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

Given that Erin the ant starts at a given corner of a hypercube (4-dimensional cube) and crawls along exactly 15 edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point, determine the number of paths that Erin can follow to meet these conditions.

Answer: $24$


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

The probability of an event happening is $\frac{1}{2}$, find the relation between this probability and the outcome of two repeated experiments.

Answer: $50\%$


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

As the Chinese New Year approaches, workers from a factory begin to go home to reunite with their families starting from Monday, January 17, 2011. If the number of workers leaving the factory each day is the same, and by January 31, 121 workers remain in the factory, while the total number of worker-days during this 15-day period is 2011 (one worker working for one day counts as one worker-day, and the day a worker leaves and any days after are not counted), with weekends (Saturdays and Sundays) being rest days and no one being absent, then by January 31, the total number of workers who have gone home for the New Year is ____.

Answer: $120$


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

Find the smallest real number  $p$  such that the inequality  $\sqrt{1^2+1}+\sqrt{2^2+1}+...+\sqrt{n^2+1} \le \frac{1}{2}n(n+p)$  holds for all natural numbers  $n$ .

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


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

Given that $21^{-1} \equiv 17 \pmod{53}$, find $32^{-1} \pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.)

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>.

The vertical coordinate of the intersection point of the new graph obtained by shifting the graph of the quadratic function $y=x^{2}+2x+1$ $2$ units to the left and then $3$ units up is ______.

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>.

Given that $\sin \alpha \cos \alpha = \frac{1}{8}$, and $\alpha$ is an angle in the third quadrant. Find $\frac{1 - \cos^2 \alpha}{\cos(\frac{3\pi}{2} - \alpha) + \cos \alpha} + \frac{\sin(\alpha - \frac{7\pi}{2}) + \sin(2017\pi - \alpha)}{\tan^2 \alpha - 1}$.

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


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

Given the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $e = \frac{\sqrt{2}}{2}$, and one of its vertices is at $(0, -1)$.

(Ⅰ) Find the equation of the ellipse $C$.

(Ⅱ) If there exist two distinct points $A$ and $B$ on the ellipse $C$ that are symmetric about the line $y = -\frac{1}{m}x + \frac{1}{2}$, find the maximum value of the area of $\triangle OAB$ ($O$ is the origin).

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>.

A certain school randomly selected several students to investigate the daily physical exercise time of students in the school. They obtained data on the daily physical exercise time (unit: minutes) and organized and described the data. Some information is as follows:
- $a$. Distribution of daily physical exercise time:
| Daily Exercise Time $x$ (minutes) | Frequency (people) | Percentage |
|-----------------------------------|--------------------|------------|
| $60\leqslant x \lt 70$            | $14$               | $14\%$     |
| $70\leqslant x \lt 80$            | $40$               | $m$        |
| $80\leqslant x \lt 90$            | $35$               | $35\%$     |
| $x\geqslant 90$                   | $n$                | $11\%$     |

- $b$. The daily physical exercise time in the group $80\leqslant x \lt 90$ is: $80$, $81$, $81$, $81$, $82$, $82$, $83$, $83$, $84$, $84$, $84$, $84$, $84$, $85$, $85$, $85$, $85$, $85$, $85$, $85$, $85$, $86$, $87$, $87$, $87$, $87$, $87$, $88$, $88$, $88$, $89$, $89$, $89$, $89$, $89$.

Based on the above information, answer the following questions:
$(1)$ In the table, $m=$______, $n=$______.
$(2)$ If the school has a total of $1000$ students, estimate the number of students in the school who exercise for at least $80$ minutes per day.
$(3)$ The school is planning to set a time standard $p$ (unit: minutes) to commend students who exercise for at least $p$ minutes per day. If $25\%$ of the students are to be commended, what value can $p$ be?

Answer: $86$


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

You know that the Jones family has five children, and the Smith family has three children.  Of the eight children you know that there are five girls and three boys.  Let  $\dfrac{m}{n}$   be the probability that at least one of the families has only girls for children.  Given that  $m$  and  $n$  are relatively prime positive integers,  find  $m+ n$ .

