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

We have learned methods to factorize a polynomial, such as the method of common factors and the application of formulas. In fact, there are other methods for factorization, such as grouping method and splitting method.
① Grouping method:
For example, $x^{2}-2xy+y^{2}-25=(x^{2}-2xy+y^{2})-25=(x-y)^{2}-5^{2}=(x-y-5)(x-y+5)$.
② Splitting method:
For example, $x^{2}+2x-3=x^{2}+2x+1-4=(x+1)^{2}-2^{2}=(x+1-2)(x+1+2)=(x-1)(x+3)$.
$(1)$ Following the above methods, factorize as required:
① Using grouping method: $9x^{2}+6x-y^{2}+1$;
② Using splitting method: $x^{2}-6x+8$;
$(2)$ Given: $a$, $b$, $c$ are the three sides of $\triangle ABC$, $a^{2}+5b^{2}+c^{2}-4ab-6b-10c+34=0$, find the perimeter of $\triangle ABC$.

Answer: $14$


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

A line passing through the vertex \( A \) of triangle \( ABC \) perpendicular to its median \( BD \) bisects this median.

Find the ratio of the sides \( AB \) and \( AC \).

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>.

If $11 = x^6 + \frac{1}{x^6}$, find the value of $x^3 + \frac{1}{x^3}$.

Answer: $\sqrt{13}$


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

Let \(A, B, C\), and \(D\) be four points that are not coplanar. A plane passes through the centroid of triangle \(ABC\) that is parallel to the lines \(AB\) and \(CD\). In what ratio does this plane divide the median drawn to the side \(CD\) of triangle \(ACD\)?

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>.

What is the least natural number that can be added to 71,382 to create a palindrome?

Answer: $35$


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

The height of a trapezoid, whose diagonals are mutually perpendicular, is 4. Find the area of the trapezoid if one of its diagonals is 5.

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


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

In triangle \( \triangle ABC \), given \( AB = 4 \), \( AC = 3 \), and \( P \) is a point on the perpendicular bisector of \( BC \), find \( \overrightarrow{BC} \cdot \overrightarrow{AP} \).

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


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

On a table, there are 210 different cards each with a number from the sequence \(2, 4, 6, \ldots, 418, 420\) (each card has exactly one number, and each number appears exactly once). In how many ways can 2 cards be chosen so that the sum of the numbers on the chosen cards is divisible by \(7\)?

Answer: $3135$


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

Find the smallest natural number with the following properties:
a) It ends in digit 6 in decimal notation;
b) If you remove the last digit 6 and place this digit 6 at the beginning of the remaining number, the result is four times the original number.

Answer: $153846$


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

On a table, there are 2004 boxes, each containing one ball. It is known that some of the balls are white, and their number is even. You are allowed to point to any two boxes and ask if there is at least one white ball in them. What is the minimum number of questions needed to guarantee the identification of a box that contains a white ball?

Answer: $2003$


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

Find the smallest natural number that starts with the digit 5, which, when this 5 is removed from the beginning of its decimal representation and appended to its end, becomes four times smaller.

Answer: $512820$


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

A chessboard of size $8 \times 8$ is considered.

How many ways are there to place 6 rooks such that no two rooks are ever on the same row or column?

Answer: $564480$


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

Determine the average daily high temperature in Addington from September 15th, 2008, through September 21st, 2008, inclusive. The high temperatures for the days are as follows: 51, 64, 60, 59, 48, 55, 57 degrees Fahrenheit. Express your answer as a decimal to the nearest tenth.

Answer: $56.3$


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>.

Using the six digits 0, 1, 2, 3, 4, 5, we can form _ _ _ _ _ different five-digit numbers that do not have repeating digits and are divisible by 5.

Answer: $216$


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

Given \(\sin \alpha + \sin (\alpha + \beta) + \cos (\alpha + \beta) = \sqrt{3}\), where \(\beta \in \left[\frac{\pi}{4}, \pi\right]\), find the value of \(\beta\).

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>.

Ten football teams played each other exactly once. As a result, each team ended up with exactly $x$ points.

What is the largest possible value of $x$? (A win earns 3 points, a draw earns 1 point, and a loss earns 0 points.)

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>.

A game board is constructed by shading three of the regions formed by the diagonals of a regular pentagon. What is the probability that the tip of the spinner will come to rest in a shaded region? Assume the spinner can land in any region with equal likelihood.

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


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

Given the sequence ${a_n}$ is an arithmetic sequence, with $a_1 \geq 1$, $a_2 \leq 5$, $a_5 \geq 8$, let the sum of the first n terms of the sequence be $S_n$. The maximum value of $S_{15}$ is $M$, and the minimum value is $m$. Determine $M+m$.

