\left(\frac{x}{3}\right) = \frac{1}{2}f(x)$;
3. $f(1-x) = 1-f(x)$.

Find the value of $f(1) + f\left(\frac{1}{2}\right) + f\left(\frac{1}{3}\right) + f\left(\frac{1}{6}\right) + f\left(\frac{1}{7}\right) + f\left(\frac{1}{8}\right)$.

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


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

Find all solutions to  $aabb=n^4-6n^3$ , where  $a$  and  $b$  are non-zero digits, and  $n$  is an integer.  ( $a$  and  $b$  are not necessarily distinct.)

Answer: $6655$


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

For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean of $x$ and $y$ equal to $12^{10}$?

Answer: $220$


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

Given the differences between the scores of 14 students in a group and the class average score of 85 are 2, 3, -3, -5, 12, 12, 8, 2, -1, 4, -10, -2, 5, 5, find the average score of this group.

Answer: $87.29$


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>.

Let  $\sigma (n)$  denote the sum and  $\tau (n)$  denote the amount of natural divisors of number  $n$  (including  $1$  and  $n$ ). Find the greatest real number  $a$  such that for all  $n>1$  the following inequality is true:   $$ \frac{\sigma (n)}{\tau (n)}\geq a\sqrt{n} $$

Answer: $\frac{3 \sqrt{2}}{4}$


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

Suppose that \( a^3 \) varies inversely with \( b^2 \). If \( a = 5 \) when \( b = 2 \), find the value of \( a \) when \( b = 8 \).

Answer: $2.5$


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

Calculate the sum of the squares of the roots of the equation \[x\sqrt{x} - 8x + 9\sqrt{x} - 3 = 0,\] given that all roots are real and nonnegative.

Answer: $46$


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

Determine for how many different values of $p<2000$, there exists a quadrilateral $ABCD$ with integer side lengths, perimeter $p$, right angles at $D$ and $C$, $AB=3$, and $CD=AD$.

Answer: $996$


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

A circle with diameter  $AB$  is drawn, and the point  $ P$  is chosen on segment  $AB$  so that  $\frac{AP}{AB} =\frac{1}{42}$  . Two new circles  $a$  and  $b$  are drawn with diameters  $AP$  and  $PB$  respectively. The perpendicular line to  $AB$  passing through  $ P$  intersects the circle twice at points  $S$  and  $T$  . Two more circles  $s$  and  $t$  are drawn with diameters  $SP$  and  $ST$  respectively. For any circle  $\omega$  let  $A(\omega)$  denote the area of the circle. What is  $\frac{A(s)+A(t)}{A(a)+A(b)}$ ?

Answer: $205/1681$


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

In the movie "The Wandering Earth 2," there are many UEG (United Earth Government) mechanical devices that are drivable, operable, and deformable, all of which are from the leading Chinese engineering machinery brand - XCMG. Many of the hardcore equipment in the movie are not special effects, but are actually designed and modified from cutting-edge domestic equipment. Many of the equipment can find prototypes in reality. A new device has been developed in a workshop of the group. The specific requirement of the group for the new device is: the product with a part inner diameter (unit: mm) within the range of $(199.82, 200.18)$ is qualified, otherwise it is defective; the part inner diameter $X$ follows a normal distribution $X \sim N(200, 0.0036)$.

$(1)$ If the workshop installed and debugged the new device and produced 5 parts for trial, with measured inner diameters (unit: mm) of $199.87$, $199.91$, $199.99$, $200.13$, $200.19$, if you are in charge of the workshop, try to determine whether this device needs further debugging based on the $3\sigma$ principle? Explain your reasoning.

$(2)$ If the device meets the production requirements of the group, and now 10,000 parts produced by this device are tracked.

① Approximately how many parts out of the 10,000 parts have an inner diameter exceeding $200.12$ mm?

② What is the most likely number of defective parts out of the 10,000 parts?

Reference data:

If a random variable $X \sim N(\mu, \sigma^2)$, then $P(\mu - \sigma < X < \mu + \sigma) \approx 0.683$, $P(\mu - 2\sigma < X < \mu + 2\sigma) \approx 0.955$, $P(\mu - 3\sigma < X < \mu + 3\sigma) \approx 0.997$, $0.997^4 \approx 0.988$, $0.997^5 \approx 0.985$.

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>.

Given two-dimensional vectors $\vec{a}$, $\vec{b}$, with $|\vec{a}|=1$, $|\vec{b}|=2$, and $\vec{a} \cdot \vec{b}=1$. If $\vec{e}$ is a two-dimensional unit vector, find the maximum value of $|\vec{a} \cdot \vec{e}| + |\vec{b} \cdot \vec{e}|$.

Answer: $\sqrt{7}$


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

In triangle $XYZ$, $\angle Y = 90^\circ$, $YZ = 4$, and $XY = 5$. What is $\tan X$?

