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

Given $f(x)$ be a differentiable function, and $\lim_{\Delta x \to 0} \frac{{f(1)-f(1-2\Delta x)}}{{\Delta x}}=-1$, determine the slope of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$.

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>.

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>.

The rodent control task force went into the woods one day and caught  $200$  rabbits and  $18$  squirrels. The next day they went into the woods and caught  $3$  fewer rabbits and two more squirrels than the day before. Each day they went into the woods and caught  $3$  fewer rabbits and two more squirrels than the day before. This continued through the day when they caught more squirrels than rabbits. Up through that day how many rabbits did they catch in all?

Answer: $5491$


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

A, B, and C are three people passing a ball to each other. The first pass is made by A, who has an equal chance of passing the ball to either of the other two people. After three passes, the probability that the ball is still with A is _______.

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


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

Given that the function $f(x)$ satisfies: $4f(x)f(y)=f(x+y)+f(x-y)$ $(x,y∈R)$ and $f(1)= \frac{1}{4}$, find $f(2014)$.

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


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

Let \( M = 35 \cdot 36 \cdot 65 \cdot 280 \). Calculate the ratio of the sum of the odd divisors of \( M \) to the sum of the even divisors of \( M \).

Answer: $1:62$


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

Let the function $f(x)=(x+a)\ln x$, $g(x)= \frac {x^{2}}{e^{x}}$, it is known that the tangent line of the curve $y=f(x)$ at the point $(1,f(1))$ is parallel to the line $2x-y=0$.
(Ⅰ) If the equation $f(x)=g(x)$ has a unique root within $(k,k+1)$ $(k\in\mathbb{N})$, find the value of $k$.
(Ⅱ) Let the function $m(x)=\min\{f(x),g(x)\}$ (where $\min\{p,q\}$ represents the smaller value between $p$ and $q$), find the maximum value of $m(x)$.

Answer: $\frac {4}{e^{2}}$


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

Given that $a>0$, the minimum value of the function $f(x) = e^{x-a} - \ln(x+a) - 1$ $(x>0)$ is 0. Determine the range of values for the real number $a$.

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


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

Calculate

(1) Use a simplified method to calculate $2017^{2}-2016 \times 2018$;

(2) Given $a+b=7$ and $ab=-1$, find the values of $(a+b)^{2}$ and $a^{2}-3ab+b^{2}$.

Answer: $54$


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

In triangle $\triangle ABC$, it is known that the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, $c$, with $a=\sqrt{6}$, $\sin ^{2}B+\sin ^{2}C=\sin ^{2}A+\frac{2\sqrt{3}}{3}\sin A\sin B\sin C$. Choose one of the following conditions to determine whether triangle $\triangle ABC$ exists. If it exists, find the area of $\triangle ABC$; if it does not exist, please explain the reason.
① The length of the median $AD$ of side $BC$ is $\frac{\sqrt{10}}{2}$
② $b+c=2\sqrt{3}$
③ $\cos B=-\frac{3}{5}$.

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


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

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>.

The minimum value of \\(f(x)=\sin x+\cos x-\sin x\cos x\\) is

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


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

A plane intersects a right circular cylinder of radius $3$ forming an ellipse. If the major axis of the ellipse is $75\%$ longer than the minor axis, the length of the major axis is:

Answer: $10.5$


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>.

Among 7 students, 6 are to be arranged to participate in social practice activities in two communities on Saturday, with 3 people in each community. How many different arrangements are there? (Answer with a number)

Answer: $140$


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

Given the actual lighthouse's cylindrical base is 60 meters high, and the spherical top's volume is approximately 150,000 liters, and the miniature model's top holds around 0.15 liters, determine the height of Lara’s model lighthouse, in centimeters.

Answer: $60$


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

How many numbers are in the list $-58, -51, -44, \ldots, 71, 78$?

Answer: $20$


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

Given the sequence $\{a_n\}$, its first term is $7$, and $a_n= \frac{1}{2}a_{n-1}+3(n\geqslant 2)$, find the value of $a_6$.

Answer: $\frac{193}{32}$


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

Regarding the value of \\(\pi\\), the history of mathematics has seen many creative methods for its estimation, such as the famous Buffon's Needle experiment and the Charles' experiment. Inspired by these, we can also estimate the value of \\(\pi\\) through designing the following experiment: ask \\(200\\) students, each to randomly write down a pair of positive real numbers \\((x,y)\\) both less than \\(1\\); then count the number of pairs \\((x,y)\\) that can form an obtuse triangle with \\(1\\) as the third side, denoted as \\(m\\); finally, estimate the value of \\(\pi\\) based on the count \\(m\\). If the result is \\(m=56\\), then \\(\pi\\) can be estimated as \_\_\_\_\_\_ (expressed as a fraction).