Answer: $600$


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>.

Point \( M \) is the midpoint of the hypotenuse \( AC \) of right triangle \( ABC \). Points \( P \) and \( Q \) on lines \( AB \) and \( BC \) respectively are such that \( AP = PM \) and \( CQ = QM \). Find the measure of angle \( \angle PQM \) if \( \angle BAC = 17^{\circ} \).

Answer: $17$


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

A regular hexagon has a side length of 8 cm. Calculate the area of the shaded region formed by connecting two non-adjacent vertices to the center of the hexagon, creating a kite-shaped region.

[asy]
size(100);

pair A,B,C,D,E,F,O;
A = dir(0); B = dir(60); C = dir(120); D = dir(180); E = dir(240); F = dir(300); O = (0,0);

fill(A--C--O--cycle,heavycyan);

draw(A--B--C--D--E--F--A);
draw(A--C--O);
[/asy]

Answer: $16\sqrt{3}$


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 difference of the logarithms of the hundreds digit and the tens digit of a three-digit number is equal to the logarithm of the difference of the same digits, and the sum of the logarithms of the hundreds digit and the tens digit is equal to the logarithm of the sum of the same digits, increased by 4/3. If you subtract the number, having the reverse order of digits, from this three-digit number, their difference will be a positive number, in which the hundreds digit coincides with the tens digit of the given number. Find this number.

Answer: $421$


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

Two identical cylindrical sheets are cut open along the dotted lines and glued together to form one bigger cylindrical sheet. The smaller sheets each enclose a volume of 100. What volume is enclosed by the larger sheet?

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>.

In triangle \( \triangle ABC \), \(\angle ABC = 50^\circ\), \(\angle ACB = 30^\circ\), \(M\) is a point inside the triangle such that \(\angle MCB = 20^\circ\), \(\angle MAC = 40^\circ\). Find the measure of \(\angle MBC\).

(Problem 1208 from Mathematical Bulletin)

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>.

Juan is measuring the diameter of a large rather ornamental plate to cover it with a decorative film. Its actual diameter is 30cm, but his measurement tool has an error of up to $30\%$. Compute the largest possible percent error, in percent, in Juan's calculated area of the ornamental plate.

Answer: $69$


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>.

In the plane rectangular coordinate system \(x O y\), the equation of the ellipse \(C\) is \(\frac{x^{2}}{9}+\frac{y^{2}}{10}=1\). Let \(F\) be the upper focus of \(C\), \(A\) be the right vertex of \(C\), and \(P\) be a moving point on \(C\) located in the first quadrant. Find the maximum area of the quadrilateral \(O A P F\).

Answer: $\frac{3}{2} \sqrt{11}$


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

In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\frac{\cos B}{b} + \frac{\cos C}{c} = \frac{\sin A}{\sqrt{3} \sin C}$.

(1) Find the value of $b$.
(2) If $\cos B + \sqrt{3} \sin B = 2$, find the maximum area of triangle $ABC$.

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


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

A cheetah takes strides of 2 meters each, while a fox takes strides of 1 meter each. The time it takes for the cheetah to run 2 strides is the same as the time it takes for the fox to run 3 strides. If the distance between the cheetah and the fox is 30 meters, how many meters must the cheetah run to catch up with the fox?

Answer: $120$


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

31 people attended a class club afternoon event, and after the program, they danced. Ági danced with 7 boys, Anikó with 8, Zsuzsa with 9, and so on, with each subsequent girl dancing with one more boy than the previously mentioned one. Finally, Márta danced with all but 3 boys. How many boys were at the club afternoon?

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>.

What is the value of $[\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + \cdots + [\sqrt{1989 \cdot 1990}] + [-\sqrt{1}] + [-\sqrt{2}] + [-\sqrt{3}] + \cdots + [-\sqrt{1989 \cdot 1990}]$?
(The 1st "Hope Cup" Mathematics Contest, 1990)

Answer: $-3956121$


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

You have 68 coins of different weights. Using 100 weighings on a balance scale without weights, find the heaviest and the lightest coin.

Answer: $100$


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

The greatest prime number that is a divisor of $16,385$ can be deduced similarly, find the sum of the digits of this greatest prime number.

Answer: $19$


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

A club has 99 members. Find the smallest positive integer $n$ such that if the number of acquaintances of each person is greater than $n$, there must exist 4 people who all know each other mutually (here it is assumed that if $A$ knows $B$, then $B$ also knows $A$).

