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

Given real numbers $x$ and $y$ satisfying $x^2+4y^2=4$, find the maximum value of $\frac {xy}{x+2y-2}$.

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


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

Given the function $f(x)= \begin{cases} 2x,& x > 0 \\ f(x+1),& x\leqslant 0 \\ \end{cases}$, find $f(- \frac {4}{3})=$\_\_\_\_\_\_ and the maximum value of the real number $x_{0}$ that satisfies $f(f(x_{0}))=2$.

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


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

The greatest common divisor of two integers is $(x+3)$ and their least common multiple is $x(x+3)$, where $x$ is a positive integer. If one of the integers is 30, what is the smallest possible value of the other one?

Answer: $162$


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

Jane Doe invested some amount of money into a savings account and mutual funds. The total amount she invested was \$320,000. If she invested 6 times as much in mutual funds as she did in the savings account, what was her total investment in mutual funds?

Answer: $274,285.74$


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

On one particular Wednesday, Jack worked \( t-2 \) hours and earned \( 3t-2 \) dollars per hour. His coworker Bob worked 1.5 times more hours than Jack but earned \( 2t-7 \) dollars per hour less than Jack. After paying a fixed tax of $10 each, they both netted the same amount of earnings. Determine the value of \( t \).

Answer: $\frac{19}{3}$


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>.

Define $n!!$ as in the original problem. Evaluate $\sum_{i=1}^{5} \frac{(2i-1)!!}{(2i)!!}$, and express the result as a fraction in lowest terms.

Answer: $\frac{437}{256}$


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

Jessica enjoys vanilla ice cream, and she visits her favorite ice cream shop every day for a week. However, sometimes the shop serves only chocolate ice cream, so each day the shop has a 3/4 chance of serving vanilla ice cream. What is the probability that the shop serves vanilla ice cream exactly 3 out of the 7 days she visits?

Answer: $\frac{945}{16384}$


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

The national security agency's wiretap recorded a conversation between two spies and found that on a 30-minute tape, starting from the 30-second mark, there was a 10-second segment of conversation containing information about the spies' criminal activities. Later, it was discovered that part of this conversation was erased by a staff member. The staff member claimed that he accidentally pressed the wrong button, causing all content from that point onwards to be erased. What is the probability that the conversation containing criminal information was partially or completely erased due to pressing the wrong button?

Answer: $\frac{1}{45}$


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

Marsha now has two numbers, $a$ and $b$. When she divides $a$ by $60$ she gets a remainder of $58$. When she divides $b$ by $90$ she gets a remainder of $84$. What remainder does she get when she divides $a+b$ by $30$?

Answer: $22$


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

Given the function $f(x) = |x^2 + bx|$ ($b \in \mathbb{R}$), when $x \in [0, 1]$, find the minimum value of the maximum value of $f(x)$.

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>.

Two spinners are divided into fifths and sixths, respectively. If each of these spinners is spun once, what is the probability that the product of the results of the two spins will be an even number? The numbers on the first spinner are 3, 5, 6, 7, and 9. The numbers on the second spinner are 2, 4, 6, 8, 9, and 11.

Answer: $\frac{11}{15}$


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>.

For a given list of three numbers, the operation "changesum" replaces each number in the list with the sum of the other two. For example, applying "changesum" to \(3,11,7\) gives \(18,10,14\). Arav starts with the list \(20,2,3\) and applies the operation "changesum" 2023 times. What is the largest difference between two of the three numbers in his final list?
A 17
B 18
C 20
D 2021
E 2023

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>.

Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\heartsuit$ and the second card dealt is a $\clubsuit$?

Answer: $\frac{13}{204}$


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

If the price of a stamp is 50 cents, what is the maximum number of stamps that could be purchased with $50? Furthermore, if a customer buys more than 80 stamps, they receive a discount of 5 cents per stamp. How many stamps would then be purchased at maximum?

Answer: $111$


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

Point P lies on the curve represented by the equation $$\sqrt {(x-5)^{2}+y^{2}}- \sqrt {(x+5)^{2}+y^{2}}=6$$. If the y-coordinate of point P is 4, then its x-coordinate is ______.

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


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>.