Answer: $\dfrac {78}{25}$


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>.

Determine the smallest integer $k$ such that $k>1$ and $k$ has a remainder of $3$ when divided by any of $11,$ $4,$ and $3.$

Answer: $135$


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

A copper cube with an edge length of $l = 5 \text{ cm}$ is heated to a temperature of $t_{1} = 100^{\circ} \text{C}$. Then, it is placed on ice, which has a temperature of $t_{2} = 0^{\circ} \text{C}$. Determine the maximum depth the cube can sink into the ice. The specific heat capacity of copper is $c_{\text{s}} = 400 \text{ J/(kg}\cdot { }^{\circ} \text{C})$, the latent heat of fusion of ice is $\lambda = 3.3 \times 10^{5} \text{ J/kg}$, the density of copper is $\rho_{m} = 8900 \text{ kg/m}^3$, and the density of ice is $\rho_{n} = 900 \text{ kg/m}^3$. (10 points)

Answer: $0.06$


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

Pentagon  $ANDD'Y$  has  $AN \parallel DY$  and  $AY \parallel D'N$  with  $AN = D'Y$  and  $AY = DN$ . If the area of  $ANDY$  is 20, the area of  $AND'Y$  is 24, and the area of  $ADD'$  is 26, the area of  $ANDD'Y$  can be expressed in the form  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Find  $m+n$ .

*Proposed by Andy Xu*

Answer: $71$


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

Two cylindrical poles, with diameters of $10$ inches and $30$ inches respectively, are placed side by side and bound together with a wire. Calculate the length of the shortest wire that will go around both poles.
**A)** $20\sqrt{3} + 24\pi$
**B)** $20\sqrt{3} + \frac{70\pi}{3}$
**C)** $30\sqrt{3} + 22\pi$
**D)** $16\sqrt{3} + 25\pi$
**E)** $18\sqrt{3} + \frac{60\pi}{3}$

Answer: $20\sqrt{3} + \frac{70\pi}{3}$


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>.

Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\diamondsuit$?

Answer: $\frac{52}{221}$


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

In triangle $\triangle ABC$, $\angle BAC = \frac{π}{3}$, $D$ is the midpoint of $AB$, $P$ is a point on segment $CD$, and satisfies $\overrightarrow{AP} = t\overrightarrow{AC} + \frac{1}{3}\overrightarrow{AB}$. If $|\overrightarrow{BC}| = \sqrt{6}$, then the maximum value of $|\overrightarrow{AP}|$ is ______.

Answer: $\sqrt{2}$


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

Let $F_{1}$ and $F_{2}$ be the two foci of the hyperbola $\dfrac {x^{2}}{4}- \dfrac {y^{2}}{b^{2}}=1$. Point $P$ is on the hyperbola and satisfies $\angle F_{1}PF_{2}=90^{\circ}$. If the area of $\triangle F_{1}PF_{2}$ is $2$, find the value of $b$.

Answer: $\sqrt {2}$


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

Given real numbers $a$, $b$, $c$, and $d$ satisfy $(b + 2a^2 - 6\ln a)^2 + |2c - d + 6| = 0$, find the minimum value of $(a - c)^2 + (b - d)^2$.

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>.

Without using a calculator, find the largest prime factor of \( 17^4 + 2 \times 17^2 + 1 - 16^4 \).

Answer: $17$


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

In the list where each integer $n$ appears $n$ times for $1 \leq n \leq 300$, find the median of the numbers.

Answer: $212$


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

$x_{n+1}= \left ( 1+\frac2n \right )x_n+\frac4n$ , for every positive integer  $n$ . If  $x_1=-1$ , what is  $x_{2000}$ ?

Answer: $2000998$


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

In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $b=2$, $c=2\sqrt{2}$, and $C=\frac{\pi}{4}$, find the area of $\Delta ABC$.

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>.

Given 50 feet of fencing, where 5 feet is used for a gate that does not contribute to the enclosure area, what is the greatest possible number of square feet in the area of a rectangular pen enclosed by the remaining fencing?