Answer: $67$


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

Given a $4 \times 4$ square grid partitioned into $16$ unit squares, each of which is painted white or black with a probability of $\frac{1}{2}$, determine the probability that the grid is entirely black after a $90^{\circ}$ clockwise rotation and any white square landing in a position previously occupied by a black square is repainted black.

Answer: $\frac{1}{65536}$


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

The largest value of the real number $k$ for which the inequality $\frac{1+\sin x}{2+\cos x} \geqslant k$ has a solution.

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


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

How many distinct five-digit positive integers are there such that the product of their digits equals 16?

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>.

How many natural numbers greater than 10 but less than 100 are relatively prime to 21?

Answer: $53$


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

Given that the sequence $\{a_n\}$ is a geometric sequence, and $a_4 = e$, if $a_2$ and $a_7$ are the two real roots of the equation $$ex^2 + kx + 1 = 0, (k > 2\sqrt{e})$$ (where $e$ is the base of the natural logarithm),
1. Find the general formula for $\{a_n\}$.
2. Let $b_n = \ln a_n$, and $S_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. When $S_n = n$, find the value of $n$.
3. For the sequence $\{b_n\}$ in (2), let $c_n = b_nb_{n+1}b_{n+2}$, and $T_n$ be the sum of the first $n$ terms of the sequence $\{c_n\}$. Find the maximum value of $T_n$ and the corresponding value of $n$.

Answer: $n = 4$


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

Given the function $f(x) = \frac{1}{3}x^3 - 4x + 4$,
(I) Find the extreme values of the function;
(II) Find the maximum and minimum values of the function on the interval [-3, 4].

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


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

The Songjiang tram project is in full swing. After the tram starts operating, it will bring convenience to the public's travel. It is known that after the opening of a certain route, the tram interval $t$ (unit: minutes) satisfies $2 \leq t \leq 20$. According to market survey calculations, the tram’s passenger capacity is related to the departure interval $t$. When $10 \leq t \leq 20$, the tram is fully loaded with a capacity of 400 people. When $2 \leq t < 10$, the passenger capacity decreases, and the number of passengers reduced is directly proportional to the square of $(10-t)$, and when the interval is 2 minutes, the passenger capacity is 272 people. Let the tram's passenger capacity be $p(t)$.
(1) Find the expression for $p(t)$ and the passenger capacity when the departure interval is 6 minutes;
(2) If the net income per minute of the line is $Q = \dfrac{6p(t)-1500}{t} - 60$ (unit: yuan), what is the departure interval that maximizes the line’s net income per minute?

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>.

A school selects 4 teachers from 8 to teach in 4 remote areas at the same time (one person per area), where A and B cannot go together, and A and C can only go together or not go at all. The total number of different dispatch plans is \_\_\_\_\_\_ (answer in numbers).

Answer: $600$


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

Consider the system of equations:
\begin{align*}
8x - 5y &= a, \\
10y - 15x &= b.
\end{align*}
If this system has a solution \((x, y)\) where both \(x\) and \(y\) are nonzero, calculate \(\frac{a}{b}\), assuming \(b\) is nonzero.

Answer: $\frac{8}{15}$


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 rectangle $ABCD$, $P$ is a point on side $\overline{BC}$ such that $BP = 20$ and $CP = 5.$ If $\tan \angle APD = 2,$ then find $AB.$

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>.

Eight students from a university are planning to carpool for a trip, with two students from each of the grades one, two, three, and four. How many ways are there to arrange the four students in car A, such that the last two students are from the same grade?

Answer: $24$


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

Given the function $f(x) = \frac {a^{x}}{a^{x}+1}$ ($a>0$ and $a \neq 1$).
- (I) Find the range of $f(x)$.
- (II) If the maximum value of $f(x)$ on the interval $[-1, 2]$ is $\frac {3}{4}$, find the value of $a$.

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 ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$) has an eccentricity of $e = \frac{2}{3}$. Points A and B lie on the ellipse and are not symmetrical with respect to the x-axis or the y-axis. The perpendicular bisector of segment AB intersects the x-axis at point P(1, 0). Let the midpoint of AB be C($x_0$, $y_0$). Find the value of $x_0$.