What is the smallest possible area, in square units, of a right triangle with two sides measuring $7$ units and $8$ units?

Answer: $\frac{7\sqrt{15}}{2}$


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 all two-digit positive integers whose squares end with the digits 25?

Answer: $495$


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

Given the function $f(x)=x^3+ax^2+bx+16$ has an extremum of $10$ at $x=1$;

(1) Find the values of $a$ and $b$;

(2) Find the maximum and minimum values of $f(x)$ on the interval $[0,2]$.

Answer: $10$


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

Given that $f(x)$ is a function defined on $[1,+\infty)$, and
$$
f(x)=\begin{cases}
1-|2x-3|, & 1\leqslant x < 2, \\
\frac{1}{2}f\left(\frac{1}{2}x\right), & x\geqslant 2,
\end{cases}
$$
then the number of zeros of the function $y=2xf(x)-3$ in the interval $(1,2015)$ is ______.

Answer: $11$


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

Given $a\in\{1,3,5\}$ and $b\in\{2,4,8\}$, find the probability that the function $y=\log_{\frac{b}{a}}{\frac{1}{x}}$ is an increasing function.

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>.

A national team needs to select 4 out of 6 sprinters to participate in the 4×100 m relay at the Asian Games. If sprinter A cannot run the first leg and sprinter B cannot run the fourth leg, there are a total of ______ ways to participate.

Answer: $252$


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

One of the eight faces of a hexagonal prism will be transformed into the base of a new pyramid. Calculate the maximum sum of the number of exterior faces, vertices, and edges of the resultant composite shape (combining the prism and pyramid) when the pyramid is added to each type of face of the prism.

Answer: $50$


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

A positive two-digit number is odd and is a multiple of 9. The product of its digits is a perfect square. What is this two-digit number?

Answer: $99$


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

Consider the set  $M=\{1,2,3,...,2020\}.$  Find the smallest positive integer  $k$  such that for any subset  $A$  of  $M$  with  $k$  elements, there exist  $3$  distinct numbers  $a,b,c$  from  $M$  such that  $a+b, b+c$  and  $c+a$  are all in  $A.$

Answer: $1011$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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. Given that $$bsin(C- \frac {π}{3})-csinB=0$$
(I) Find the value of angle C;
(II) If $a=4$, $c=2 \sqrt {7}$, find the area of $\triangle ABC$.

Answer: $2 \sqrt {3}$


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

Calculate:
$(1)\left(-12\right)-5+\left(-14\right)-\left(-39\right)$;
$(2)-2^{2}\times 5-\left(-12\right)\div 4-4$.

Answer: $-21$


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

A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered.

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>.

A green point and a purple point are chosen at random on the number line between 0 and 1. What is the probability that the purple point is greater than the green point but less than three times the green point?

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>.

Let $P$ be a point not on line $XY$, and $Q$ a point on line $XY$ such that $PQ \perp XY.$ Meanwhile, $R$ is a point on line $PY$ such that $XR \perp PY.$ If $XR = 3$, $PQ = 6$, and $XY = 7$, then what is the length of $PY?$

Answer: $14$


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

The number $m$ is a prime number between 30 and 50. If you divide $m$ by 12, the remainder is 7. What is the value of $m$?

Answer: $43$


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

In $\triangle ABC$, if $\sin B= \sqrt {3}\sin A$, $BC= \sqrt {2}$, and $C= \frac {\pi}{6}$, then the height to side $AC$ is ______.

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


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

Set A has 30 elements, and set B has 25 elements. Set C has 10 elements. Calculate the smallest possible number of elements in the union A ∪ B ∪ C.

Answer: $30$


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

In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive $x$-axis as the polar axis, the parametric equation of curve $C_1$ is
$$
\begin{cases}
x=2+ \sqrt {3}\cos \theta \\
y= \sqrt {3}\sin \theta
\end{cases}
(\theta \text{ is the parameter}),
$$
and the polar equation of curve $C_2$ is $\theta= \frac {\pi}{6} (\rho \in \mathbb{R})$.