Answer: $126.5625$


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

A conference hall is setting up seating for a workshop. Each row must contain $13$ chairs, and initially, there are $169$ chairs in total. The organizers expect $95$ participants to attend the workshop. To ensure all rows are completely filled with minimal empty seats, how many chairs should be removed?

Answer: $65$


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

Suppose that  $N$  is a three digit number divisible by  $7$  such that upon removing its middle digit, the remaining two digit number is also divisible by  $7$ . What is the minimum possible value of  $N$ ?

*2019 CCA Math Bonanza Lightning Round #3.1*

Answer: $154$


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

For the inequality system about $y$ $\left\{\begin{array}{l}{2y-6≤3(y-1)}\\{\frac{1}{2}a-3y＞0}\end{array}\right.$, if it has exactly $4$ integer solutions, then the product of all integer values of $a$ that satisfy the conditions is ______.

Answer: $720$


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

(1) Given $\sin \left( \frac{\pi }{3}-\alpha \right)=\frac{1}{2}$, find the value of $\cos \left( \frac{\pi }{6}+\alpha \right)$;

(2) Given $\cos \left( \frac{5\pi }{12}+\alpha \right)=\frac{1}{3}$ and $-\pi < \alpha < -\frac{\pi }{2}$, find the value of $\cos \left( \frac{7\pi }{12}-\alpha \right)+\sin \left( \alpha -\frac{7\pi }{12} \right)$:

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


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

Given that point $(a, b)$ moves on the line $x + 2y + 3 = 0$, find the maximum or minimum value of $2^a + 4^b$.

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>.

How many different positive three-digit integers can be formed using only the digits in the set $\{1, 3, 4, 4, 7, 7, 7\}$ if no digit may be used more times than it appears in the given set of available digits?

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>.

Distribute 5 students into two dormitories, A and B, with each dormitory accommodating at least 2 students. Find the number of distinct arrangements.

Answer: $20$


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

Given that $m$ is an integer and $0<3m<27$, what is the sum of all possible integer values of $m$?

Answer: $36$


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

In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases} & x=3\cos \alpha \\ & y=\sin \alpha \end{cases}$ ($\alpha$ is the parameter), and in the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of line $l$ is $\rho \sin (\theta -\dfrac{\pi }{4})=\sqrt{2}$.

$(1)$ Find the general equation of curve $C$ and the inclination angle of line $l$;

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

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


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>.

Given plane vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ satisfy $\langle \overrightarrow{a}, \overrightarrow{b} \rangle = 60^\circ$, and $\{|\overrightarrow{a}|, |\overrightarrow{b}|, |\overrightarrow{c}|\} = \{1, 2, 3\}$, calculate the maximum value of $|\overrightarrow{a}+ \overrightarrow{b}+ \overrightarrow{c}|$.

Answer: $\sqrt{7}+3$


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 symbol \( R_{k} \) represents an integer whose decimal representation consists of \( k \) consecutive 1s. For example, \( R_{3} = 111 \), \( R_{5} = 11111 \), and so on. If \( R_{4} \) divides \( R_{24} \), the quotient \( Q = \frac{R_{24}}{R_{4}} \) is an integer, and its decimal representation contains only the digits 1 and 0. How many zeros are in \( Q \)?

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>.

Find the value of the algebraic cofactor of element $a$ in the determinant \\( \begin{vmatrix} 3 & a & 5 \\\\ 0 & -4 & 1 \\\\ -2 & 1 & 3\\end{vmatrix} \\).

Answer: $-2$


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

Given the quadratic function $f(x)=ax^{2}+(2b+1)x-a-2 (a,b \in R, a \neq 0)$ has at least one root in the interval $[3,4]$, calculate the minimum value of $a^{2}+b^{2}$.

Answer: $\frac{1}{100}$


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

The new PUMaC tournament hosts  $2020$  students, numbered by the following set of labels  $1, 2, . . . , 2020$ . The students are initially divided up into  $20$  groups of  $101$ , with each division into groups equally likely. In each of the groups, the contestant with the lowest label wins, and the winners advance to the second round. Out of these  $20$  students, we chose the champion uniformly at random. If the expected value of champion’s number can be written as  $\frac{a}{b}$ , where  $a, b$  are relatively prime integers, determine  $a + b$ .

Answer: $2123$


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

In $\triangle ABC$, $2\sin 2A\cos A-\sin 3A+\sqrt{3}\cos A=\sqrt{3}$.
(1) Find the measure of angle $A$;
(2) Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, if $a=1$ and $\sin A+\sin (B-C)=2\sin 2C$, find the area of $\triangle ABC$.