Answer: $\frac{9}{4}$


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

Let \( p, q, r, s, \) and \( t \) be the roots of the polynomial
\[ x^5 + 10x^4 + 20x^3 + 15x^2 + 6x + 3 = 0. \]
Find the value of
\[ \frac{1}{pq} + \frac{1}{pr} + \frac{1}{ps} + \frac{1}{pt} + \frac{1}{qr} + \frac{1}{qs} + \frac{1}{qt} + \frac{1}{rs} + \frac{1}{rt} + \frac{1}{st}. \]

Answer: $\frac{20}{3}$


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>.

What is the maximum number of checkers that can be placed on a $6 \times 6$ board such that no three checkers (specifically, the centers of the cells they occupy) lie on the same straight line (at any angle)?

Answer: $12$


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

A 10x10 arrangement of alternating black and white squares has a black square $R$ in the second-bottom row and a white square $S$ in the top-most row. Given that a marker is initially placed at $R$ and can move to an immediately adjoining white square on the row above either to the left or right, and the path must consist of exactly 8 steps, calculate the number of valid paths from $R$ to $S$.

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 an isosceles triangle \( \triangle AMC \), \( AM = AC \), the median \( MV = CU = 12 \), and \( MV \perp CU \) at point \( P \). What is the area of \( \triangle AMC \)?

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>.

In a certain competition, the rules are as follows: among the 5 questions preset by the organizer, if a contestant can answer two consecutive questions correctly, they will stop answering and advance to the next round. Assuming the probability of a contestant correctly answering each question is 0.8, and the outcomes of answering each question are independent of each other, then the probability that the contestant will exactly answer 4 questions before advancing to the next round is

Answer: $0.128$


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

Find the area of the region bounded by a function  $y=-x^4+16x^3-78x^2+50x-2$  and the tangent line which is tangent to the curve at exactly two distinct points.

Proposed by Kunihiko Chikaya

Answer: $1296/5$


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>.

Write a twelve-digit number that is not a perfect cube.

Answer: $100000000000$


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

Find the smallest positive real number  $\lambda$  such that for every numbers  $a_1,a_2,a_3 \in \left[0, \frac{1}{2} \right]$  and  $b_1,b_2,b_3 \in (0, \infty)$  with  $\sum\limits_{i=1}^3a_i=\sum\limits_{i=1}^3b_i=1,$  we have   $$ b_1b_2b_3 \le \lambda (a_1b_1+a_2b_2+a_3b_3). $$

Answer: $1/8$


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

From a deck of cards marked with 1, 2, 3, and 4, two cards are drawn consecutively. The probability of drawing the card with the number 4 on the first draw, the probability of not drawing it on the first draw but drawing it on the second, and the probability of drawing the number 4 at any point during the drawing process are, respectively:

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


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

Given the sets $M={x|m\leqslant x\leqslant m+ \frac {3}{4}}$ and $N={x|n- \frac {1}{3}\leqslant x\leqslant n}$, both of which are subsets of ${x|0\leqslant x\leqslant 1}$, what is the minimum "length" of the set $M\cap N$? (Note: The "length" of a set ${x|a\leqslant x\leqslant b}$ is defined as $b-a$.)

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>.

When three standard dice are tossed, the numbers $x, y, z$ are obtained. Find the probability that $xyz = 72$.

Answer: $\frac{7}{216}$


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

Let \( D \) be a point inside \( \triangle ABC \) such that \( \angle BAD = \angle BCD \) and \( \angle BDC = 90^\circ \). If \( AB = 5 \), \( BC = 6 \), and \( M \) is the midpoint of \( AC \), find the length of \( DM \).

