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

Given $\sqrt{99225}=315$, $\sqrt{x}=3.15$, then $x=(\ )$.

Answer: $9.9225$


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

Evaluate $\cos \frac {\pi}{7}\cos \frac {2\pi}{7}\cos \frac {4\pi}{7}=$ ______.

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


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

In the diagram, \(\triangle ABC\) is right-angled at \(C\). Point \(D\) is on \(AC\) so that \(\angle ABC = 2 \angle DBC\). If \(DC = 1\) and \(BD = 3\), determine the length of \(AD\).

Answer: $\frac{9}{7}$


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>.

Find the sum of the squares of the solutions to
\[\left| x^2 - x + \frac{1}{2010} \right| = \frac{1}{2010}.\]

Answer: $\frac{2008}{1005}$


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

Find the length of \(PQ\) in the triangle below, where \(PQR\) is a right triangle with \( \angle RPQ = 45^\circ \) and the length \(PR\) is \(10\).

Answer: $10\sqrt{2}$


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 sequence $\{a_n\}$ satisfies $a_{n+1}+(-1)^{n}a_{n}=2n-1$. Find the sum of the first $60$ terms of $\{a_n\}$.

Answer: $1830$


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

Determine the number of revolutions a wheel, with a fixed center and with an outside diameter of 8 feet, would require to cause a point on the rim to travel one mile.

Answer: $\frac{660}{\pi}$


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

Given that $\sin(a + \frac{\pi}{4}) = \sqrt{2}(\sin \alpha + 2\cos \alpha)$, determine the value of $\sin 2\alpha$.

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


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

Given seven positive integers from a list of eleven positive integers are \(3, 5, 6, 9, 10, 4, 7\). What is the largest possible value of the median of this list of eleven positive integers if no additional number in the list can exceed 10?

Answer: $10$


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

Given that $S_{n}$ is the sum of the first $n$ terms of the sequence $\{a_{n}\}$, and ${S}_{n}=\frac{{n}^{2}+3n}{2}$.
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;
$(2)$ Let $T_{n}$ be the sum of the first $n$ terms of the sequence $\{\frac{1}{{a}_{n}{a}_{n+1}}\}$. If $\lambda T_{n}\leqslant a_{n+1}$ holds for all $n\in \mathbb{N}^{*}$, find the maximum value of the real number $\lambda$.

Answer: $16$


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

In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $(\sqrt{3}b-c)\cos A=a\cos C$, find the value of $\cos A$.

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>.

If $0 < \alpha < \frac{\pi}{2}$ and $\tan \alpha = 2$, then $\frac{\sin 2\alpha + 1}{\cos^4 \alpha - \sin^4 \alpha} = \_\_\_\_\_\_$.

Answer: $-3$


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

How many  $6$ -digit positive integers have their digits in nondecreasing order from left to right? Note that  $0$  cannot be a leading digit.

Answer: $3003$


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

Given the function $f(x)=a\ln(x+1)+bx+1$
$(1)$ If the function $y=f(x)$ has an extremum at $x=1$, and the tangent line to the curve $y=f(x)$ at the point $(0,f(0))$ is parallel to the line $2x+y-3=0$, find the value of $a$;
$(2)$ If $b= \frac{1}{2}$, discuss the monotonicity of the function $y=f(x)$.

Answer: $-4$


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>.

Shuffle the 12 cards of the four suits of A, K, Q in a deck of playing cards.
(1) If a person draws 2 cards at random, what is the probability that both cards are Aces?
(2) If the person has already drawn 2 Kings without replacement, what is the probability that another person draws 2 Aces?

Answer: $\frac{2}{15}$


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

In triangle $ABC$, we have $\angle A = 90^\circ$, $BC = 20$, and $\tan C = 3\cos B$. What is $AB$?

Answer: $\frac{40\sqrt{2}}{3}$


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

Of the thirteen members of the volunteer group, Hannah selects herself, Tom Morris, Jerry Hsu, Thelma Paterson, and Louise Bueller to teach the September classes.  When she is done, she decides that it's not necessary to balance the number of female and male teachers with the proportions of girls and boys at the hospital  $\textit{every}$  month, and having half the women work while only  $2$  of the  $7$  men work on some months means that some of the women risk getting burned out.  After all, nearly all the members of the volunteer group have other jobs.

Hannah comes up with a plan that the committee likes.  Beginning in October, the comittee of five volunteer teachers will consist of any five members of the volunteer group, so long as there is at least one woman and at least one man teaching each month.  Under this new plan, what is the least number of months that  $\textit{must}$  go by (including October when the first set of five teachers is selected, but not September) such that some five-member comittee  $\textit{must have}$  taught together twice (all five members are the same during two different months)?

Answer: $1261$


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

Larry can swim from Harvard to MIT (with the current of the Charles River) in  $40$  minutes, or back (against the current) in  $45$  minutes. How long does it take him to row from Harvard to MIT, if he rows the return trip in  $15$  minutes? (Assume that the speed of the current and Larry’s swimming and rowing speeds relative to the current are all constant.) Express your answer in the format mm:ss.