Answer: $\frac{\sqrt{3}}{6}$


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

Given that $α∈(0,π)$, and $\sin α + \cos α = \frac{\sqrt{2}}{2}$, find the value of $\sin α - \cos α$.

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


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

There is a solid iron cone with a base radius of $3cm$ and a slant height of $5cm$. After melting it at high temperature and casting it into a solid iron sphere (without considering any loss), the radius of this iron sphere is _______ $cm$.

Answer: $\sqrt[3]{9}$


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

If a worker receives a 30% cut in wages, calculate the percentage raise he needs to regain his original pay.

Answer: $42.857\%$


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

Calculate and simplify $\sqrt[4]{3^5 \cdot 5^3}$ to the form $c\sqrt[4]{d}$, where $c$ and $d$ are positive integers. What is $c+d$?

Answer: $378$


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

Given the function $f(x)=\frac{1}{3}x^{3}+ax^{2}+bx$ $(a,b\in \mathbb{R})$ attains a maximum value of $9$ at $x=-3$.

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

$(II)$ Find the maximum and minimum values of the function $f(x)$ in the interval $[-3,3]$.

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


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

A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known that at \( x = 2020 \), all the resulting irreducible polynomials take the value -3. Find \( P(2020) \).

Answer: $243$


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

Calculate using your preferred method:
(1) $42.67-(12.95-7.33)$
(2) $\left[8.4-8.4\times(3.12-3.7)\right]\div0.42$
(3) $5.13\times0.23+8.7\times0.513-5.13$
(4) $6.66\times222+3.33\times556$

Answer: $3330$


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

How many ways are there to put 7 balls in 2 boxes if the balls are distinguishable but the boxes are not?

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>.

A triangle is divided into 1000 triangles. What is the maximum number of distinct points that can be vertices of these triangles?

Answer: $1002$


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

Deﬁne the determinant  $D_1$  =  $|1|$ , the determinant  $D_2$  =      $|1 1|$   $|1 3|$
, and the determinant  $D_3=$
|1 1 1|
|1 3 3|
|1 3 5|
.
In general, for positive integer n, let the determinant  $D_n$  have 1s in every position of its ﬁrst row and ﬁrst
column, 3s in the remaining positions of the second row and second column, 5s in the remaining positions of the third row and third column, and so forth. Find the least n so that  $D_n$   $\geq$  2015.

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>.

Calculate: $|1-\sqrt{2}|+(\frac{1}{2})^{-2}-\left(\pi -2023\right)^{0}$.

Answer: $\sqrt{2} + 2$


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 $f(x)=x^{2}-2ax+{b}^{2}$, where $a$ and $b$ are real numbers.
(1) If $a$ is taken from the set $\{0,1,2,3\}$ and $b$ is taken from the set $\{0,1,2\}$, find the probability that the equation $f(x)=0$ has two distinct real roots.
(2) If $a$ is taken from the interval $[0,2]$ and $b$ is taken from the interval $[0,3]$, find the probability that the equation $f(x)=0$ has no real roots.

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


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

In the Cartesian coordinate system $xOy$, a moving line $l$: $y=x+m$ intersects the parabola $C$: $x^2=2py$ ($p>0$) at points $A$ and $B$, and $\overrightarrow {OA}\cdot \overrightarrow {OB}=m^{2}-2m$.

1. Find the equation of the parabola $C$.
2. Let $P$ be the point where the line $y=x$ intersects $C$ (and $P$ is different from the origin), and let $D$ be the intersection of the tangent line to $C$ at $P$ and the line $l$. Define $t= \frac {|PD|^{2}}{|DA|\cdot |DB|}$. Is $t$ a constant value? If so, compute its value; otherwise, explain why it's not constant.

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


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

Nine lines parallel to the base of a triangle divide the other sides each into  $10$  equal segments and the area into  $10$  distinct parts. If the area of the largest of these parts is  $38$, calculate the area of the original triangle.

Answer: $200$


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

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

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


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

How many ways are there to put 7 balls in 2 boxes if the balls are distinguishable but the boxes are not?

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>.

The diagram shows a large triangle divided into squares and triangles. Let \( S \) be the number of squares of any size in the diagram and \( T \) be the number of triangles of any size in the diagram. What is the value of \( S \times T \)?