Answer: $66$


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

If a sequence $\{a_n\}$ satisfies $$\frac {1}{a_{n+1}}- \frac {1}{a_{n}}=d$$ (where $n\in\mathbb{N}^*$, $d$ is a constant), then the sequence $\{a_n\}$ is called a "harmonic sequence". Given that the sequence $\{\frac {1}{x_{n}}\}$ is a "harmonic sequence", and $x_1+x_2+\ldots+x_{20}=200$, the maximum value of $x_3x_{18}$ is \_\_\_\_\_\_.

Answer: $100$


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

Vasya, Petya, and Kolya are in the same class. Vasya always lies in response to any question, Petya alternates between lying and telling the truth, and Kolya lies in response to every third question but tells the truth otherwise. One day, each of them was asked six consecutive times how many students are in their class. The responses were "Twenty-five" five times, "Twenty-six" six times, and "Twenty-seven" seven times. Can we determine the actual number of students in their class based on their answers?

Answer: $27$


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

Given that the equation \(2x^3 - 7x^2 + 7x + p = 0\) has three distinct roots, and these roots form a geometric progression. Find \(p\) and solve this equation.

Answer: $-2$


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

In the triangle $\triangle ABC$, $\angle A = 60^{\circ}$ and $\angle B = 45^{\circ}$. A line $DE$, with $D$ on $AB$ and $E$ on $BC$, such that $\angle ADE =75^{\circ}$, divides $\triangle ABC$ into two pieces of equal area. Determine the ratio $\frac{AD}{AB}$.
A) $\frac{1}{2}$
B) $\frac{1}{\sqrt{3}}$
C) $\frac{1}{\sqrt{6}}$
D) $\frac{1}{3}$
E) $\frac{1}{4}$

Answer: $\frac{1}{\sqrt{6}}$


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

The value of $999 + 999$ is

Answer: $1998$


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

A father and son were walking one after the other along a snow-covered road. The father's step length is $80 \mathrm{~cm}$, and the son's step length is $60 \mathrm{~cm}$. Their steps coincided 601 times, including at the very beginning and at the end of the journey. What distance did they travel?

Answer: $1440$


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

Given $0 \leq a_k \leq 1$ for $k=1,2,\ldots,2020$, and defining $a_{2021}=a_1, a_{2022}=a_2$, find the maximum value of $\sum_{k=1}^{2020}\left(a_{k}-a_{k+1} a_{k+2}\right)$.

Answer: $1010$


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

On side \(AD\) of rectangle \(ABCD\), a point \(E\) is marked. On segment \(EC\) there is a point \(M\) such that \(AB = BM\) and \(AE = EM\). Find the length of side \(BC\), given that \(ED = 16\) and \(CD = 12\).

Answer: $20$


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

Given complex numbers \( x \) and \( y \), find the maximum value of \(\frac{|3x+4y|}{\sqrt{|x|^{2} + |y|^{2} + \left|x^{2}+y^{2}\right|}}\).

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


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

On the complex plane, consider the parallelogram formed by the points 0, $z,$ $\frac{1}{z},$ and $z + \frac{1}{z}$ where the area of the parallelogram is $\frac{24}{25}.$ If the real part of $z$ is positive, determine the smallest possible value of $\left| z + \frac{1}{z} \right|.$ Compute the square of this value.

Answer: $\frac{36}{25}$


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>.

In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. If $c\cos B + b\cos C = 2a\cos A$, $M$ is the midpoint of $BC$, and $AM=1$, find the maximum value of $b+c$.

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


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

$48n$ is the smallest positive integer that satisfies the following conditions:
1. $n$ is a multiple of 75;
2. $n$ has exactly 75 positive divisors (including 1 and itself).

Find $\frac{n}{75}$.

Answer: $432$


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

Find the natural number that is divisible by 9 and 5 and has 14 distinct divisors.

Answer: $3645$


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

If the graph of the function $f(x) = \sin \omega x \cos \omega x + \sqrt{3} \sin^2 \omega x - \frac{\sqrt{3}}{2}$ ($\omega > 0$) is tangent to the line $y = m$ ($m$ is a constant), and the abscissas of the tangent points form an arithmetic sequence with a common difference of $\pi$.

(Ⅰ) Find the values of $\omega$ and $m$;
(Ⅱ) Find the sum of all zeros of the function $y = f(x)$ in the interval $x \in [0, 2\pi]$.

Answer: $\frac{11\pi}{3}$


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>.

Convert the binary number $101111011_{(2)}$ to its decimal equivalent.

Answer: $379$


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

Calculate the limit of the function:

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

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 sphere with radius $r$ is inside a cone, whose axial section is an equilateral triangle with the sphere inscribed in it. The ratio of the total surface area of the cone to the surface area of the sphere is \_\_\_\_\_\_.