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


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

Three students, A, B, and C, are playing badminton with the following rules:
The player who loses two games in a row will be eliminated. Before the game, two players are randomly selected to play against each other, while the third player has a bye. The winner of each game will play against the player with the bye in the next round, and the loser will have the bye in the next round. This process continues until one player is eliminated. After one player is eliminated, the remaining two players will continue playing until one of them is eliminated, and the other player will be the ultimate winner of the game. A and B are selected to play first, while C has the bye. The probability of winning for each player in a game is $\frac{1}{2}$.
$(1)$ Find the probability of A winning four consecutive games;
$(2)$ Find the probability of needing a fifth game to be played;
$(3)$ Find the probability of C being the ultimate winner.

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


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

What is the least positive multiple of 45 for which the product of its digits is also a positive multiple of 45?

Answer: $945$


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

The diagonals of a convex quadrilateral \(ABCD\) intersect at point \(E\). Given that \(AB = AD\) and \(CA\) is the bisector of angle \(C\), with \(\angle BAD = 140^\circ\) and \(\angle BEA = 110^\circ\), find angle \(CDB\).

Answer: $50$


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

Given a triangle $ABC$ with sides $a$, $b$, $c$, and area $S$ satisfying $S=a^{2}-(b-c)^{2}$, and $b+c=8$.
$(1)$ Find $\cos A$;
$(2)$ Find the maximum value of $S$.

Answer: $\frac{64}{17}$


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>.

Given Ms. Grace Swift leaves her house at 7:30 AM to reach her office, she arrives 5 minutes late at an average speed of 50 miles per hour, and 5 minutes early at an average speed of 70 miles per hour, determine the average speed she should maintain to arrive exactly on time.

Answer: $58$


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

Determine how many 4-digit numbers are mountain numbers, where mountain numbers are defined as having their middle two digits larger than any other digits in the number. For example, 3942 and 5732 are mountain numbers.

Answer: $240$


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

Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, and $S_n=n^2$ ($n\in\mathbb{N}^*$).
1. Find $a_n$;
2. The function $f(n)$ is defined as $$f(n)=\begin{cases} a_{n} & \text{, $n$ is odd} \\ f(\frac{n}{2}) & \text{, $n$ is even}\end{cases}$$, and $c_n=f(2^n+4)$ ($n\in\mathbb{N}^*$), find the sum of the first $n$ terms of the sequence ${c_n}$, denoted as $T_n$.
3. Let $\lambda$ be a real number, for any positive integers $m$, $n$, $k$ that satisfy $m+n=3k$ and $m\neq n$, the inequality $S_m+S_n>\lambda\cdot S_k$ always holds, find the maximum value of the real number $\lambda$.

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


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

Consider an ordinary $6$-sided die, with numbers from $1$ to $6$. How many ways can I paint three faces of a die so that the sum of the numbers on the painted faces is not exactly $9$?

Answer: $18$


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

Given the set  $X=\left\{1,2,3,4\right\}$, consider a function  $f:X\to X$ where  $f^1=f$  and  $f^{k+1}=\left(f\circ f^k\right)$  for  $k\geq1$. Determine the number of functions  $f$  that satisfy  $f^{2014}\left(x\right)=x$  for all  $x$  in  $X$.

Answer: $13$


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

Two distinct numbers are selected simultaneously and at random from the set $\{1, 2, 3, 6, 9\}$. What is the probability that the smaller one divides the larger one? Express your answer as a common fraction.

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


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

Calculate: \( 8.0 \dot{\dot{1}} + 7.1 \dot{2} + 6.2 \dot{3} + 5.3 \dot{4} + 4.4 \dot{5} + 3.5 \dot{6} + 2.6 \dot{7} + 1.7 \dot{8} = \)

Answer: $39.2$


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

Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $\overrightarrow{a} \cdot (\overrightarrow{a} - 2\overrightarrow{b}) = 3$, and $|\overrightarrow{a}| = 1$, $\overrightarrow{b} = (1,1)$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

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


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

While there do not exist pairwise distinct real numbers  $a,b,c$  satisfying  $a^2+b^2+c^2 = ab+bc+ca$ , there do exist complex numbers with that property. Let  $a,b,c$  be complex numbers such that  $a^2+b^2+c^2 = ab+bc+ca$  and  $|a+b+c| = 21$ . Given that  $|a-b| = 2\sqrt{3}$ ,  $|a| = 3\sqrt{3}$ , compute  $|b|^2+|c|^2$ .
<details><summary>Clarifications</summary>