A) 30

B) 35

C) 48

D) 70

E) 100

Answer: $70$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 $(xOy)$, there is a circle $C_{1}$: $(x+ \sqrt {3})^{2}+y^{2}=4$, and a curve $C_{2}$ with parametric equations $\begin{cases}x=2+2\cos θ \\\\ y=2\sin θ\end{cases} (θ \text{ is the parameter})$. Establish a polar coordinate system with $O$ as the pole and the positive $x$-axis as the polar axis.

1. Write the polar equation of $C_{1}$ and convert $C_{2}$ to a Cartesian equation.
2. If the polar equation of the line $C_{3}$ is $θ= \dfrac {π}{3}(ρ∈R)$, and $C_{2}$ intersects with $C_{3}$ at points $A$ and $B$, find the area of $△ABC_{1}$ ($C_{1}$ is the center of circle $C_{1}$).

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


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>.

For any positive integer $n$, the value of $n!$ is the product of the first $n$ positive integers. Calculate the greatest common divisor of $8!$ and $10!$.

Answer: $40320$


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

The coefficient of $x^3$ in the expansion of $(2x-1)\left( \frac {1}{x}+x\right)^{6}$ is \_\_\_\_\_\_.

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>.

Find the 6-digit repetend in the decimal representation of $\frac{7}{29}$.

Answer: $241379$


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

There are  $2017$  distinct points in the plane. For each pair of these points, construct the midpoint of the segment joining the pair of points. What is the minimum number of distinct midpoints among all possible ways of placing the points?

Answer: $2016$


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

Let \( g(x) = x^2 - 4x \). How many distinct real numbers \( c \) satisfy \( g(g(g(g(c)))) = 2 \)?

Answer: $16$


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

Given a finite arithmetic sequence \( a_{1}, a_{2}, \cdots a_{k} \), and the conditions:
$$
\begin{array}{c}
a_{4}+a_{7}+a_{10}=17, \\
a_{4}+a_{5}+a_{6}+a_{7}+a_{8}+a_{9}+a_{10}+a_{11}+a_{12}+a_{13}+a_{14}=77 .
\end{array}
$$
If \( a_{k}=13 \), then \( k \) 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>.

In triangle \( \triangle ABC \), point \( E \) is on side \( AB \) with \( AE = 1 \) and \( EB = 2 \). Suppose points \( D \) and \( F \) are on sides \( AC \) and \( BC \) respectively, and \( DE \parallel BC \) and \( EF \parallel AC \). What is the ratio of the area of quadrilateral \( CDEF \) to the area of triangle \( \triangle ABC \)?

Answer: $4: 9$


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

Given the presence of hereditary traits determined by paired genes $A$ and $a$ in peas, where $A$ is dominant and $a$ is recessive, and the offspring $Aa$ is yellow, determine the probability that the offspring will produce yellow peas after self-crossing.

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>.

Given an ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with a vertex coordinate at $B(0,1)$, and the eccentricity of the ellipse being $\frac{\sqrt{3}}{2}$,
$(I)$ find the equation of the ellipse;
$(II)$ let point $Q$ be a point on ellipse $C$ below the x-axis, and $F_1$, $F_2$ are the left and right foci of the ellipse, respectively. If the slope of line $QF_1$ is $\frac{\pi}{6}$, find the area of $\triangle QF_1F_2$.

Answer: $\frac{\sqrt{3}}{7}$


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

Luke wants to fence a rectangular piece of land with an area of at least 450 square feet. The length of the land is 1.5 times the width. What should the width of the rectangle be if he wants to use the least amount of fencing?

Answer: $10\sqrt{3}$


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

Let $x, y, a \in \mathbb{R}^*$, and when $x + 2y = 1$, the minimum value of $\frac{3}{x} + \frac{a}{y}$ is $6\sqrt{3}$. Determine the minimum value of $3x + ay$ when $\frac{1}{x} + \frac{2}{y} = 1$.

Answer: $6\sqrt{3}$


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>.

Xiaoming forgot how to write $egg$ and only remembered there were three letters $e$, $g$, $g$. He randomly wrote one of them. Calculate the probability that he wrote it correctly.

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>.

The number of triangles with all sides being positive integers and the longest side being 11 can be expressed as a combination problem, calculate the number of such triangles.

Answer: $36$


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

In parallelogram ABCD, AB=2AD=4, ∠BAD=60°, E is the midpoint of BC, calculate the dot product of vectors BD and AE.

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>.