Answer: $9:4$


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

A group of 8 boys and 8 girls was paired up randomly. Find the probability that there is at least one pair with two girls. Round your answer to the nearest hundredth.

Answer: $0.98$


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

The calculation result of the expression \(143 \times 21 \times 4 \times 37 \times 2\) is $\qquad$.

Answer: $888888$


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

Compute the least possible value of  $ABCD - AB \times CD$ , where  $ABCD$  is a 4-digit positive integer, and  $AB$  and  $CD$  are 2-digit positive integers.  (Here  $A$ ,  $B$ ,  $C$ , and  $D$  are digits, possibly equal.  Neither  $A$  nor  $C$  can be zero.)

Answer: $109$


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

In a football tournament, each team is supposed to play one match against each of the other teams. However, during the tournament, half of the teams were disqualified and did not participate further. As a result, a total of 77 matches were played, and the disqualified teams managed to play all their matches against each other, with each disqualified team having played the same number of matches. How many teams were there at the beginning of the tournament?

Answer: $14$


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

In a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with a side length of 1, points $E$ and $F$ are located on $A A_{1}$ and $C C_{1}$ respectively, such that $A E = C_{1} F$. Determine the minimum area of the quadrilateral $E B F D_{1}$.

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


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

Given a sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$. The sequence satisfies the conditions $a\_1=23$, $a\_2=-9$, and $a_{n+2}=a\_n+6\times(-1)^{n+1}-2$ for all $n \in \mathbb{N}^*$.
(1) Find the general formula for the terms of the sequence $\{a\_n\}$;
(2) Find the value of $n$ when $S\_n$ reaches its maximum.

Answer: $11$


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

If $α$ is an acute angle, and $\sin (α - \frac{π}{4})= \frac{3}{5}$, then $\cos 2α$ equals ______.

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


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

Find all natural numbers having exactly six divisors, the sum of which equals 3500.

Answer: $1996$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{5}}$ with an accuracy of $10^{-3}$.

Answer: $0.973$


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

Trapezoid \(ABCD\) is inscribed in the parabola \(y = x^2\) such that \(A = (a, a^2)\), \(B = (b, b^2)\), \(C = (-b, b^2)\), and \(D = (-a, a^2)\) for some positive reals \(a, b\) with \(a > b\). If \(AD + BC = AB + CD\), and \(AB = \frac{3}{4}\), what is \(a\)?

Answer: $\frac{27}{40}$


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>.

Three flower beds, X, Y, and Z, contain X = 600, Y = 480, and Z = 420 plants, respectively. The plants are shared as follows: Beds X and Y share 60 plants, Beds Y and Z share 70 plants, and Beds X and Z share 80 plants. Also, there are 30 plants common to all three beds. Find the total number of unique plants.

Answer: $1320$


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

Given the hyperbola $$\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1$$ ($a>0$, $b>0$) with its right focus at $F(c, 0)$.
(1) If one of the asymptotes of the hyperbola is $y=x$ and $c=2$, find the equation of the hyperbola;
(2) With the origin $O$ as the center and $c$ as the radius, draw a circle. Let the intersection of the circle and the hyperbola in the first quadrant be $A$. Draw the tangent line to the circle at $A$, with a slope of $-\sqrt{3}$. Find the eccentricity of the hyperbola.

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>.

The reciprocal of $\frac{2}{3}$ is ______, the opposite of $-2.5$ is ______.

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 function $f(x)=e^{x}$, for real numbers $m$, $n$, $p$, it is known that $f(m+n)=f(m)+f(n)$ and $f(m+n+p)=f(m)+f(n)+f(p)$. Determine the maximum value of $p$.

Answer: $2\ln2-\ln3$


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

Given that m > 0, p: 0 < x < m, q: x(x - 1) < 0, if p is a sufficient but not necessary condition for q, then the value of m can be _______. (Only one value of m that satisfies the condition is needed)

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>.

Solve the equation \( 2 \sqrt{2} \sin ^{3}\left(\frac{\pi x}{4}\right) = \sin \left(\frac{\pi}{4}(1+x)\right) \).

How many solutions of this equation satisfy the condition: \( 2000 \leq x \leq 3000 \)?

Answer: $250$


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

On the International Space Station, there was an electronic clock displaying time in the HH:MM format. Due to an electromagnetic storm, the device started malfunctioning, causing each digit on the display to either increase by 1 or decrease by 1. What was the actual time when the storm occurred, if immediately after the storm the clock showed 09:09?

Answer: $18:18$


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

Given an ellipse $C: \frac{x^{2}}{4} + y^{2} = 1$, with $O$ being the origin of coordinates, and a line $l$ intersects the ellipse $C$ at points $A$ and $B$, and $\angle AOB = 90^{\circ}$.