Answer: $14:24$


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>.

A shooter's probabilities of hitting the 10, 9, 8, 7 rings, and below 7 rings in a shooting are 0.24, 0.28, 0.19, 0.16, and 0.13, respectively. Calculate the probability that the shooter in a single shot:
(1) Hits the 10 or 9 rings,
(2) Hits at least the 7 ring,
(3) Hits less than 8 rings.

Answer: $0.29$


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

Find  $f(2)$  given that  $f$  is a real-valued function that satisfies the equation  $$ 4f(x)+\left(\frac23\right)(x^2+2)f\left(x-\frac2x\right)=x^3+1. $$

Answer: $\frac{19}{12}$


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 the solution set of the inequality about $x$, $|x-2| < a$ ($a \in \mathbb{R}$), be $A$, and $\frac{3}{2} \in A$, $-\frac{1}{2} \notin A$.
(1) For any $x \in \mathbb{R}$, the inequality $|x-1| + |x-3| \geq a^2 + a$ always holds true, and $a \in \mathbb{N}$. Find the value of $a$.
(2) If $a + b = 1$, and $a, b \in \mathbb{R}^+$, find the minimum value of $\frac{1}{3b} + \frac{b}{a}$, and indicate the value of $a$ when the minimum is attained.

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


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

If \(\lceil{\sqrt{x}}\rceil=20\), how many possible integer values of \(x\) are there?

Answer: $39$


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

Let  $ABC$  be an equilateral triangle with  $AB=1.$  Let  $M$  be the midpoint of  $BC,$  and let  $P$  be on segment  $AM$  such that  $AM/MP=4.$  Find  $BP.$

Answer: $\frac{\sqrt{7}}{5}$


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 triangle $\bigtriangleup ABC$ has vertices lying on the parabola defined by $y = x^2 + 4$. Vertices $B$ and $C$ are symmetric about the $y$-axis and the line $\overline{BC}$ is parallel to the $x$-axis. The area of $\bigtriangleup ABC$ is $100$. $A$ is the point $(2,8)$. Determine the length of $\overline{BC}$.

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>.

Simplify first, then evaluate.
$(1) 3x^{3}-[x^{3}+(6x^{2}-7x)]-2(x^{3}-3x^{2}-4x)$, where $x=-1$;
$(2) 2(ab^{2}-2a^{2}b)-3(ab^{2}-a^{2}b)+(2ab^{2}-2a^{2}b)$, where $a=2$, $b=1$.

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>.

A regular polygon has interior angles of 160 degrees, and each side is 4 units long. How many sides does the polygon have, and what is its perimeter?

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>.

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

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


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

Given that the random variable $\xi$ follows a normal distribution $N(4, 6^2)$, and $P(\xi \leq 5) = 0.89$, find the probability $P(\xi \leq 3)$.

Answer: $0.11$


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>.

Set  $S = \{1, 2, 3, ..., 2005\}$ . If among any  $n$  pairwise coprime numbers in  $S$  there exists at least a prime number, find the minimum of  $n$ .

Answer: $15$


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

Randomly split 2.5 into the sum of two non-negative numbers, and round each number to its nearest integer. What is the probability that the sum of the two resulting integers is 3?

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


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

A group consists of 4 male students and 3 female students. From this group, 4 people are selected to complete three different tasks, with the condition that at least two of the selected individuals must be female, and each task must have at least one person assigned to it. The number of different ways to select and assign these individuals is ____.

Answer: $792$


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

The sampling group size is 10.

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>.

From the numbers 0, 1, 2, 3, and 4, select three different digits to form a three-digit number and calculate the total number of such numbers that are odd.

Answer: $18$


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

Given a complex number $z$ satisfying $z+ \bar{z}=6$ and $|z|=5$.

$(1)$ Find the imaginary part of the complex number $z$;

$(2)$ Find the real part of the complex number $\dfrac{z}{1-i}$.

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


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

Observe the following set of equations, and based on the equations above, guess that ${{S}_{2n-1}}=\left( 4n-3 \right)\left( an+b \right)$, then ${{a}^{2}}+{{b}^{2}}=$ ____.

${{S}_{1}}=1$

${{S}_{2}}=2+3+4=9$,

${{S}_{3}}=3+4+5+6+7=25$,

${{S}_{4}}=4+5+6+7+8+9+10=49$, (......)

Answer: $25$


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

A farmer contracted several acres of fruit trees. This year, he invested 13,800 yuan, and the total fruit yield was 18,000 kilograms. The fruit sells for a yuan per kilogram in the market and b yuan per kilogram when sold directly from the orchard (b < a). The farmer transports the fruit to the market for sale, selling an average of 1,000 kilograms per day, requiring the help of 2 people, paying each 100 yuan per day, and the transportation cost of the agricultural vehicle and other taxes and fees average 200 yuan per day.