During the National Day military parade, the marching sequences of three formations, namely A, B, and C, pass by the viewing stand in a certain order. If the order is randomly arranged, the probability that B passes before both A and C is ( ).

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


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

In a class of 35 students, everyone participates in physics and chemistry. Nine students received an A in chemistry and 15 received an A in physics, including five that received an A in both subjects. How many students did not receive an A in either of these two subjects?

Answer: $16$


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

If $P = \sqrt{1988 \cdot 1989 \cdot 1990 \cdot 1991 + 1} + \left(-1989^{2}\right)$, calculate the value of $P$.

Answer: $1988$


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

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

Answer: $96$


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

Given the function $$f(x)=\cos 2x + 2\sqrt{3}\sin x\cos x$$
(1) Find the range of the function $f(x)$ and write down the intervals where $f(x)$ is monotonically increasing;
(2) If $$0 < \theta < \frac{\pi}{6}$$ and $$f(\theta) = \frac{4}{3}$$, calculate the value of $\cos 2\theta$.

Answer: $\frac{\sqrt{15} + 2}{6}$


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

There are five positive integers that are divisors of each number in the list $$60, 120, -30, 180, 240$$. Find the sum of these five positive integers.

Answer: $17$


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

At a robot racing competition, a certain number of mechanisms were presented. The robots were paired to race the same distance. The protocol recorded the differences in the finishing times between the winner and the loser in each of the races. All the differences were distinct: 1 sec, 2 sec, 3 sec, 4 sec, 5 sec, 6 sec, 7 sec, 8 sec, 9 sec, 13 sec. It is known that during the races, each robot competed against each other robot exactly once, and each robot always ran at the same speed. Determine the time of the slowest mechanism if the best time to complete the distance was 50 seconds.

Answer: $63$


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

Given vectors $\overrightarrow {a} = (1, -3)$ and $\overrightarrow {b} = (-2, m)$, and $\overrightarrow {a}$ is perpendicular to ($\overrightarrow {a} - \overrightarrow {b}$).
(1) Find the value of the real number $m$;
(2) When $k\overrightarrow {a} + \overrightarrow {b}$ is parallel to $\overrightarrow {a} - \overrightarrow {b}$, find the value of the real number $k$.

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>.

Let $T$ denote the sum of all four-digit positive integers where each digit is distinct and ranges from 1 to 9. Compute the remainder when $T$ is divided by $1000$.

Answer: $320$


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

The equation of curve $C$ is $\frac{x^2}{m^2} + \frac{y^2}{n^2} = 1$, where $m$ and $n$ are the numbers obtained by rolling a die twice in succession. Let event $A$ be "The equation $\frac{x^2}{m^2} + \frac{y^2}{n^2} = 1$ represents an ellipse with foci on the $x$-axis". Then, the probability of event $A$ occurring, $P(A)=$ .

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


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

Rodney is trying to guess a secret number based on the following clues:
1. It is a two-digit integer.
2. The tens digit is odd.
3. The units digit is one of the following: 2, 4, 6, 8.
4. The number is greater than 50.

What is the probability that Rodney will guess the correct number? Express your answer as a common fraction.

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


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

For how many integer values of $b$ does the equation $$x^2 + bx + 12b = 0$$ have integer solutions for $x$?

Answer: $16$


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

Given that the sequence $\{a_{n}\}$ is an arithmetic sequence, $a_{9}+a_{12} \lt 0$, $a_{10}\cdot a_{11} \lt 0$, and the sum of the first $n$ terms of the sequence $\{a_{n}\}$, denoted as $S_{n}$, has a maximum value, determine the maximum value of $n$ when $S_{n} \gt 0$.

Answer: $19$


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

A unit purchased a vacant lot for 21.6 million yuan, planning to build a building of at least 10 floors, with each floor being 2000 square meters. It is calculated that if the building is built to $x$ ($x \geqslant 10$) floors, then the average construction cost per square meter is $560+48x$ (unit: yuan). To minimize the average comprehensive cost per square meter of the building, how many floors should the building have?

(Note: The average comprehensive cost = average construction cost + average land purchase cost, where the average land purchase cost = $\dfrac{\text{total land purchase cost}}{\text{total construction area}}$)

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>.

The sum of the first n terms of a geometric sequence $\{a_n\}$ is $S_n$. If for any positive integer $k$, it holds that $a_k = \lim_{n\to\infty}(S_n - S_k)$, then the common ratio $q = \boxed{\frac{1}{2}}$.