(Ⅰ) If the line $l$ is parallel to the x-axis, find the area of $\triangle AOB$;
(Ⅱ) If the line $l$ is always tangent to the circle $x^{2} + y^{2} = r^{2} (r > 0)$, find the value of $r$.

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


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

Calculate the value of $\dfrac{\sqrt[3]{81}}{\sqrt[4]{81}}$ in terms of 81 raised to what power?

Answer: $\frac{1}{12}$


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

Given the function $y=\cos (x+\frac{π}{3})$, determine the horizontal shift of the graph of the function $y=\sin x$.

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


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

Given that connecting all the vertices of a polygon from a point on one of the edges results in 2022 triangles, determine the number of sides of this polygon.

Answer: $2023$


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>.

A high school basketball team has 16 players, including a set of twins named Bob and Bill, and another set of twins named Tom and Tim. How many ways can we choose 5 starters if no more than one player from each set of twins can be chosen?

Answer: $3652$


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

In square ABCD, an isosceles triangle AEF is inscribed; point E lies on side BC, point F lies on side CD, and AE = AF. The tangent of angle AEF is 3. Find the cosine of angle FAD.

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


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

A circle made of wire and a rectangle are arranged in such a way that the circle passes through two vertices $A$ and $B$ and touches the side $CD$. The length of side $CD$ is 32.1. Find the ratio of the sides of the rectangle, given that its perimeter is 4 times the radius of the circle.

Answer: $4:1$


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>.

For \( n=1,2,3, \ldots \), let \( a_{n}=n^{2}+100 \), and let \( d_{n} \) denote the greatest common divisor of \( a_{n} \) and \( a_{n+1} \). Find the maximum value of \( d_{n} \) as \( n \) ranges over all positive integers.

Answer: $401$


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

In triangle \( ABC \) with sides \( AB = 8 \), \( AC = 4 \), \( BC = 6 \), the angle bisector \( AK \) is drawn, and on the side \( AC \) a point \( M \) is marked such that \( AM : CM = 3 : 1 \). Point \( N \) is the intersection point of \( AK \) and \( BM \). Find \( AN \).

Answer: $\frac{18\sqrt{6}}{11}$


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

Find the sum of all real roots \( x \) of the equation \( \left(2^{x}-4\right)^{3}+\left(4^{x}-2\right)^{3}=\left(4^{x}+2^{x}-6\right)^{3} \).

Answer: $3.5$


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

In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it; this grid's *score* is the sum of all numbers recorded this way. Deyuan shades each square in a blank   $n\times n$  grid with probability  $k$ ; he notices that the expected value of the score of the resulting grid is equal to  $k$ , too! Given that  $k > 0.9999$ , find the minimum possible value of  $n$ .

*Proposed by Andrew Wu*

Answer: $51$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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. Let vector \\( \overrightarrow{m}=(\cos A+ \sqrt {2},\sin A)\\) and vector \\( \overrightarrow{n}=(-\sin A,\cos A)\\). If \\(| \overrightarrow{m}+ \overrightarrow{n}|=2\\),
\\((1)\\) find the magnitude of angle \\(A\\);
\\((2)\\) if \\(b=4 \sqrt {2}\\) and \\(c= \sqrt {2}a\\), find the area of \\(\triangle ABC\\).

Answer: $16$


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

Given the function $f(x)=2\cos \left(2x+ \frac{2\pi}{3}\right)+ \sqrt{3}\sin 2x$.

$(1)$ Find the smallest positive period and the maximum value of the function $f(x)$;

$(2)$ Let $\triangle ABC$ have internal angles $A$, $B$, and $C$ respectively. If $f\left(\frac{C}{2}\right)=- \frac{1}{2}$ and $AC=1$, $BC=3$, find the value of $\sin A$.

Answer: $\frac{3 \sqrt{21}}{14}$


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>.

Consider the quadratic equation $2x^2 - 5x + m = 0$. Find the value of $m$ such that the sum of the roots of the equation is maximized while ensuring that the roots are real and rational.

Answer: $\frac{25}{8}$


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

The graph of the function $y=g(x)$ is given. For all $x > 5$, it holds that $g(x) > 0.5$. The function $g(x)$ is defined as $g(x) = \frac{x^2}{Ax^2 + Bx + C}$ where $A$, $B$, and $C$ are integers. The vertical asymptotes of $g$ are at $x = -3$ and $x = 4$, and the horizontal asymptote is such that $y = 1/A < 1$. Find $A + B + C$.