$(1)$ Find the general equation of curve $C_1$ and the Cartesian coordinate equation of curve $C_2$;

$(2)$ Curves $C_1$ and $C_2$ intersect at points $A$ and $B$. Given point $P(3, \sqrt {3})$, find the value of $||PA|-|PB||$.

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>.

Larry now only likes numbers that end with two digits making the whole number divisible by 4. He has some favorite numbers like 120, 1156, and 504. How many different pairs of the last two digits are possible in numbers that Larry likes?

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>.

What is the value of \(a + b + c + d\) if

$$
\begin{gathered}
6a + 2b = 3848 \\
6c + 3d = 4410 \\
a + 3b + 2d = 3080
\end{gathered}
$$

Answer: $1986$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 $f(x)=\sin (\omega x+\frac{π}{3})$ ($\omega\ \ \gt 0$) is shifted to the left by $\frac{π}{2}$ units to obtain the curve $C$. If $C$ is symmetric about the $y$-axis, then find the minimum value of $\omega$.

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>.

Let $\{a_n\}$ be an arithmetic sequence. If we select any 4 different numbers from $\{a_1, a_2, a_3, \ldots, a_{10}\}$ such that these 4 numbers still form an arithmetic sequence, then the maximum number of such arithmetic sequences is $\boxed{24}$.

Answer: $24$


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

John has 8 green marbles and 7 purple marbles. He chooses a marble at random, records its color, and then does not put the marble back. He repeats this process 6 times. What is the probability that he chooses exactly three green marbles?

Answer: $\frac{392}{1001}$


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

The sides of rhombus \( EFGH \) are the hypotenuses of the isosceles right triangles \( EAF, FDG, GCH, \) and \( HBE \), and all these triangles have common interior points with the rhombus \( EFGH \). The sum of the areas of quadrilateral \( ABCD \) and rhombus \( EFGH \) is 12. Find \( GH \).

Answer: $2\sqrt{3}$


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

The given arithmetic sequences $\{a_{n}\}$ and $\{b_{n}\}$ have respective sums of the first $n$ terms, denoted by $S_{n}$ and $T_{n}$. The ratio $\frac{S_{n}}{T_{n}} = \frac{3n + 1}{n + 3}$. Determine the value of $\frac{a_{2} + a_{20}}{b_{7} + b_{15}}$.

Answer: $\frac{8}{3}$


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 distinct arrangements of the letters in the word "balloon" are there?

Answer: $1260$


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

Given $\tan 2\alpha= \frac {3}{4}$, $\alpha\in(- \frac {\pi}{2}, \frac {\pi}{2})$, $f(x)=\sin (x+\alpha)+\sin (\alpha-x)-2\sin \alpha$, and for any $x\in\mathbb{R}$, it always holds that $f(x)\geqslant 0$, find the value of $\sin (\alpha- \frac {\pi}{4})$.

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


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

Let $T = (2+i)^{20} - (2-i)^{20}$, where $i = \sqrt{-1}$. Find $|T|$.

Answer: $19531250$


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

Given $\sin\theta + \cos\theta = \frac{3}{4}$, where $\theta$ is an angle of a triangle, find the value of $\sin\theta - \cos\theta$.

Answer: $\frac{\sqrt{23}}{4}$


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

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

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


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

The following is the process of simplifying fractions by Xiaobai. Please read carefully and complete the corresponding tasks.

Solution: $(\frac{3x+4}{x^2-1}-\frac{2}{x-1})÷\frac{x+2}{x^2-2x+1}$

$=[\frac{3x+4}{(x+1)(x-1)}-\frac{2}{x-1}]÷\frac{x+2}{(x-1)^2}\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots $Step 1

$=[\frac{3x+4}{(x+1)(x-1)}-\frac{2(x+1)}{(x+1)(x-1)}]\div \frac{x+2}{(x-1)^{2}}\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots $Step 2

$=\frac{3x+4-2x+2}{(x+1)(x-1)}×\frac{(x-1)^2}{x+2}\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots $Step 3

$=\frac{x+6}{x+1}×\frac{x-1}{x+2}\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots $Step 4

$=\frac{x^2+5x-6}{x^2+3x+2}\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots $Step 5

Tasks:

(1) Fill in the blanks:

① In the simplification steps above, the ______ step is to find a common denominator for the fractions, based on the property of ______.