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 coordinates of the 3 vertices of triangle are \( P(-8, 5) \), \( Q(-15, -19) \), and \( R(1, -7) \). The equation of the angle bisector of \(\angle P\) can be written as \(a x + b y + c = 0\), where \(a, b, c \in \mathbf{Z}^{+}\) and \((a, b, c)=1\). What is the value of \(a + c\)?

Answer: $89$


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

For a geometric sequence $\{b_n\}$ with a common ratio of $4$, if $T_n$ is the product of the first $n$ terms of the sequence $\{b_n\}$, then $\frac{T_{20}}{T_{10}}$, $\frac{T_{30}}{T_{20}}$, $\frac{T_{40}}{T_{30}}$ also form a geometric sequence with a common ratio of $4^{100}$. Similarly, in an arithmetic sequence $\{a_n\}$ with a common difference of $3$, if $S_n$ is the sum of the first $n$ terms of $\{a_n\}$, then \_\_\_\_\_\_ also form an arithmetic sequence with a common difference of \_\_\_\_\_\_.

Answer: $300$


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

Two distinct numbers a and b are chosen randomly from the set $\{3, 3^2, 3^3, ..., 3^{15}\}$. What is the probability that $\mathrm{log}_a b$ is an integer?
A) $\frac{1}{10}$
B) $\frac{2}{7}$
C) $\frac{1}{7}$
D) $\frac{1}{5}$

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


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

In a box, there are 10 balls of the same size, among which 3 are labeled with 1, 4 are labeled with 2, and 3 are labeled with 5. First, a ball is randomly drawn from the box and then put back. After that, another ball is randomly drawn (assuming the probability of drawing each ball is the same). Let the sum of the labels of the balls drawn in the first and second draws be $\epsilon$. Find the distribution and the expected value $E\epsilon$ of the random variable $\epsilon$.

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>.

$(1)$ Solve the system of inequalities: $\left\{\begin{array}{l}{2x-1＜5,①}\\{\frac{4-x}{2}≥1,②}\end{array}\right.$<br/>$(2)$ Simplify first, then find the value: $\frac{1}{a-1}÷(\frac{{a}^{2}+a}{{a}^{2}-1}+\frac{1}{a-1})$, where $a=(-2023)^0+(\frac{1}{2})^{-1}⋅$

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


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

A cuboid has an integer volume. Three of the faces have different areas, namely  $7, 27$ , and  $L$ . What is the smallest possible integer value for  $L$ ?

Answer: $21$


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

Given $\cos \alpha= \frac {1}{7}$, $\cos (\alpha+\beta)=- \frac {11}{14}$, and $\alpha\in(0, \frac {\pi}{2})$, $\alpha+\beta\in( \frac {\pi}{2},\pi)$, find the value of $\cos \beta$.

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>.

While entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards.  Let  $a$  be the probability that the cards are of different ranks.  Compute  $\lfloor 1000a\rfloor$ .

Answer: $941$


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

Given two parallel lines \\(l_{1}\\) and \\(l_{2}\\) passing through points \\(P_{1}(1,0)\\) and \\(P_{2}(0,5)\\) respectively, and the distance between \\(l_{1}\\) and \\(l_{2}\\) is \\(5\\), then the slope of line \\(l_{1}\\) is \_\_\_\_\_\_.

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


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

In a directed graph with  $2013$  vertices, there is exactly one edge between any two vertices and for every vertex there exists an edge outwards this vertex. We know that whatever the arrangement of the edges, from every vertex we can reach  $k$  vertices using at most two edges. Find the maximum value of  $k$ .

Answer: $2012$


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

Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $| \overrightarrow {a} |=5$, $| \overrightarrow {b} |=3$, and $( \overrightarrow {a} - \overrightarrow {b} )(2 \overrightarrow {a} + 3 \overrightarrow {b} )=13$.

1. Find the cosine value of the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$.
2. Find the magnitude of $| \overrightarrow {a} + 2 \overrightarrow {b} |$.

Answer: $\sqrt{21}$


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

A right pyramid has a square base where each side measures 15 cm. The height of the pyramid, measured from the center of the base to the peak, is 15 cm. Calculate the total length of all edges of the pyramid.

Answer: $60 + 4\sqrt{337.5}$


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>.

When the square root of \( x \) is cubed, the result is 100. What is the value of \( x \)?