- The problem should read  $|a+b+c| = 21$ .  An earlier version of the test read  $|a+b+c| = 7$ ; that value is incorrect.
- $|b|^2+|c|^2$  should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers  $m$  and  $n$ .  Find  $m+n$ .''
</details>

*Ray Li*

Answer: $132$


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

In triangle $ABC$, side $AC = 900$ and side $BC = 600$. Points $K$ and $L$ are located on segment $AC$ and segment $AB$ respectively, such that $AK = CK$, and $CL$ is the angle bisector of $\angle C$. Let point $P$ be the intersection of line segments $BK$ and $CL$. Point $M$ is located on line $BK$ such that $K$ is the midpoint of segment $PM$. If $AM = 360$, find $LP$.

Answer: $144$


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

Let  $ a $  ,  $ b $  ,  $ c $  ,  $ d $  ,  $ (a + b + c + 18 + d) $  ,  $ (a + b + c + 18 - d) $  ,  $ (b + c) $  , and  $ (c + d) $  be distinct prime numbers such that  $ a + b + c = 2010 $ ,  $ a $ ,  $ b $ ,  $  c $ ,  $ d \neq 3 $  , and  $ d \le 50 $ . Find the maximum value of the diﬀerence between two of these prime numbers.

Answer: $2067$


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

The mode and median of the data $9.30$, $9.05$, $9.10$, $9.40$, $9.20$, $9.10$ are ______ and ______, respectively.

Answer: $9.15$


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>.

A trapezoid \(ABCD\) (\(AD \parallel BC\)) and a rectangle \(A_{1}B_{1}C_{1}D_{1}\) are inscribed in a circle \(\Omega\) with a radius of 13 such that \(AC \parallel B_{1}D_{1}\) and \(BD \parallel A_{1}C_{1}\). Find the ratio of the areas of \(ABCD\) and \(A_{1}B_{1}C_{1}D_{1}\), given that \(AD = 24\) and \(BC = 10\).

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>.

Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails?

Answer: $499$


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

Two chords \( PQ \) and \( PR \) are drawn in a circle with diameter \( PS \). The point \( T \) lies on \( PR \) and \( QT \) is perpendicular to \( PR \). The angle \(\angle QPR = 60^\circ\), \( PQ = 24 \text{ cm} \), and \( RT = 3 \text{ cm} \). What is the length of the chord \( QS \) in cm?

A) \( \sqrt{3} \)
B) 2
C) 3
D) \( 2\sqrt{3} \)
E) \( 3\sqrt{2} \)

Answer: $2\sqrt{3}$


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

In triangle $PQR$, $PQ = 8$, $QR = 15$, $PR = 17$, and $QS$ is the angle bisector. Find the length of $QS$.

Answer: $\sqrt{87.04}$


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

Let $a_0 = 3,$ $b_0 = 4,$ and
\[a_{n + 1} = \frac{a_n^2}{b_n} \quad \text{and} \quad b_{n + 1} = \frac{b_n^2}{a_n}\] for all $n \ge 0.$ Calculate $b_7 = \frac{4^p}{3^q}$ for some integers $p$ and $q.$

Answer: $(1094,1093)$


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

Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______.

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


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

Given \(0 \leqslant x \leqslant 2\), the function \(y=4^{x-\frac{1}{2}}-3 \cdot 2^{x}+5\) reaches its minimum value at?

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


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

Given Erin has 4 sisters and 6 brothers, determine the product of the number of sisters and the number of brothers of her brother Ethan.

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>.

A sequence begins with 3, and each subsequent term is triple the sum of all preceding terms. Determine the first term in the sequence that exceeds 15000.