Answer: $-24$


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

With the rapid development of the "Internet + transportation" model, "shared bicycles" have appeared successively in many cities. In order to understand the satisfaction of users in a certain area with the services provided, a certain operating company randomly surveyed 10 users and obtained satisfaction ratings of 92, 84, 86, 78, 89, 74, 83, 77, 89.
$(1)$ Calculate the sample mean $\overline{x}$ and variance $s^{2}$;
$(2)$ Under condition (1), if the user's satisfaction rating is between $({\overline{x}-s,\overline{x}+s})$, then the satisfaction level is "$A$ grade". Estimate the percentage of users in the area whose satisfaction level is "$A$ grade".
Reference data: $\sqrt{30}≈5.48, \sqrt{33}≈5.74, \sqrt{35}≈5.92$.

Answer: $50\%$


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

What is the smallest positive integer \( n \) such that \( 5n \equiv 105 \pmod{24} \)?

Answer: $21$


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

Given that the sum of the first $n$ terms of a geometric sequence ${a_{n}}$ is $S_{n}$, if $a_{3}=4$, $S_{3}=12$, find the common ratio.

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


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

Given that \( A \), \( B \), and \( C \) are any three non-collinear points on a plane, and point \( O \) is inside \( \triangle ABC \) such that:
\[
\angle AOB = \angle BOC = \angle COA = 120^\circ.
\]
Find the maximum value of \( \frac{OA + OB + OC}{AB + BC + CA} \).

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


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

In a class of 50 students, the math scores $\xi$ ($\xi \in \mathbb{N}$) follow a normal distribution $N(100, 10^2)$. It is known that $P(90 \leq \xi \leq 100) = 0.3$. Estimate the number of students whose math scores are above 110.

Answer: $10$


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

Given the sets \( A = \{(x, y) \mid ax + y = 1, x, y \in \mathbb{Z}\} \), \( B = \{(x, y) \mid x + ay = 1, x, y \in \mathbb{Z}\} \), and \( C = \{(x, y) \mid x^2 + y^2 = 1\} \), find the value of \( a \) when \( (A \cup B) \cap C \) is a set with four elements.

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>.

In triangle \( \triangle ABC \), \( M \) is the midpoint of side \( AC \), \( D \) is a point on side \( BC \) such that \( AD \) is the angle bisector of \( \angle BAC \), and \( P \) is the point of intersection of \( AD \) and \( BM \). Given that \( AB = 10 \, \text{cm} \), \( AC = 30 \, \text{cm} \), and the area of triangle \( \triangle ABC \) is \( 100 \, \text{cm}^2 \), calculate the area of triangle \( \triangle ABP \).

Answer: $20$


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

Given the set \( A = \{1, 2, 3, \ldots, 1002\} \), Petya and Vasya play a game. Petya names a number \( n \), and Vasya then selects a subset of \( A \) consisting of \( n \) elements. Vasya wins if there are no two coprime numbers in the subset he selects; otherwise, Petya wins. What is the minimum \( n \) that Petya must name to guarantee a win?

Answer: $502$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 $\frac{3b^{-1} - \frac{b^{-1}}{3}}{b^2}$ when $b = \tfrac{1}{3}$.

Answer: $72$


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

Given $a\ln a=be^{b}$, where $b > 0$, find the maximum value of $\frac{b}{{{a^2}}}$

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


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

Given \( x_{1}, x_{2}, x_{3} \in [0, 12] \),
\[ x_{1} x_{2} x_{3} = \left(\left(12 - x_{1}\right)\left(12 - x_{2}\right)\left(12 - x_{3}\right)\right)^{2}. \]

Find the maximum value of \( f = x_{1} x_{2} x_{3} \).

Answer: $729$


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

Calculate the limit of the numerical sequence:

$$\lim _{n \rightarrow \infty} \frac{\sqrt[4]{2+n^{5}}-\sqrt{2 n^{3}+3}}{(n+\sin n) \sqrt{7 n}}$$

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


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>.

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 the reason.

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 problem involves finding the value of the expressions $\lg 2 + \lg 5$ and $4(-100)^4$.

Answer: $400000000$


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

Let the function \( f(x) = x^2 - x + 1 \). Define \( f^{(n)}(x) \) as follows:
$$
f^{(1)}(x) = f(x), \quad f^{(n)}(x) = f\left(f^{(n-1)}(x)\right).
$$

Let \( r_{n} \) be the arithmetic mean of all the roots of \( f^{(n)}(x) = 0 \). Find \( r_{2015} \).

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>.

A modified sign pyramid with five levels, where a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. If a "-" is to be at the top of the pyramid, calculate the number of possible ways to fill the five cells in the bottom row.