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


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

**
How many non-similar regular 500-pointed stars are there?

**

Answer: $99$


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

There are 10 mountaineers, divided equally into two groups. Among them, 4 are familiar with the trails. Each group needs 2 people who are familiar with the trails. The number of different ways to distribute them is:

Answer: $60$


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

Let $z_1$, $z_2$, $z_3$, $\dots$, $z_{8}$ be the 8 zeros of the polynomial $z^{8} - 16^8$. For each $j$, let $w_j$ be either $z_j$, $-z_j$, or $iz_j$. Find the maximum possible value of the real part of
\[\sum_{j = 1}^{8} w_j.\]

Answer: $32 + 32 \sqrt{2}$


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

Given real numbers $x$, $y$, $z$ satisfying $2x-y-2z-6=0$, and $x^2+y^2+z^2\leq4$, calculate the value of $2x+y+z$.

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


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

A natural number $n$ is called a "good number" if the column addition of $n$, $n+1$, and $n+2$ does not produce any carry-over. For example, 32 is a "good number" because $32+33+34$ does not result in a carry-over; however, 23 is not a "good number" because $23+24+25$ does result in a carry-over. The number of "good numbers" less than 1000 is \_\_\_\_\_\_.

Answer: $48$


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

The diagram shows a rectangle $AEFJ$ inside a regular decagon $ABCDEFGHIJ$. What is the ratio of the area of the rectangle to the area of the decagon?

Answer: $2:5$


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

Given the integers \( 1, 2, 3, \ldots, 40 \), find the greatest possible sum of the positive differences between the integers in twenty pairs, where the positive difference is either 1 or 3.

Answer: $58$


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

If the function \( y = \sin(w x) \) with \( w > 0 \) attains its maximum value at least 50 times in the interval \([0,1]\), what is the minimum value of \( w \)?

Answer: $100 \pi$


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 3 by 2 rectangle is split into four congruent right-angled triangles. Those four triangles are rearranged to form a rhombus. What is the ratio of the perimeter of the rectangle to the perimeter of the rhombus?

Answer: $1: 1$


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

If real numbers \(x\) and \(y\) satisfy \(x^2 + y^2 = 20\), what is the maximum value of \(xy + 8x + y\)?

Answer: $42$


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

Suppose  $\overline{AB}$  is a segment of unit length in the plane. Let  $f(X)$  and  $g(X)$  be functions of the plane such that  $f$  corresponds to rotation about  $A$   $60^\circ$  counterclockwise and  $g$  corresponds to rotation about  $B$   $90^\circ$  clockwise. Let  $P$  be a point with  $g(f(P))=P$ ; what is the sum of all possible distances from  $P$  to line  $AB$ ?

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


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

All positive integers whose digits add up to 14 are listed in increasing order. What is the eleventh number in that list?

Answer: $194$


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

In rectangle $ABCD$, $AB = 4$ cm, $BC = 10$ cm, and $DE = DF$. The area of triangle $DEF$ is one-fourth the area of rectangle $ABCD$. What is the length in centimeters of segment $EF$? Express your answer in simplest radical form.

Answer: $2\sqrt{10}$


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

Given the function $f(x) = 2\sin x\cos x - 2\sin^2 x$
(1) Find the smallest positive period of the function $f(x)$.
(2) Let $\triangle ABC$ have internal angles $A$, $B$, $C$ opposite sides $a$, $b$, $c$, respectively, and satisfy $f(A) = 0$, $c = 1$, $b = \sqrt{2}$. Find the area of $\triangle ABC$.

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>.

ABCDE is a regular pentagon. The star ACEBD has an area of 1. AC and BE meet at P, BD and CE meet at Q. Find the area of APQD.

Answer: $1/2$


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 $c$ and $d$ be real numbers such that
\[\frac{c}{d} + \frac{c}{d^3} + \frac{c}{d^6} + \dots = 9.\]
Find
\[\frac{c}{c + 2d} + \frac{c}{(c + 2d)^2} + \frac{c}{(c + 2d)^3} + \dotsb.\]

Answer: $\frac{9}{11}$


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

Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors and satisfy $\overrightarrow{a} \cdot \overrightarrow{b} = 0$, find the maximum value of $(\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}) \cdot (\overrightarrow{a} + \overrightarrow{c})$.

Answer: $2 + \sqrt{5}$


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>.

Over the summer, a one-room apartment increased in price by 21%, a two-room apartment by 11%, and the total cost of both apartments by 15%. How many times cheaper is the one-room apartment compared to the two-room apartment?

Answer: $1.5$


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

\( z_{1}, z_{2}, z_{3} \) are the three roots of the polynomial
\[ P(z) = z^{3} + a z + b \]
and satisfy the condition
\[ \left|z_{1}\right|^{2} + \left|z_{2}\right|^{2} + \left|z_{3}\right|^{2} = 250 \]
Moreover, the three points \( z_{1}, z_{2}, z_{3} \) in the complex plane form a right triangle. Find the length of the hypotenuse of this right triangle.

Answer: $5\sqrt{15}$


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>.

Three of the four vertices of a rectangle are \((1, 7)\), \((14, 7)\), and \((14, -4)\). What is the area of the intersection of this rectangular region and the region inside the graph of the equation \((x - 1)^2 + (y + 4)^2 = 16\)?

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>.