(1) Use algebraic expressions involving a and b to represent the income from selling the fruit in both ways.
(2) If a = 4.5 yuan, b = 4 yuan, and all the fruit is sold out within the same period using both methods, calculate which method of selling is better.
(3) If the farmer strengthens orchard management, aiming for a net income of 72,000 yuan next year, and uses the better selling method from (2), what is the growth rate of the net income (Net income = Total income - Total expenses)?

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 non-negative numbers $x$, $y$, and $z$ such that $x+y+z=2$, determine the minimum value of $\frac{1}{3}x^3+y^2+z$.

Answer: $\frac{13}{12}$


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>.

Calculate the sum of the first five prime numbers that have a units digit of 3.

Answer: $135$


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

Given the variance of a sample is $$s^{2}= \frac {1}{20}[(x_{1}-3)^{2}+(x_{2}-3)^{2}+\ldots+(x_{n}-3)^{2}]$$, then the sum of this set of data equals \_\_\_\_\_\_.

Answer: $60$


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

In trapezoid EFGH, sides EF and GH are equal. It is known that EF = 12 units and GH = 10 units. Additionally, each of the non-parallel sides forms a right-angled triangle with half of the difference in lengths of EF and GH and a given leg of 6 units. Determine the perimeter of trapezoid EFGH.

Answer: $22 + 2\sqrt{37}$


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

Consider a sequence $\{a_n\}$ where the sum of the first $n$ terms, $S_n$, satisfies $S_n = 2a_n - a_1$, and $a_1$, $a_2 + 1$, $a_3$ form an arithmetic sequence.

(1) Find the general formula for the sequence $\{a_n\}$.
(2) Let $b_n = \log_2 a_n$ and $c_n = \frac{3}{b_nb_{n+1}}$. Denote the sum of the first $n$ terms of the sequence $\{c_n\}$ as $T_n$. If $T_n < \frac{m}{3}$ holds for all positive integers $n$, determine the smallest possible positive integer value of $m$.

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>.

Bonnie constructs the frame of a cuboid using wire pieces. She uses eight pieces each of 8 inches for the length, and four pieces each of 10 inches for the width and height. Roark, on the other hand, uses 2-inch-long wire pieces to construct frames of smaller cuboids, all of the same size with dimensions 1 inch by 2 inches by 2 inches. The total volume of Roark's cuboids is the same as Bonnie's cuboid. What is the ratio of the total length of Bonnie's wire to the total length of Roark's wire?

Answer: $\frac{9}{250}$


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

Given a vertical wooden pillar, a rope is tied to its top, with 4 feet of the rope hanging down to the ground. Additionally, when pulling the rope, it runs out when 8 feet away from the base of the pillar. Determine the length of the rope.

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>.

If  $\phi$  is the Golden Ratio, we know that  $\frac1\phi = \phi - 1$ .  Define a new positive real number, called  $\phi_d$ , where  $\frac1{\phi_d} = \phi_d - d$  (so  $\phi = \phi_1$ ).  Given that  $\phi_{2009} = \frac{a + \sqrt{b}}{c}$ ,  $a, b, c$  positive integers, and the greatest common divisor of  $a$  and  $c$  is 1, find  $a + b + c$ .

Answer: $4038096$


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

You roll a fair 12-sided die repeatedly. The probability that all the prime numbers show up at least once before seeing any of the other numbers can be expressed as a fraction \( \frac{p}{q} \) in lowest terms. What is \( p+q \)?

Answer: $793$


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

The value of \( 333 + 33 + 3 \) is:

Answer: $369$


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

Given vectors $\overrightarrow{a}=(\sin x, \frac{3}{2})$ and $\overrightarrow{b}=(\cos x,-1)$.
(1) When $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $2\cos ^{2}x-\sin 2x$.
(2) Find the maximum value of $f(x)=( \overrightarrow{a}+ \overrightarrow{b}) \cdot \overrightarrow{b}$ on $\left[-\frac{\pi}{2},0\right]$.

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


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

Calculate:
$(1)(\frac{5}{7})×(-4\frac{2}{3})÷1\frac{2}{3}$;
$(2)(-2\frac{1}{7})÷(-1.2)×(-1\frac{2}{5})$.

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


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

$$
\text{Consider the system of inequalities:}
\begin{cases}
x + 2y \leq 6 \\
3x + y \geq 3 \\
x \leq 4 \\
y \geq 0
\end{cases}
$$
Determine the number of units in the length of the longest side of the polygonal region formed by this system. Express your answer in simplest radical form.