② Errors start to appear in step ______. The reason for the error in this step is ______.

(2) Please write down the correct simplification process.

(3) When $x=2$, find the value of the fraction.

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 a point P $(x, y)$ on the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$, find the minimum distance from point P to the line $2x + y - 10 = 0$.

Answer: $\sqrt{5}$


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>.

A circle has a radius of 6. What is the area of the smallest square that can entirely contain this circle, and what is the circumference of the circle?

Answer: $12\pi$


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

If $4 \in \{a^2-3a, a\}$, then the value of $a$ equals ____.

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 that $\tan \alpha = 2$, where $\alpha$ is an angle in the first quadrant, find the value of $\sin 2\alpha + \cos \alpha$.

Answer: $\dfrac{4 + \sqrt{5}}{5}$


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

Cara is sitting at a circular table with her seven friends. How many different possible pairs of people could Cara be sitting between?

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>.

If \(a, c, d\) are integers, \(b\) is a positive integer, and they satisfy \(a + b = c\), \(b + c = d\), and \(c + d = a\), then the maximum value of \(a + b + c + d\) is:

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 that three balls are randomly and independently tossed into bins numbered with the positive integers such that for each ball, the probability that it is tossed into bin i is $3^{-i}$ for i = 1,2,3,..., find the probability that all balls end up in consecutive bins.

Answer: $1/702$


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

In 2005, the ages of a brother and sister were 16 and 10 years old, respectively. In which year was the brother's age twice that of the sister's?

Answer: $2001$


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

The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio?

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


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

Given the function $f(x)=-\frac{1}{3}x^{3}+bx^{2}+cx+bc$ has an extreme value of $-\frac{4}{3}$ at $x=1$, find the value of $b$.

Answer: $-1$


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

Convert the binary number $101101110_{(2)}$ to a decimal number and then to an octal number $({\ }\_{(8)})$.

Answer: $556_{(8)}$


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>.

Quantities $a$ and $b$ vary inversely. When $a$ is $800$, $b$ is $0.5$. If the product of $a$ and $b$ increases by $200$ when $a$ is doubled, what is $b$ when $a$ is $1600$?

Answer: $0.375$


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

Given $( \sqrt {x}+ \dfrac {2}{x^{2}})^{n}$, the ratio of the coefficient of the fifth term to the coefficient of the third term in its expansion is $56:3$.
(Ⅰ) Find the constant term in the expansion;
(Ⅱ) When $x=4$, find the term with the maximum binomial coefficient in the expansion.

Answer: $\dfrac {63}{256}$


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

Estimate the number of primes among the first thousand primes divide some term of the sequence
    \[2^0+1,2^1+1,2^2+1,2^3+1,\ldots.\]
    An estimate of  $E$  earns  $2^{1-0.02|A-E|}$  points, where  $A$  is the actual answer.

*2021 CCA Math Bonanza Lightning Round #5.4*

Answer: $10$


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

Given the equation of the parabola $y^{2}=4x$, and a line $l$ passing through its focus $F$ intersecting the parabola at points $A$ and $B$. If $S_{\triangle AOF}=3S_{\triangle BOF}$ (where $O$ is the origin), calculate the length of $|AB|$.

Answer: $\dfrac {16}{3}$


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 $a$ and $b$ are two positive numbers, and the three numbers $a$, $b$, and $-4$ can be rearranged to form both an arithmetic sequence and a geometric sequence, then the value of $a+b$ is __________.

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>.

In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $2c-2a\cos B=b$.
$(1)$ Find the size of angle $A$;
$(2)$ If the area of $\triangle ABC$ is $\frac{\sqrt{3}}{4}$, and $c^{2}+ab\cos C+a^{2}=4$, find $a$.

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


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>.

Calculate the value of the polynomial $f(x) = 3x^6 + 4x^5 + 5x^4 + 6x^3 + 7x^2 + 8x + 1$ at $x=0.4$ using Horner's method, and then determine the value of $v_1$.

Answer: $5.2$


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>.