Answer: $16$


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

Using an electric stove with a power of $P=500 \mathrm{W}$, a certain amount of water is heated. When the electric stove is turned on for $t_{1}=1$ minute, the water temperature increases by $\Delta T=2^{\circ} \mathrm{C}$, and after turning off the heater, the temperature decreases to the initial value in $t_{2}=2$ minutes. Determine the mass of the heated water, assuming the thermal power losses are constant. The specific heat capacity of water is $c_{B}=4200$ J/kg$\cdot{ }^{\circ} \mathrm{C}$.

Answer: $2.38$


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

Convert from kilometers to miles. In the problem 3.125, the Fibonacci numeral system was introduced as being useful when converting distances from kilometers to miles or vice versa.

Suppose we want to find out how many miles are in 30 kilometers. For this, we represent the number 30 in the Fibonacci numeral system:

$$
30=21+8+1=F_{8}+F_{6}+F_{2}=(1010001)_{\mathrm{F}}
$$

Now we need to shift each number one position to the right, obtaining

$$
F_{7}+F_{5}+F_{1}=13+5+1=19=(101001)_{\mathrm{F}}
$$

So, the estimated result is 19 miles. (The correct result is approximately 18.46 miles.) Similarly, conversions from miles to kilometers are done.

Explain why this algorithm works. Verify that it gives a rounded number of miles in $n$ kilometers for all $n \leqslant 100$, differing from the correct answer by less than $2 / 3$ miles.

Answer: $19$


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

In the expansion of $(x+y)(x-y)^{5}$, the coefficient of $x^{2}y^{4}$ is ____ (provide your answer as a number).

Answer: $-5$


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

A shipbuilding company has an annual shipbuilding capacity of 20 ships. The output function of building $x$ ships is $R(x) = 3700x + 45x^2 - 10x^3$ (unit: ten thousand yuan), and the cost function is $C(x) = 460x + 5000$ (unit: ten thousand yuan). In economics, the marginal function $Mf(x)$ of a function $f(x)$ is defined as $Mf(x) = f(x+1) - f(x)$.
(1) Find the profit function $P(x)$ and the marginal profit function $MP(x)$; (Hint: Profit = Output - Cost)
(2) How many ships should be built annually to maximize the company's annual profit?

Answer: $12$


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

Let $S$ and $S_{1}$ respectively be the midpoints of edges $AD$ and $B_{1}C_{1}$. A rotated cube is denoted by $A^{\prime}B^{\prime}C^{\prime}D^{\prime}A_{1}^{\prime}B_{1}^{\prime}C_{1}^{\prime}D_{1}^{\prime}$. The common part of the original cube and the rotated one is a polyhedron consisting of a regular quadrilateral prism $EFGHE_{1}F_{1}G_{1}H_{1}$ and two regular quadrilateral pyramids $SEFGH$ and $S_{1}E_{1}F_{1}G_{1}H_{1}$.

The side length of the base of each pyramid is 1, and its height is $\frac{1}{2}$, making its volume $\frac{1}{6}$. The volume of the prism is $\sqrt{2}-1$.

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


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>.

In the expansion of $({\frac{1}{x}-\sqrt{x}})^{10}$, determine the coefficient of $x^{2}$.

Answer: $45$


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

Richard and Shreyas are arm wrestling against each other. They will play  $10$  rounds, and in each round, there is exactly one winner. If the same person wins in consecutive rounds, these rounds are considered part of the same “streak”. How many possible outcomes are there in which there are strictly more than  $3$  streaks? For example, if we denote Richard winning by  $R$  and Shreyas winning by  $S,$   $SSRSSRRRRR$  is one such outcome, with  $4$  streaks.

Answer: $932$


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

A circle with center $O$ and equation $x^2 + y^2 = 1$ passes through point $P(-1, \sqrt{3})$. Two tangents are drawn from $P$ to the circle, touching the circle at points $A$ and $B$ respectively. Find the length of the chord $|AB|$.

Answer: $\sqrt{3}$


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

Let \( S = \{1, 2, \cdots, 2005\} \), and \( A \subseteq S \) with \( |A| = 31 \). Additionally, the sum of all elements in \( A \) is a multiple of 5. Determine the number of such subsets \( A \).

Answer: $\frac{1}{5} \binom{2005}{31}$


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

The ratio of the magnitudes of two angles of a triangle is 2, and the difference in lengths of the sides opposite these angles is 2 cm; the length of the third side of the triangle is 5 cm. Calculate the area of the triangle.

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


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

Let \(x\), \(y\), and \(z\) be complex numbers such that:
\[
xy + 3y = -9, \\
yz + 3z = -9, \\
zx + 3x = -9.
\]

Find all possible values of \(xyz\).