Answer: $2\sqrt{5}$


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

Let \[p(x, y) =
\begin{cases}
xy &\quad \text{if } x \ge 0 \text{ and } y \ge 0, \\
x - 2y &\quad \text{if } x < 0 \text{ and } y < 0, \\
2x + 3y &\quad \text{if } x \ge 0 \text{ and } y < 0, \\
x + 3y &\quad \text{if } x < 0 \text{ and } y \ge 0 \\
3x + y &\quad \text{otherwise}.
\end{cases}
\]What is $p(p(2,-3),p(-1,4))$?

Answer: $28$


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

Determine the ratio of the shorter side to the longer side of the rectangular park, given that by taking a shortcut along the diagonal, a boy saved a distance equal to $\frac{1}{3}$ of the longer side of the park.

Answer: $\frac{5}{12}$


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>.

Calculate $(3^5 \cdot 6^5)^2$.

Answer: $3570467226624$


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

A train starts its journey, then stops after 1 hour due to an incident and remains halted for half an hour. After that, it continues at $\frac{3}{4}$ of its original speed, resulting in a delay of $3 \frac{1}{2}$ hours upon reaching its destination. If the incident had occurred 90 miles further ahead, the train would have arrived only 3 hours late. What is the total distance of the whole journey in miles?

(Problem from the 5th Annual American High School Mathematics Exam, 1954)

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>.

How many digits does the number \(2^{100}\) have? What are its last three digits? (Give the answers without calculating the power directly or using logarithms!) If necessary, how could the power be quickly calculated?

Answer: $376$


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

A regular hexagon is inscribed in an equilateral triangle. If the hexagon has an area of 12, what is the area of the equilateral triangle?

Answer: $18$


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

For some constants \( c \) and \( d \), let \[ g(x) = \left\{
\begin{array}{cl}
cx + d & \text{if } x < 3, \\
10 - 2x & \text{if } x \ge 3.
\end{array}
\right.\] The function \( g \) has the property that \( g(g(x)) = x \) for all \( x \). What is \( c + d \)?

Answer: $\frac{9}{2}$


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

What is the sum of the ten terms in the arithmetic sequence $-3, 4, \dots, 40$?

Answer: $285$


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

Given that $a$, $b$, $c \in R^{+}$ and $a + b + c = 1$, find the maximum value of $\sqrt{4a + 1} + \sqrt{4b + 1} + \sqrt{4c + 1}$.

Answer: $\sqrt{21}$


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>.

Find the area bounded by the graph of  $ y\equal{}\sqrt[3]{x\plus{}\sqrt{x^2\plus{}1}}\plus{}\sqrt[3]{x\minus{}\sqrt{x^2\plus{}1}}$ , the line  $ x\minus{}y\minus{}1\equal{}0$  and the  $ x$  axis.

Answer: $5/8$


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>.

Select two distinct numbers simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6\}$. What is the probability that the smaller one divides the larger one and both numbers are either both even or both odd?

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


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

Given that $2$ boys and $4$ girls are lined up, calculate the probability that the boys are neither adjacent nor at the ends.

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


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

Given that $ \sin \left( \frac{\pi}{3} - \frac{\alpha}{2} \right) = -\frac{\sqrt{3}}{2} $, find the value of $ \cos \left( \frac{\pi}{3} + \alpha \right) $.

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


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

Three circles with radii 2, 3, and 3 touch each other. What is the area of the triangle formed by joining the centers of these circles?

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>.

Five cards have the numbers 101, 102, 103, 104, and 105 on their fronts. On the reverse, each card has one of five different positive integers: \(a, b, c, d\), and \(e\) respectively. We know that:

1. \(c = b \cdot e\)
2. \(a + b = d\)
3. \(e - d = a\)

Frankie picks up the card which has the largest integer on its reverse. What number is on the front of Frankie's card?

Answer: $103$


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

Let $PQRS$ be a convex quadrilateral, and let $H_P,$ $H_Q,$ $H_R,$ $H_S$ denote the centroids of triangles $QRS,$ $PRS,$ $PQS,$ and $PQR,$ respectively. Calculate $\frac{[H_P H_Q H_R H_S]}{[PQRS]}.$

Answer: $\frac{1}{9}$


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

In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is given that $\frac{-b + \sqrt{2}c}{\cos B} = \frac{a}{\cos A}$,
$(I)$ Find the size of angle $A$;
$(II)$ If $a=2$, find the maximum value of the area $S$.

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>.

Linda is tasked with writing a report on extracurricular activities at her school. The school offers two clubs: Robotics and Science. Linda has a list of 30 students who are members of at least one club. She knows that 22 students are in the Robotics club and 24 students are in the Science club. If Linda picks two students randomly from her list to interview, what is the probability that she will be able to gather information about both clubs? Express your answer as a fraction in simplest form.

Answer: $\frac{392}{435}$


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

Given the function $$f(x)=(2-a)\ln x+ \frac {1}{x}+2ax \quad (a\leq0)$$.
(Ⅰ) When $a=0$, find the extreme value of $f(x)$;
(Ⅱ) When $a<0$, discuss the monotonicity of $f(x)$.