Given plane vectors $\overrightarrow{a}=(-1,2)$ and $\overrightarrow{b}=(1,-4)$.
$(1)$ If $4\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to $k\overrightarrow{a}-\overrightarrow{b}$, find the value of the real number $k$.
$(2)$ If $\theta$ is the angle between $4\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}+\overrightarrow{b}$, find the value of $\tan \theta$.

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


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

Mark rolls 5 fair 8-sided dice. What is the probability that at least three of the dice show the same number?

Answer: $\frac{1052}{8192}$


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

A school has between 130 and 210 students enrolled. Every afternoon, all the students gather to participate in a singing session. The students are divided into eight distinct groups. If two students are absent from school, the groups can all have the same number of students. What is the sum of all possible numbers of students enrolled at the school?

Answer: $1870$


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

If $\tan(\alpha-\beta) = \frac{1}{2}$ and $\tan(\alpha+\beta) = \frac{1}{3}$, calculate the value of $\tan 2\beta$.

Answer: $- \frac{1}{7}$


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 an ellipse with foci at $(1,0)$ and $(-1,0)$, it intersects with the line $y=x-2$. Determine the maximum value of the eccentricity of this ellipse.

Answer: $\frac{\sqrt{10}}{5}$


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

Let \( A \) and \( B \) be points on the curve \( xy = 1 \) (where \( x > 0 \) and \( y > 0 \)) in the Cartesian coordinate system \( xOy \). Given the vector \( \vec{m} = (1, |OA|) \), find the minimum value of the dot product \( \vec{m} \cdot \overrightarrow{OB} \).

Answer: $2 \sqrt[4]{2}$


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>.

For the power function $y = (m^2 - m - 1)x^{2m+1}$, if it is a decreasing function for $x \in (0, +\infty)$, then the value of the real number $m$ is ______.

Answer: $-1$


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

The positive integers $m$ and $n$ satisfy $8m + 9n = mn + 6$. Find the maximum value of $m$.

Answer: $75$


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

What is the value of \(6102 - 2016\)?

Answer: $4086$


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

During the "Cool Summer Happy Shopping" promotion held in a certain shopping mall, Xiao Yang bought $m$ items of type A goods priced at $5$ yuan each, and $n$ items of type B goods priced at $17 yuan each, spending a total of $203$ yuan. Then the maximum value of $m+n$ is ______.

Answer: $31$


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

Given a decreasing arithmetic sequence $\{a_n\}$, if $a_1 + a_{100} = 0$, find the value of $n$ when the sum of the first $n$ terms, $S_n$, is maximized.

Answer: $50$


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

A quarry wants to sell a large pile of gravel. At full price, the gravel would sell for  $3200$  dollars. But during the first week the quarry only sells  $60\%$  of the gravel at full price. The following week the quarry drops the price by  $10\%$ , and, again, it sells  $60\%$  of the remaining gravel. Each week, thereafter, the quarry reduces the price by another  $10\%$  and sells  $60\%$  of the remaining gravel. This continues until there is only a handful of gravel left. How many dollars does the quarry collect for the sale of all its gravel?

Answer: $3000$


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

How many different four-digit numbers, divisible by 4, can be made from the digits 1, 2, 3, and 4,

a) if each digit can be used only once?

b) if each digit can be used multiple times?

Answer: $64$


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

In $\triangle ABC$, $a=5$, $b=8$, $C=60^{\circ}$, the value of $\overrightarrow{BC}\cdot \overrightarrow{CA}$ is $\_\_\_\_\_\_$.

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>.

The inclination angle of the line $x+ \sqrt {3}y+c=0$ is \_\_\_\_\_\_.

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


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

A store offers an initial discount of 25% on all its merchandise. Subsequently, an additional discount of 10% is applied to the already reduced prices. The store advertises that the final price of the goods is 35% less than the original price. What is the actual percentage difference between the store's claimed discount and the true discount from the original price?