Answer: $27$


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

If the graph of the function $f(x)=\sin 2x+\cos 2x$ is translated to the left by $\varphi (\varphi > 0)$ units, and the resulting graph is symmetric about the $y$-axis, then find the minimum value of $\varphi$.

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


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

The product of two of the four roots of the quartic equation \( x^4 - 18x^3 + kx^2 + 200x - 1984 = 0 \) is -32. Determine the value of \( k \).

Answer: $86$


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

Find the minimum value of the expression \(\frac{13 x^{2}+24 x y+13 y^{2}+16 x+14 y+68}{\left(9-x^{2}-8 x y-16 y^{2}\right)^{5 / 2}}\). Round the answer to the nearest hundredth if needed.

Answer: $0.26$


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

Given an arithmetic sequence $\{a_{n}\}$ with a common difference of $\frac{{2π}}{3}$, let $S=\{\cos a_{n}|n\in N^{*}\}$. If $S=\{a,b\}$, find the value of $ab$.

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>.

Vitya and Masha were born in the same year in June. Find the probability that Vitya is at least one day older than Masha.

Answer: $29/60$


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>.

A container with a capacity of 100 liters is filled with pure alcohol. After pouring out a portion of the alcohol, the container is filled with water. The mixture is then stirred thoroughly, and an amount of liquid equal to the first portion poured out is poured out again. The container is filled with water once more. At this point, the volume of water in the container is three times the volume of pure alcohol. How many liters of pure alcohol were poured out the first time?

Answer: $50$


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

If \( a(x+1)=x^{3}+3x^{2}+3x+1 \), find \( a \) in terms of \( x \).

If \( a-1=0 \), then the value of \( x \) is \( 0 \) or \( b \). What is \( b \) ?

If \( p c^{4}=32 \), \( p c=b^{2} \), and \( c \) is positive, what is the value of \( c \) ?

\( P \) is an operation such that \( P(A \cdot B) = P(A) + P(B) \).

\( P(A) = y \) means \( A = 10^{y} \). If \( d = A \cdot B \), \( P(A) = 1 \) and \( P(B) = c \), find \( d \).

Answer: $1000$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 $a$, $b$, $c$ be the lengths of the sides opposite to angles $A$, $B$, $C$ respectively, and it is given that $b = a \cos C + \frac{\sqrt{3}}{3} c \sin A$.
(i) Find the measure of angle $A$.
(ii) If the area of $\triangle ABC$ is $\sqrt{3}$ and the median to side $AB$ is $\sqrt{2}$, find the lengths of sides $b$ and $c$.

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
$$
\frac{1}{1+\frac{1}{1+\frac{1}{1+\ddots-\frac{1}{1}}}}=\frac{m}{n}
$$
where \(m\) and \(n\) are coprime natural numbers, and there are 1988 fraction lines on the left-hand side of the equation. Calculate the value of \(m^2 + mn - n^2\).

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>.

In a division equation, the dividend is 2016 greater than the divisor, the quotient is 15, and the remainder is 0. What is the dividend?

Answer: $2160$


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

Given an equilateral triangle \( \triangle ABC \) with a side length of 1,
\[
\overrightarrow{AP} = \frac{1}{3}(\overrightarrow{AB} + \overrightarrow{AC}), \quad \overrightarrow{AQ} = \overrightarrow{AP} + \frac{1}{2}\overrightarrow{BC}.
\]

Find the area of \( \triangle APQ \).

Answer: $\frac{\sqrt{3}}{12}$


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

Given \( f(x) = x^{5} + a_{1} x^{4} + a_{2} x^{3} + a_{3} x^{2} + a_{4} x + a_{5} \), and \( f(m) = 2017m \) for \( m = 1, 2, 3, 4 \), calculate \( f(10) - f(-5) \).

Answer: $75615$


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

Find the area of trapezoid \(ABCD\) with a side \(BC = 5\), where the distances from vertices \(A\) and \(D\) to the line \(BC\) are 3 and 7 respectively.

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>.

Six consecutive prime numbers have sum \( p \). Given that \( p \) is also a prime, determine all possible values of \( p \).

Answer: $41$


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

Calculate: $-\sqrt{4}+|-\sqrt{2}-1|+(\pi -2013)^{0}-(\frac{1}{5})^{0}$.

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>.

The constant term in the expansion of $( \sqrt {x}+ \frac {2}{x^{2}})^{n}$ is \_\_\_\_\_\_ if only the sixth term of the binomial coefficient is the largest.

Answer: $180$


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

Given that the focus of the parabola $y=x^{2}$ is $F$, a line passing through point $F$ intersects the parabola at points $A$ and $B$. If $|AB|=4$, find the distance from the midpoint of chord $AB$ to the $x$-axis.

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