Gamma and Delta both participated in a two-day science quiz. Each attempted questions totaling 600 points after the second day. On the first day, Gamma scored 210 points out of 350 points attempted, and on the second day scored 150 points out of 250 points attempted. Delta, who also did not attempt 350 points on the first day, scored a positive integer number of points on each of the two days, and Delta's daily success ratio (points scored divided by points attempted) on each day was less than Gamma's on that day. Gamma's overall success ratio for the two days was $360/600 = 3/5$.

Find the largest possible overall success ratio that Delta could have achieved.

Answer: $\frac{359}{600}$


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

Form five-digit numbers without repeating digits using the numbers \\(0\\), \\(1\\), \\(2\\), \\(3\\), and \\(4\\).

\\((\\)I\\()\\) How many of these five-digit numbers are even?

\\((\\)II\\()\\) How many of these five-digit numbers are less than \\(32000\\)?

Answer: $54$


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

What is the probability that $abc + ab + a$ is divisible by $4$, where positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3, \dots, 1024\}$.

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


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

Given the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a,b>0)\) with left and right foci as \(F_{1}\) and \(F_{2}\), a line passing through \(F_{2}\) with an inclination angle of \(\frac{\pi}{4}\) intersects the hyperbola at a point \(A\). If the triangle \(\triangle F_{1}F_{2}A\) is an isosceles right triangle, calculate the eccentricity of the hyperbola.

Answer: $\sqrt{2}+1$


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

Arrange numbers  $ 1,\ 2,\ 3,\ 4,\ 5$  in a line. Any arrangements are equiprobable. Find the probability such that the sum of the numbers for the first, second and third equal to the sum  of that of the third, fourth and fifth. Note that in each arrangement each number are used one time without overlapping.

Answer: $1/15$


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

In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled, with $\angle AEB=\angle BEC = \angle CED = 45^\circ$, and $AE=28$. Find the length of $CE$, given that $CE$ forms the diagonal of a square $CDEF$.

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>.

Given that the angle between the unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is obtuse, the minimum value of $|\overrightarrow{b} - t\overrightarrow{a}| (t \in \mathbb{R})$ is $\frac{\sqrt{3}}{2}$, and $(\overrightarrow{c} - \overrightarrow{a}) \cdot (\overrightarrow{c} - \overrightarrow{b}) = 0$, find the maximum value of $\overrightarrow{c} \cdot (\overrightarrow{a} + \overrightarrow{b})$.

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


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

Find the number of ordered quadruples  $(a, b, c, d)$  where each of  $a, b, c,$  and  $d$  are (not necessarily distinct) elements of  $\{1, 2, 3, 4, 5, 6, 7\}$  and  $3abc + 4abd + 5bcd$  is even. For example,  $(2, 2, 5, 1)$  and  $(3, 1, 4, 6)$  satisfy the conditions.

Answer: $2017$


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

Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of maximum length?

Answer: $1236$


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

In rectangle $JKLM$, $P$ is a point on $LM$ so that $\angle JPL=90^{\circ}$. $UV$ is perpendicular to $LM$ with $LU=UP$, as shown. $PL$ intersects $UV$ at $Q$. Point $R$ is on $LM$ such that $RJ$ passes through $Q$. In $\triangle PQL$, $PL=25$, $LQ=20$ and $QP=15$. Find $VD$. [asy]
size(7cm);defaultpen(fontsize(9));
real vd = 7/9 * 12;
path extend(pair a, pair b) {return a--(10 * (b - a));}

// Rectangle
pair j = (0, 0); pair l = (0, 16); pair m = (24 + vd, 0); pair k = (m.x, l.y);
draw(j--l--k--m--cycle);
label("$J$", j, SW);label("$L$", l, NW);label("$K$", k, NE);label("$M$", m, SE);

// Extra points and lines
pair q = (24, 7); pair v = (q.x, 0); pair u = (q.x, l.y);
pair r = IP(k--m, extend(j, q));
pair p = (12, l.y);
draw(q--j--p--m--r--cycle);draw(u--v);
label("$R$", r, E); label("$P$", p, N);label("$Q$", q, 1.2 * NE + 0.2 * N);label("$V$", v, S); label("$U$", u, N);

// Right angles and tick marks
markscalefactor = 0.1;
draw(rightanglemark(j, l, p)); draw(rightanglemark(p, u, v)); draw(rightanglemark(q, v, m));draw(rightanglemark(j, p, q));
add(pathticks(l--p, 2, spacing=3.4, s=10));add(pathticks(p--u, 2, spacing=3.5, s=10));

// Number labels
label("$16$", midpoint(j--l), W); label("$25$", midpoint(j--p), NW); label("$15$", midpoint(p--q), NE);
label("$20$", midpoint(j--q), 0.8 * S + E);
[/asy]

Answer: $\dfrac{28}{3}$


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

Given a finite arithmetic sequence \(\left\{a_{n}\right\}\) with the first term equal to 1 and the last term \(a_{n} = 1997\) (where \(n > 3\)), and the common difference being a natural number, find the sum of all possible values of \(n\).

Answer: $3501$