Answer: $2.5\%$


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

The circumference of a circle is 100. The diameter of this circle is equal to:

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


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

Given the function $f(x)= \begin{vmatrix} x & \frac {1}{x} \\ -2 & 1\end{vmatrix}$ ($x>0$), its range is set $A$,
(1) If the universal set $U=\mathbb{R}$, find $C_UA$;
(2) For any $x \in (0, \frac {1}{2}]$, the inequality $f(x)+a \geq 0$ always holds, find the range of the real number $a$;
(3) Let $P$ be any point on the graph of the function $f(x)$, and perpendiculars are drawn from point $P$ to the line $y=x$ and the y-axis, with the feet of the perpendiculars being $A$ and $B$ respectively, find the value of $\overrightarrow {PA} \cdot \overrightarrow {PB}$.

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>.

A line $l$ passing through the focus of the parabola $y=4x^2$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. If $y_1+y_2=2$, then the length of segment $AB$ equals \_\_\_\_\_\_.

Answer: $\frac{17}{8}$


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>.

Molly and Jake play a game where Molly picks an integer between 1 and 720 inclusive, and Jake divides 720 by that integer and states whether the quotient is an integer. How many integers could Molly pick such that Jake’s quotient is an integer?

Answer: $30$


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

Add 74.6893 to 23.152 and round to the nearest hundredth.

Answer: $97.84$


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

In how many ways can the digits of $45,\!502,\!2$ be arranged to form a 6-digit number? (Remember, numbers cannot begin with 0.)

Answer: $150$


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

In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. Given vectors $\overrightarrow{m}=(\sin B+\sin C, \sin A-\sin B)$ and $\overrightarrow{n}=(\sin B-\sin C, \sin(B+C))$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
(1) Find the magnitude of angle $C$;
(2) If $\sin A= \frac{4}{5}$, find the value of $\cos B$.

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


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

If \(a\), \(b\), \(c\), \(d\), \(e\), and \(f\) are integers for which \(729x^3+64 = (ax^2 + bx + c)(dx^2 + ex + f)\) for all \(x\), then what is \(a^2+b^2+c^2+d^2+e^2+f^2\)?

Answer: $8210$


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

Given the polynomial function $f(x) = 2x^5 - 5x^4 - 4x^3 + 3x^2 - 6x + 7$, using the Horner's method when $x = 5$, we can obtain $v_2 = \_$.

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>.

Find the greatest possible value of  $pq + r$ , where p, q, and r are (not necessarily distinct) prime numbers satisfying  $pq + qr + rp = 2016$ .

Answer: $1008$


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

It is known that Xiao Li has a probability of hitting the bullseye of $40\%$ each time he shoots. Now, the method of random simulation is used to estimate the probability that Xiao Li hits the bullseye exactly twice in three shots. First, a calculator is used to generate random integers between $0$ and $9$, with $0$, $1$, $2$, $3$ representing hitting the bullseye, and $4$, $5$, $6$, $7$, $8$, $9$ representing missing the bullseye. Then, every three random numbers are grouped together to represent the results of three shots. After random simulation, the following $20$ groups of random numbers were generated:

$321$　$421$　$191$　$925$　$271$　$932$　$800$　$478$　$589$　$663$

$531$　$297$　$396$　$021$　$546$　$388$　$230$　$113$　$507$　$965$

Based on this, estimate the probability that Xiao Li hits the bullseye exactly twice in three shots.

Answer: $0.30$


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

Given that point $A(-2,α)$ lies on the terminal side of angle $α$ and $\sin α=- \dfrac{ \sqrt{5}}{5}$.

(1) Find the values of $α$, $\cos α$, and $\tan α$.

(2) Find the value of $\dfrac{\cos ( \dfrac{π}{2}+α)\sin (-π-α)}{\cos ( \dfrac{11π}{2}-α)\sin ( \dfrac{9π}{2}+α)}$.

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


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

Determine the sum of all real numbers $x$ that are not in the domain of the function $$g(x) = \frac{1}{2+\frac{1}{3+\frac{1}{x}}}.$$

Answer: $-\frac{13}{21}$


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>.

Let the function $f(x)=\begin{cases} 2^{x}+\int_{0}^{1}{3(\sqrt{x}-x^{2})dx} & (x\geqslant 4) \\ f(x+2) & (x < 4) \end{cases}$. Evaluate the expression $f(\log_{2}3)$.