Answer: $10^{\frac{4}{3}}$


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

A river boat travels at a constant speed from point A to point B. Along the riverbank, there is a road. The boat captain observes that every 30 minutes, a bus overtakes the boat from behind, and every 10 minutes, a bus approaches from the opposite direction. Assuming that the buses depart from points A and B uniformly and travel at a constant speed, what is the interval time (in minutes) between each bus departure?

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>.

The equation $x^2 - kx - 24 = 0$ has only integer solutions for certain positive integers $k$. What is the sum of all such values of $k$?

Answer: $40$


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

A high school's 11th-grade class 1 has 45 male students and 15 female students. The teacher uses stratified sampling to form a 4-person extracurricular interest group. Calculate the probability of a student being selected for this group and the number of male and female students in the extracurricular interest group.
After a month of study and discussion, this interest group decides to select two students to conduct an experiment. The method is to first select one student from the group to conduct the experiment, and after that student has completed it, select another student from the remaining group members to conduct the experiment. Calculate the probability that among the two selected students, exactly one is a female student.
After the experiment, the first student obtained experimental data of 68, 70, 71, 72, 74. The second student obtained experimental data of 69, 70, 70, 72, 74. Which student's experiment is more stable? And explain why.

Answer: $0.5$


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

In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a=3$, $b=\sqrt{3}$, and $A=\dfrac{\pi}{3}$, find the measure of angle $B$.

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>.

Given positive numbers \(a,b\) satisfying \(2ab=\dfrac{2a-b}{2a+3b},\) then the maximum value of \(b\) is \_\_\_\_\_

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


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 $44$-gon $Q_1$ is constructed in the Cartesian plane, and the sum of the squares of the $x$-coordinates of the vertices equals $176$. The midpoints of the sides of $Q_1$ form another $44$-gon, $Q_2$. Finally, the midpoints of the sides of $Q_2$ form a third $44$-gon, $Q_3$. Find the sum of the squares of the $x$-coordinates of the vertices of $Q_3$.

Answer: $44$


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

How many times does the digit 9 appear in the list of all integers from 1 to 1000?

Answer: $300$


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

Find the smallest solution to the equation \[\lfloor x^2 \rfloor - \lfloor x \rfloor^2 = 19.\]

Answer: $\sqrt{119}$


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

Four steel balls, each with a radius of 1, are completely packed into a container in the shape of a regular tetrahedron. The minimum height of this regular tetrahedron is:

Answer: $2 + \frac{2 \sqrt{6}}{3}$


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

Integers  $1, 2, 3, ... ,n$ , where  $n > 2$ , are written on a board. Two numbers  $m, k$  such that  $1 < m < n, 1 < k < n$  are removed and the average of the remaining numbers is found to be  $17$ . What is the maximum sum of the two removed numbers?

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>.

What is the result when $\frac{2}{3}$ is subtracted from $2\frac{1}{4}$?

Answer: $1\frac{7}{12}$


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 number from the set $\{30, 31, 32, \ldots, 500\}$ is randomly selected. What is the probability that the number is greater than 100 but less than or equal to 200? Express your answer as a common fraction.

Answer: $\frac{100}{471}$


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

John is tasked with creating a special mixture in his Science class, consisting of 0.05 liters of Compound X and 0.01 liters of Compound Y. He determined that each liter of this mixture has a specific ratio of Compound Y. Now, John needs to prepare 0.90 liters of this mixture. How much Compound Y will he require?

Answer: $0.15$


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

In triangle $ABC$, $AB=2$, $AC=5$, and $\cos A= \frac {4}{5}$. A point $P$ is chosen randomly inside triangle $ABC$. The probability that the area of triangle $PAB$ is greater than $1$ and less than or equal to $2$ is __________.

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>.

If a worker receives a $20$% cut in wages, calculate the exact percentage or dollar amount of a raise needed for the worker to regain their original pay.

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>.

How many odd divisors does $10!$ have?

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>.

Let \\(\alpha\\) be an acute angle. If \\(\cos (\alpha+ \dfrac {\pi}{6})= \dfrac {3}{5}\\), then \\(\sin (\alpha- \dfrac {\pi}{6})=\\) \_\_\_\_\_\_.

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


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

Given that $x\_1$ is a root of the equation $x \ln x = 2006$ and $x\_2$ is a root of the equation $xe^{x} = 2006$, find the value of $x\_1 \cdot x\_2$.

Answer: $2006$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 chosen from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. What is the probability that their greatest common factor is one? Express your answer as a common fraction.

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