Answer: $2-2\ln2$


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

Two teams, Team A and Team B, participate in a table tennis group match. The ratio of their strength is $3:2$. Assuming both teams play to their normal levels, find the probability that Team A wins after playing 4 games in a best-of-5 series (3 wins out of 5 games).

Answer: $\frac{162}{625}$


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

My friend and I decided to dine at a new Mediterranean restaurant. I am a vegan and also avoid soy. The restaurant features six dishes that are vegan-friendly. These vegan dishes constitute one-fourth of the entire menu. However, five of these vegan dishes contain soy. What fraction of the menu can I eat?

Answer: $\frac{1}{24}$


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>.

In right triangle $ABC$ with $\angle B = 90^\circ$, sides $AB=1$ and $BC=3$. The bisector of $\angle BAC$ meets $\overline{BC}$ at $D$. Calculate the length of segment $BD$.
A) $\frac{1}{2}$
B) $\frac{3}{4}$
C) $1$
D) $\frac{5}{4}$
E) $2$

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


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

The ratio of the sums of the first n terms of the arithmetic sequences {a_n} and {b_n} is given by S_n / T_n = (2n) / (3n + 1). Find the ratio of the fifth terms of {a_n} and {b_n}.

Answer: $\dfrac{9}{14}$


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>.

If the universal set $U = \{-1, 0, 1, 2\}$, and $P = \{x \in \mathbb{Z} \,|\, -\sqrt{2} < x < \sqrt{2}\}$, determine the complement of $P$ in $U$.

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>.

A factory produces two types of products, A and B, with profits P and Q (in ten thousand yuan), respectively. The relationship between the profits and the invested capital m (in ten thousand yuan) follows the empirical formulas P = (1/3)m + 65 and Q = 76 + 4√m. Now, 150 ten thousand yuan of capital will be invested in producing products A and B, with a minimum investment of 25 ten thousand yuan for each product.

1. Let x represent the capital (in ten thousand yuan) invested in product B. Derive the function y (in ten thousand yuan) representing the total profit in terms of x and determine its domain.
2. How should the capital be allocated to maximize the total profit? What is the maximum profit?

Answer: $203$


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

Higher Secondary P4

If the fraction  $\dfrac{a}{b}$  is greater than  $\dfrac{31}{17}$  in the least amount while  $b<17$ , find  $\dfrac{a}{b}$ .

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


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

Let  $P$  be a  $2019-$ gon, such that no three of its diagonals concur at an internal point. We will call each internal intersection point of diagonals of  $P$  a knot. What is the greatest number of knots one can choose, such that there doesn't exist a cycle of chosen knots? ( Every two adjacent knots in a cycle must be on the same diagonal and on every diagonal there are at most two knots from a cycle.)

Answer: $2018$


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

What is the area, in square units, of a trapezoid bounded by the lines $y = x + 2$, $y = 12$, $y = 3$, and the $y$-axis? Express your answer as a decimal to the nearest tenth.

Answer: $49.5$


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

If $\vec{e}_1$ and $\vec{e}_2$ are unit vectors with an angle of $\frac{\pi}{3}$ between them, and $\vec{a}=2\vec{e}_1+ \vec{e}_2$, $\vec{b}=-3\vec{e}_1+2\vec{e}_2$, calculate the value of $\vec{a}\cdot \vec{b}$.

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>.

Given the function $f(x)=|\log_{x}|$, $a > b > 0$, $f(a)=f(b)$, calculate the minimum value of $\frac{a^2+b^2}{a-b}$.

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>.

How many positive integers less than $201$ are multiples of either $4$ or $9$, but not both at once?

Answer: $62$


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

Given that the sequence $\{a\_n\}$ satisfies $\frac{1}{a_{n+1}} - \frac{1}{a_n} = d (n \in \mathbb{N}^*, d$ is a constant$)$, it is called a harmonic sequence. It is known that the sequence $\{\frac{1}{x\_n}\}$ is a harmonic sequence and $x\_1 + x\_2 + ... + x_{20} = 200$. Find the value of $x\_5 + x_{16}$.

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>.

Two cards are chosen consecutively without replacement from a standard 52-card deck. What is the probability that the first card is a face card (Jack, Queen, or King) and the second card is a number card (2 through 10) with the two cards totaling to 15?

Answer: $\frac{4}{221}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 $f(x) = \log_2 x$ is shifted 1 unit to the left, and then the part below the $x$-axis is reflected across the $x$-axis to obtain the graph of function $g(x)$. Suppose real numbers $m$ and $n$ ($m < n$) satisfy $g(m) = g\left(-\frac{n+1}{n+2}\right)$ and $g(10m+6n+21) = 4\log_2 2$. Find the value of $m-n$.