Answer: $49$


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

Calculate the value of $n$ such that
\[(1 + \tan 1^\circ)(1 + \tan 2^\circ)(1 + \tan 3^\circ) \dotsm (1 + \tan 30^\circ) = 2^n.\]

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>.

Determine the smallest positive integer $n$ such that $n^2$ is divisible by 50 and $n^3$ is divisible by 294.

Answer: $210$


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

The Fibonacci numbers are defined by  $F_1=F_2=1$  and  $F_n=F_{n-1}+F_{n-2}$  for  $n>2$ .  It is well-known that the sum of any  $10$  consecutive Fibonacci numbers is divisible by  $11$ .  Determine the smallest integer  $N$  so that the sum of any  $N$  consecutive Fibonacci numbers is divisible by  $12$ .

Answer: $24$


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

Define a function $f(x)$ on $\mathbb{R}$ that satisfies $f(x+6)=f(x)$. When $x \in [-3,3)$, $f(x)=\begin{cases} -(x+2)^2, & -3 \leqslant x < -1 \\ x, & -1 \leqslant x < 3 \end{cases}$. Find $f(4)$ and the value of $f(1)+f(2)+f(3)+\ldots+f(2016)+f(2017)$.

Answer: $337$


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

A farmer buys 600 cows. He sells 500 of them for the price he paid for all 600 cows. The remaining 100 cows are sold for 10% more per cow than the price of the 500 cows. Calculate the percentage gain on the entire transaction.

Answer: $22\%$


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

A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?

Answer: $\sqrt{55}$


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

Given the curve $y=\sin x$ ($0 \leq x \leq \pi$) and the line $y= \frac {1}{2}$, calculate the area of the enclosed shape formed by these two functions.

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


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

A pawn moves on a $6 \times 6$ chessboard. It starts from the bottom-left square. At each step, it can either jump to the square directly to its right or the square directly above it. It must reach the top-right square, such that it never lies strictly above the diagonal connecting the starting square and the destination square. Determine the number of possible paths.

Answer: $132$


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

Arnaldo claimed that one billion is the same as one million millions. Professor Piraldo corrected him and said, correctly, that one billion is the same as one thousand millions. What is the difference between the correct value of one billion and Arnaldo's assertion?
(a) 1000
(b) 999000
(c) 1000000
(d) 999000000
(e) 999000000000

Answer: $999000000000$


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

Suppose (4-4): Coordinate Systems and Parametric Equations

In the rectangular coordinate system $xOy$, the parametric equations of the curve $C$ are $\begin{cases} x=3\cos \alpha \\ y=\sin \alpha \end{cases}$, where $\alpha$ is the parameter. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of the line $l$ is $\rho \sin (\theta - \frac{\mathrm{\pi }}{4})=\sqrt{2}$.

(I) Find the ordinary equation of $C$ and the slope angle of the line $l$;

(II) Let point $P(0,2)$, and $l$ intersects $C$ at points $A$ and $B$. Find $|PA|+|PB|$.

Answer: $\frac{18\sqrt{2}}{5}$


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>.

The minimum value of the polynomial $x^2 + y^2 - 6x + 8y + 7$ is ______.

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>.

Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 7\}$. What is the probability that the greatest common divisor of these two numbers is one? Express your answer as a common fraction.

Answer: $\frac{17}{21}$


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>.

What is the value of $\sqrt{64 \times \sqrt{49}}$?

Answer: $8\sqrt{7}$


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

An isosceles triangle with a base of $\sqrt{2}$ has medians intersecting at a right angle. Calculate the area of this triangle.

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


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

In the 2016 art exam of a certain high school, there were 6 contestants, including 3 females and 3 males. Now, these six contestants are to perform their talents in sequence. If any two of the three males cannot perform consecutively, and the female contestant A cannot be the first to perform, then calculate the number of possible sequences for the contestants to perform.

Answer: $132$


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

Griffin and Hailey run for $45$ minutes on a circular track. Griffin runs counterclockwise at $260 m/min$ and uses the outer lane with a radius of $50$ meters. Hailey runs clockwise at $310 m/min$ and uses the inner lane with a radius of $45$ meters, starting on the same radial line as Griffin. Determine how many times do they pass each other.