Answer: $36864$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 line $l$ is $\begin{cases} x=- \dfrac { \sqrt {3}}{2}t \\ y=1+ \dfrac {1}{2}t\end{cases}$ (where $t$ is the parameter), with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar coordinate system is established. The equation of curve $C_{1}$ is $\rho= \dfrac {36}{4 \sqrt {3}\sin \theta-12\cos \theta-\rho}$. Let $M(6,0)$ be a fixed point, and $N$ be a moving point on curve $C_{1}$, with $Q$ being the midpoint of $MN$.
$(1)$ Find the Cartesian coordinate equation of the trajectory $C_{2}$ of point $Q$;
$(2)$ Given that the intersection of line $l$ with the $x$-axis is point $P$, and its intersections with curve $C_{2}$ are points $A$ and $B$. If the midpoint of $AB$ is $D$, find the length of $|PD|$.

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


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

Given the function $f(x)=\frac{{1-x}}{{2x+2}}$.
$(1)$ Find the solution set of the inequality $f(2^{x})-2^{x+1}+2 \gt 0$;
$(2)$ If the function $g(x)$ satisfies $2f(2^{x})\cdot g(x)=2^{-x}-2^{x}$, and for any $x$ $(x\neq 0)$, the inequality $g(2x)+3\geqslant k\cdot [g(x)-2]$ always holds, find the maximum value of the real number $k$.

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


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

For how many real numbers $a$ does the quadratic equation $x^2 + ax + 12a = 0$ have only integer roots for $x$?

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>.

Let $a, b, c$ be positive real numbers such that $3a + 4b + 2c = 3$. Find the minimum value of
\[
\frac{1}{2a + b} + \frac{1}{a + 3c} + \frac{1}{4b + c}.
\]

Answer: $1.5$


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

Cory has $4$ apples, $2$ oranges, and $1$ banana. If Cory eats one piece of fruit per day for a week, and must consume at least one apple before any orange, how many different orders can Cory eat these fruits? The fruits within each category are indistinguishable.

Answer: $105$


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

The base radius of a circular television tower is 15 meters. Now, it is planned to plant a 5-meter-wide ring-shaped lawn around it.
(1) How many square meters of lawn are needed?
(2) If each square meter of lawn costs 500 yuan, how much money is needed to plant this lawn at least?

Answer: $274750$


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

Evaluate the first three digits to the right of the decimal point in the decimal representation of $\left(10^{987} + 1\right)^{8/3}$ using the Binomial Expansion.

Answer: $666$


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

A line with a slope of $-3$ intersects the positive $x$-axis at $A$ and the positive $y$-axis at $B$. A second line intersects the $x$-axis at $C(10,0)$ and the $y$-axis at $D$. The lines intersect at $E(5,5)$. What is the area of the shaded quadrilateral $OBEC$?

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>.

Find distinct digits to replace the letters \(A, B, C, D\) such that the following division in the decimal system holds:

$$
\frac{ABC}{BBBB} = 0,\overline{BCDB \, BCDB \, \ldots}
$$

(in other words, the quotient should be a repeating decimal).

Answer: $219$


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

How many positive integers less than $1000$ are either a perfect cube or a perfect square?

Answer: $37$


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

When f(t)=0.1, solve for t in the function model: $f(t)=\frac{1}{{1+{e^{-0.22(3t-40)}}}$.

Answer: $10$


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

Given the function $f(x)=x^{3}+ax^{2}+bx+c$, it reaches a maximum value of $7$ when $x=-1$, and it reaches a minimum value when $x=3$. Find the values of $a$, $b$, $c$, and this minimum value.

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>.

Let $m=n^{4}+x$, where $n \in \mathbf{N}$ and $x$ is a two-digit positive integer. Which value of $x$ ensures that $m$ is always a composite number?

Answer: $64$


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

Let $\min \{a, b\}$ denote the smaller value between $a$ and $b$. When the positive numbers $x$ and $y$ vary, $t = \min \left\{ x, \frac{y}{x^{2}+y^{2}} \right\}$ also varies. What is the maximum value of $t$?

Answer: $1/2$


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

Given that the area of $\triangle ABC$ is $\frac{1}{2}$, $AB=1$, $BC=\sqrt{2}$, determine the value of $AC$.