Answer: $-\frac{1}{15}$


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

On a two-lane highway where both lanes are single-directional, cars in both lanes travel at different constant speeds. The speed of cars in the left lane is 10 kilometers per hour higher than in the right lane. Cars follow a modified safety rule: the distance from the back of the car ahead to the front of the car in the same lane is one car length for every 10 kilometers per hour of speed or fraction thereof. Suppose each car is 5 meters long, and a photoelectric eye at the side of the road detects the number of cars that pass by in one hour. Determine the whole number of cars passing the eye in one hour if the speed in the right lane is 50 kilometers per hour. Calculate $M$, the maximum result, and find the quotient when $M$ is divided by 10.

Answer: $338$


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

In the redesigned version of Mathland's licensing system, all automobile license plates still contain four symbols. The first is a vowel (A, E, I, O, U), the second a non-vowel (any consonant), and the last two symbols are now digits (0 through 9). What is the probability that a randomly chosen license plate will read "AR55"?
A) $\frac{1}{21,000}$
B) $\frac{1}{10,500}$
C) $\frac{1}{5,250}$
D) $\frac{1}{42,000}$
E) $\frac{1}{2,100}$

Answer: $\frac{1}{10,500}$


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

Consider the ellipse $\frac{x^2}{16} + \frac{y^2}{12} = 1$ whose left and right intersection points are $F_1$ and $F_2$, respectively. Let point $P$ be on the ellipse and satisfy $\vec{PF_1} \cdot \vec{PF_2} = 9$. Find the value of $|\vec{PF_1}| \cdot |\vec{PF_2}|$.

Answer: $15$


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

Given the odd function $f(x)$ is increasing on the interval $[3, 7]$ and its minimum value is 5, determine the nature of $f(x)$ and its minimum value on the interval $[-7, -3]$.

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>.

Given sandwiches cost $4 each, sodas cost $1.50 each, and fries cost $2.50 each, and you buy 4 sandwiches, 6 sodas, and 3 orders of fries, calculate the total cost before the $5 discount, and then subtract the total cost after applying the discount.

Answer: $27.50$


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

Let  $ABC$  be a triangle with  $|AB|=18$ ,  $|AC|=24$ , and  $m(\widehat{BAC}) = 150^\circ$ . Let  $D$ ,  $E$ ,  $F$  be points on sides  $[AB]$ ,  $[AC]$ ,  $[BC]$ , respectively, such that  $|BD|=6$ ,  $|CE|=8$ , and  $|CF|=2|BF|$ . Let  $H_1$ ,  $H_2$ ,  $H_3$  be the reflections of the orthocenter of triangle  $ABC$  over the points  $D$ ,  $E$ ,  $F$ , respectively. What is the area of triangle  $H_1H_2H_3$ ?

Answer: $96$


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

A new solid $T$ is defined by the set of all points $(x, y, z)$ in space such that $|x| + |y| + |z| \leq 2$. Find the volume of solid $T$.

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


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

The smallest 9-digit integer that can be divided by 11 is ____.

Answer: $100000010$


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

In right triangle $PQR$, $PQ=15$, $QR=8$, and angle $R$ is a right angle. A semicircle is inscribed in the triangle such that it touches $PQ$ and $QR$ at their midpoints and the hypotenuse $PR$. What is the radius of the semicircle?
A) $\frac{24}{5}$

B) $\frac{12}{5}$

C) $\frac{17}{4}$

D) $\frac{15}{3}$

Answer: $\frac{24}{5}$


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 triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $c\sin\frac{A+C}{2}=b\sin C$.
$(1)$ Find angle $B$;
$(2)$ Let $BD$ be the altitude from $B$ to side $AC$, and $BD=1$, $b=\sqrt{3}$. Find the perimeter of $\triangle ABC$.

Answer: $3 + \sqrt{3}$


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

A decorative garden is designed with a rectangular lawn with semicircles of grass at either end. The ratio of the length of the rectangle to its width is 5:4, and the total length including the semicircles is 50 feet. Calculate the ratio of the area of the rectangle to the combined area of the semicircles.

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


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

Given $A=\{-3,-2,-1,0,1,2,3\}$, $a$ and $b$ are elements of $A$. How many cases are there where $|a| \lt |b|$?

Answer: $18$


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

A line is drawn through the left focus $F_1$ of a hyperbola at an angle of $30^{\circ}$, intersecting the right branch of the hyperbola at point P. If a circle with diameter PF_1 passes through the right focus of the hyperbola, calculate the eccentricity of the hyperbola.

Answer: $\sqrt{3}$


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

Given positive integers \( x, y, z \) that satisfy the condition \( x y z = (14 - x)(14 - y)(14 - z) \), and \( x + y + z < 28 \), what is the maximum value of \( x^2 + y^2 + z^2 \)?

Answer: $219$


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

In triangle \( ABC \), \( AC = 3 AB \). Let \( AD \) bisect angle \( A \) with \( D \) lying on \( BC \), and let \( E \) be the foot of the perpendicular from \( C \) to \( AD \). Find \( \frac{[ABD]}{[CDE]} \). (Here, \([XYZ]\) denotes the area of triangle \( XYZ \)).