Answer: $86$


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

Given that point $P$ is a moving point on the parabola $y^{2}=4x$, the minimum value of the sum of the distance from point $P$ to line $l$: $2x-y+3=0$ and the $y$-axis is ___.

Answer: $\sqrt{5}-1$


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 that point $P$ is a moving point on circle $C$: $x^{2}+y^{2}-2x-4y+1=0$, the maximum distance from point $P$ to a certain line $l$ is $6$. If a point $A$ is taken arbitrarily on line $l$ to form a tangent line $AB$ to circle $C$, with $B$ being the point of tangency, then the minimum value of $AB$ is _______.

Answer: $2\sqrt{3}$


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

Given that the function $f\left(x\right)$ is an even function on $R$, and $f\left(x+2\right)$ is an odd function. If $f\left(0\right)=1$, then $f\left(1\right)+f\left(2\right)+\ldots +f\left(2023\right)=\_\_\_\_\_\_$.

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>.

Samantha has 10 green marbles and 5 purple marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 7 marbles. What is the probability that exactly four of the marbles that she removes are green? Express your answer as a decimal.

Answer: $0.256$


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

Find the sum of the first eight prime numbers that have a units digit of 3.

Answer: $404$


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

Let $T$ be the sum of all the real coefficients of the expansion of $(1 + ix)^{2018}$. What is $\log_2(T)$?

Answer: $1009$


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

Evaluate the expression $-20 + 15 \times (4^{\div -1} \times 2)$.

Answer: $-12.5$


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

The length of the longer side of rectangle \(R'\) is \(15\%\) more than the length of a side of square \(S'\). The length of the shorter side of rectangle \(R'\) is \(5\%\) less than the length of a side of square \(S'\). What is the ratio of the area of rectangle \(R'\) to the area of square \(S'\)? Express your answer as a common fraction.

Answer: $\frac{10925}{10000}$


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

The graph of $y = ax^2 + bx + c$ has a maximum value of 75, and passes through the points $(-3,0)$ and $(3,0)$. Find the value of $a + b + c$ at $x = 2$.

Answer: $\frac{125}{3}$


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 \( x_{i}=\frac{i}{101} \), then the value of \( S=\sum_{i=0}^{101} \frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1} \) is

Answer: $51$


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

A dodecahedron consists of two pentagonal-based pyramids glued together along their pentagonal bases, forming a polyhedron with 12 faces. Consider an ant at the top vertex of one of the pyramids, selecting randomly one of the five adjacent vertices, designated as vertex A. From vertex A, the ant then randomly selects one of the five adjacent vertices, called vertex B. What is the probability that vertex B is the bottom vertex opposite the top one?

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>.

If \(\sum_{n = 0}^{\infty} \cos^{2n} \theta = \frac{9}{2}\), what is the value of \(\cos{4\theta}\)?

Answer: $-\frac{31}{81}$


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

Given non-zero plane vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}-\overrightarrow{c}|=1$. If the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{π}{3}$, calculate the minimum value of $|\overrightarrow{a}-\overrightarrow{c}|$.

Answer: $\sqrt{3} - 1$


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

In the rectangular coordinate system $xOy$, the parametric equations of line $l$ are given by $\begin{cases} x=2t \\ y=-2-t \end{cases}$ (where $t$ is the parameter). In the polar coordinate system (using the same length unit as the rectangular coordinate system and with the origin $O$ as the pole and the polar axis coinciding with the non-negative half of the $x$-axis), the equation of circle $C$ is given by $ρ=4\sqrt{2}\cos(θ+\frac{π}{4})$. Find the length of the chord cut off by line $l$ on circle $C$.

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


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

Given the function $f(x)=\sqrt{3}\sin x\cos x+\cos^{2}x$.

(1) Find the value of $f(\frac{\pi }{24})$;

(2) If the function $f(x)$ is monotonically increasing in the interval $[-m,m]$, find the maximum value of the real number $m$.

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


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

If $A = 3009 \div 3$, $B = A \div 3$, and $Y = A - 2B$, then what is the value of $Y$?

Answer: $335$


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

What is the value of $\frac{3}{5} + \frac{2}{3} + 1\frac{1}{15}$?

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