Answer: $\sqrt{5}$


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

The Chinese mathematician Qin Jiushao (circa 1202-1261) from the Southern Song Dynasty proposed Qin Jiushao's algorithm for polynomial evaluation in his work "Mathematical Book in Nine Chapters." The provided diagram illustrates an example of using Qin Jiushao's algorithm to evaluate a polynomial. If the input values are $n=5, v=1, x=2$, what does the flowchart calculate?

Answer: $2^{5}+2^{4}+2^{3}+2^{2}+2+1$


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

Find the smallest positive integer \( n \) such that:
1. \( n \) has exactly 144 distinct positive divisors.
2. There are ten consecutive integers among the positive divisors of \( n \).

Answer: $110880$


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

What is the sum of all the integers between -20.5 and -1.5?

Answer: $-210$


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

Determine the value of \(a\) if \(a\) and \(b\) are integers such that \(x^3 - x - 1\) is a factor of \(ax^{19} + bx^{18} + 1\).

Answer: $2584$


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

Let $\overline{AB}$ be a diameter in a circle of radius $10$. Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=4$ and $\angle AEC = 30^{\circ}.$ Compute the sum $CE^2 + DE^2$.

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>.

The "Tuning Day Method" is a procedural algorithm for seeking precise fractional representations of numbers. Suppose the insufficient approximation and the excessive approximation of a real number $x$ are $\dfrac{b}{a}$ and $\dfrac{d}{c}$ ($a,b,c,d \in \mathbb{N}^*$) respectively, then $\dfrac{b+d}{a+c}$ is a more accurate insufficient approximation or excessive approximation of $x$. Given that $\pi = 3.14159…$, and the initial values are $\dfrac{31}{10} < \pi < \dfrac{16}{5}$, determine the more accurate approximate fractional value of $\pi$ obtained after using the "Tuning Day Method" three times.

Answer: $\dfrac{22}{7}$


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

Given that the polar coordinate equation of curve $C$ is $ρ=2\cos θ$, and the polar coordinate equation of line $l$ is $ρ\sin (θ+ \frac {π}{6})=m$. If line $l$ and curve $C$ have exactly one common point, find the value of the real number $m$.

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


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

Evaluate the sum $$\lceil\sqrt{5}\rceil + \lceil\sqrt{6}\rceil + \lceil\sqrt{7}\rceil + \cdots + \lceil\sqrt{39}\rceil$$
Note: For a real number $x,$ $\lceil x \rceil$ denotes the smallest integer that is greater than or equal to $x.$

Answer: $175$


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

A group of n  friends wrote a math contest consisting of eight short-answer problem  $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$ , and four full-solution problems  $F_1, F_2, F_3, F_4$ . Each person in the group correctly solved exactly 11 of the  12 problems. We create an 8 x 4 table. Inside the square located in the  $i$ th row and  $j$ th column, we write down the number of people who correctly solved both problem  $S_i$  and  $F_j$ . If the 32 entries in the table sum to 256, what is the value of n?

Answer: $32$


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>.

In a local election based on proportional representation, each voter is given a ballot with 10 candidate names. The voter must mark No. 1 against the candidate for whom they cast their first vote, and they may optionally fill out No. 2 through No. 10 for their subsequent preferences.

How many different ways can a voter mark their ballot?

Answer: $3628800$


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

In the diagram, three circles of radius 2 with centers $X$, $Y$, and $Z$ are tangent to one another and to the sides of $\triangle DEF$. Each circle touches two sides of the triangle.

Find the perimeter of triangle $DEF$.

Answer: $12\sqrt{3}$


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 $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 2,$ $\|\mathbf{b}\| = 3,$ and $\|\mathbf{c}\| = 6,$ and
\[\mathbf{a} + 2\mathbf{b} + \mathbf{c} = \mathbf{0}.\]
Compute $\mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c}.$

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>.

Given that $|\vec{a}| = 1$ and $\vec{b} = (1, \sqrt{3})$, and that $(\vec{b} - \vec{a}) \perp \vec{a}$, find the angle between vector $\vec{a}$ and vector $\vec{b}$.