Answer: $1/3$


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

Express $0.5\overline{023}$ as a common fraction.

Answer: $\frac{1045}{1998}$


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>.

In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given vectors $\overrightarrow{m} = (c-2b, a)$ and $\overrightarrow{n} = (\cos A, \cos C)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.

(1) Find the magnitude of angle $A$;
(2) If $\overrightarrow{AB} \cdot \overrightarrow{AC} = 4$, find the minimum value of side $BC$.

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>.

Given $f(\alpha)= \frac {\sin (\pi-\alpha)\cos (2\pi-\alpha)\tan (-\alpha+ \frac {3\pi}{2})}{\cot (-\alpha-\pi)\sin (-\pi-\alpha)}$.
$(1)$ Simplify $f(\alpha)$;
$(2)$ If $\alpha$ is an angle in the third quadrant, and $\cos (\alpha- \frac {3\pi}{2})= \frac {1}{5}$, find the value of $f(\alpha)$.

Answer: $\frac {2 \sqrt {6}}{5}$


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

Given the function $f(x)=\sin (\omega x-\varphi)$ $(\omega > 0,|\varphi| < \frac {\pi}{2})$ whose graph intersects with the x-axis at points that are a distance of $\frac {\pi}{2}$ apart, and it passes through the point $(0,- \frac {1}{2})$
$(1)$ Find the analytical expression of the function $f(x)$;
$(2)$ Let the function $g(x)=f(x)+2\cos ^{2}x$, find the maximum and minimum values of $g(x)$ in the interval $\left[0, \frac {\pi}{2}\right]$.

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


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

In the rectangular coordinate system xOy, the equation of the curve C₁ is: x²+y²=4. The curve C₁ undergoes a scaling transformation $$\begin{cases} x′= \frac {1}{2}x \\ y′=y\end{cases}$$ to obtain the curve C₂.
(I) Find the parametric equation of the curve C₂.
(II) Establish a polar coordinate system with the origin O as the pole and the positive semi-axis of the x-axis as the polar axis. The equation of the line l is ρsin(θ+$$\frac {π}{4}$$) = -3$$\sqrt {2}$$. If P and Q are moving points on the curve C₂ and the line l respectively, find the minimum value of |PQ|.

Answer: $3 \sqrt {2}- \frac { \sqrt {10}}{2}$


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 $S_n$ be the sum of the first $n$ terms of the difference sequence $\{a_n\}$, given that $a_2 + a_{12} = 24$ and $S_{11} = 121$.
(1) Find the general term formula for $\{a_n\}$.
(2) Let $b_n = \frac {1}{a_{n+1}a_{n+2}}$, and $T_n = b_1 + b_2 + \ldots + b_n$. If $24T_n - m \geq 0$ holds for all $n \in \mathbb{N}^*$, find the maximum value of the real number $m$.

Answer: $m = \frac {3}{7}$


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

Given the function $f(x)=4\sin x\cos \left(x- \frac {\pi}{3}\right)- \sqrt {3}$.
(I) Find the smallest positive period and zeros of $f(x)$.
(II) Find the maximum and minimum values of $f(x)$ in the interval $\left[ \frac {\pi}{24}, \frac {3\pi}{4}\right]$.

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>.

Given the function $f(x)=\log_{a} \frac{2-x}{b+x} (0 < a < 1)$ is an odd function, when $x \in (-2,2a)$, the range of the function $f(x)$ is $(-\infty,1)$, then the sum of the real numbers $a+b=$ __________.

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>.

Calculate:
$(1)\frac{\sqrt{8}-\sqrt{24}}{\sqrt{2}}+|1-\sqrt{3}|$;
$(2)(\frac{1}{2})^{-1}-2cos30°+|2-\sqrt{3}|-(2\sqrt{2}+1)^{0}$.

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


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

Let the function be $$f(x)= \frac {2^{x}}{2^{x}+ \sqrt {2}}$$. Calculate the sum of $f(-2016) + f(-2015) + \ldots + f(0) + f(1) + \ldots + f(2017)$.

Answer: $2017$


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

For every  $x \ge -\frac{1}{e}\,$ , there is a unique number  $W(x) \ge -1$  such that
\[
  W(x) e^{W(x)} = x.
\]
The function  $W$  is called Lambert's  $W$  function.  Let  $y$  be the unique positive number such that
\[
  \frac{y}{\log_{2} y} = - \frac{3}{5} \, .
\]
The value of  $y$  is of the form  $e^{-W(z \ln 2)}$  for some rational number  $z$ .  What is the value of  $z$ ?

Answer: $5/3$


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 $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $r$ be a real number. Two of the roots of $f(x)$ are $r + 2$ and $r + 8$. Two of the roots of $g(x)$ are $r + 5$ and $r + 11$, and
\[f(x) - g(x) = 2r\] for all real numbers $x$. Find $r$.

Answer: $20.25$


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

Given $x,y \in (0, +\infty)$, and satisfying $\frac{1}{x} + \frac{1}{2y} = 1$, determine the minimum value of $x+4y$.

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


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

There are three spheres and a cube. The first sphere is tangent to each face of the cube, the second sphere is tangent to each edge of the cube, and the third sphere passes through each vertex of the cube. Then, the ratio of the surface areas of these three spheres is ______.

Answer: $1:2:3$


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>.

During the festive season when the moon is full and the country is celebrating together, a supermarket plans to reduce the selling price of grapes that cost $16$ yuan per kilogram. Through statistical analysis, it was found that when the selling price is $26$ yuan per kilogram, $320$ kilograms can be sold per day. If the selling price is reduced by $1$ yuan per kilogram, the daily sales volume will increase by $80$ kilograms. If the supermarket wants to make a daily profit of $3600$ yuan from sales and also provide customers with more benefits, what should be the selling price of grapes per kilogram?

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 $a\in R$, $b\in R$, if the set $A=\{a, \frac{b}{a}, 1\}$, $B=\{a^{2}, a+b, 0\}$, $A\subseteq B$, and $B\subseteq A$, then the value of $a^{2023}+b^{2023}$ is ______.

Answer: $-1$


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

Given the equation of an ellipse is $\dfrac {x^{2}}{a^{2}} + \dfrac {y^{2}}{b^{2}} = 1 (a > b > 0)$, a line passing through the right focus of the ellipse and perpendicular to the $x$-axis intersects the ellipse at points $P$ and $Q$. The directrix of the ellipse on the right intersects the $x$-axis at point $M$. If $\triangle PQM$ is an equilateral triangle, then the eccentricity of the ellipse equals \_\_\_\_\_\_.

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


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

Given the ellipse C₁: $$\frac {x^{2}}{a^{2}}$$+ $$\frac {y^{2}}{b^{2}}$$\=1 (a＞b＞0) with one focus coinciding with the focus of the parabola C₂: y2\=4 $$\sqrt {2}$$x, and the eccentricity of the ellipse is e= $$\frac { \sqrt {6}}{3}$$.
(I) Find the equation of C₁.
(II) A moving line l passes through the point P(0, 2) and intersects the ellipse C₁ at points A and B. O is the origin. Find the maximum area of △OAB.

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


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

8 singers are participating in a festival and are scheduled to perform \( m \) times, with 4 singers performing in each show. Design a plan to minimize the number of performances \( m \) so that any two singers perform together the same number of times.

Answer: $14$


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

Given a runner who is 30 years old, and the maximum heart rate is found by subtracting the runner's age from 220, determine the adjusted target heart rate by calculating 70% of the maximum heart rate and then applying a 10% increase.

Answer: $146$


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

In trapezoid $PQRS$, the lengths of the bases $PQ$ and $RS$ are 10 and 20, respectively. The height of the trapezoid from $PQ$ to $RS$ is 6 units. The legs of the trapezoid are extended beyond $P$ and $Q$ to meet at point $T$. What is the ratio of the area of triangle $TPQ$ to the area of trapezoid $PQRS$?

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


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

Given that $7^{-1} \equiv 55 \pmod{101}$, find $49^{-1} \pmod{101}$, as a residue modulo 101 (Give an answer between 0 and 100, inclusive).

Answer: $25$


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

Given that $α,β$ satisfy $\frac{\sin α}{\sin (α +2β)}=\frac{2018}{2019}$, find the value of $\frac{\tan (α +β)}{\tan β}$.

Answer: $4037$


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

In triangle $ABC$, we have $\angle C = 90^\circ$, $AB = 13$, and $BC = 5$. What is $\tan A$?

Answer: $\frac{5}{12}$


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>.

You have 2 six-sided dice. One is a normal fair die, while the other has 2 ones, 2 threes, and 2 ﬁves. You pick a die and roll it. Because of some secret magnetic attraction of the unfair die, you have a 75% chance of picking the unfair die and a 25% chance of picking the fair die. If you roll a three, what is the probability that you chose the fair die?

Answer: $1/7$


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 a sequence $\{a_{n}\}$ with the sum of the first $n$ terms denoted as $S_{n}$, $a_{1}=3$, $\frac{{S}_{n+1}}{{S}_{n}}=\frac{{3}^{n+1}-1}{{3}^{n}-1}$, $n\in N^{*}$.
$(1)$ Find $S_{2}$, $S_{3}$, and the general formula for $\{a_{n}\}$;
$(2)$ Let $b_n=\frac{a_{n+1}}{(a_n-1)(a_{n+1}-1)}$, the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is denoted as $T_{n}$. If $T_{n}\leqslant \lambda (a_{n}-1)$ holds for all $n\in N^{*}$, find the minimum value of $\lambda$.

Answer: $\frac{9}{32}$