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


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

Let $\mathbf{a} = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -7 \\ 3 \\ 1 \end{pmatrix}.$ Find the area of the triangle with vertices $\mathbf{0},$ $\mathbf{a},$ and $\mathbf{b}.$

Answer: $13$


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

How many of the 729 smallest positive integers written in base 9 use 7 or 8 (or both) as a digit?

Answer: $386$


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

The numbers from 1 to 9 are arranged in the cells of a $3 \times 3$ table such that the sum of the numbers on one diagonal is 7, and on the other diagonal, it is 21. What is the sum of the numbers in the five shaded cells?

![Table with shaded cells](https://cdn.mathpix.com/cropped/2024_05_06_ff369b3e8ca7495bdf12g-28.jpg?height=416&width=428&top_left_y=105&top_left_x=513)

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>.

Two semicircles of diameters 2 and 4 are placed such that the smaller is inside the larger one and their diameters are along the same line. A square of side 1 is inscribed in the larger semicircle such that one of its sides is along the diameter of the semicircles. Calculate the area of the lune formed by the space in the smaller semicircle that is outside the larger semicircle and subtract the area covered by the square.

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


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

Given the function $f(x)=\sin \left( \omega x- \frac{\pi }{6} \right)+\sin \left( \omega x- \frac{\pi }{2} \right)$, where $0 < \omega < 3$. It is known that $f\left( \frac{\pi }{6} \right)=0$.

(1) Find $\omega$;

(2) Stretch the horizontal coordinates of each point on the graph of the function $y=f(x)$ to twice its original length (the vertical coordinates remain unchanged), then shift the resulting graph to the left by $\frac{\pi }{4}$ units to obtain the graph of the function $y=g(x)$. Find the minimum value of $g(x)$ on $\left[ -\frac{\pi }{4},\frac{3\pi }{4} \right]$.

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>.

Given the function $f(x)=2m\sin x-2\cos ^{2}x+ \frac{m^{2}}{2}-4m+3$, and the minimum value of the function $f(x)$ is $(-7)$, find the value of the real number $m$.

Answer: $10$


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

If $$\sin\theta= \frac {3}{5}$$ and $$\frac {5\pi}{2}<\theta<3\pi$$, then $$\sin \frac {\theta}{2}$$ equals \_\_\_\_\_\_.

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


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>.

Through points \( A(0, 14) \) and \( B(0, 4) \), two parallel lines are drawn. The first line, passing through point \( A \), intersects the hyperbola \( y = \frac{1}{x} \) at points \( K \) and \( L \). The second line, passing through point \( B \), intersects the hyperbola \( y = \frac{1}{x} \) at points \( M \) and \( N \).

What is the value of \( \frac{AL - AK}{BN - BM} \)?

Answer: $3.5$


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

Given circle $M$: $(x+1)^{2}+y^{2}=1$, and circle $N$: $(x-1)^{2}+y^{2}=9$, a moving circle $P$ is externally tangent to circle $M$ and internally tangent to circle $N$. The trajectory of the center of circle $P$ is curve $C$.
$(1)$ Find the equation of $C$.
$(2)$ Let $l$ be a line tangent to both circle $P$ and circle $M$, and $l$ intersects curve $C$ at points $A$ and $B$. When the radius of circle $P$ is the longest, find $|AB|$.

Answer: $\dfrac {18}{7}$


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

In triangle $\triangle ABC$, $A+B=5C$, $\sin \left(A-C\right)=2\sin B$.
$(1)$ Find $A$;
$(2)$ If $CM=2\sqrt{7}$ and $M$ is the midpoint of $AB$, find the area of $\triangle ABC$.

Answer: $4\sqrt{3}$


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

A workshop has 11 workers, of which 5 are fitters, 4 are turners, and the remaining 2 master workers can act as both fitters and turners. If we need to select 4 fitters and 4 turners to repair a lathe from these 11 workers, there are __ different methods for selection.

Answer: $185$


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

Given the sets \( M = \{x, y, \lg(xy)\} \) and \( N = \{0, |x|, y\} \) with \( M = N \), find the value of \(\left(x + \frac{1}{y}\right) + \left(x^{2} + \frac{1}{y^{2}}\right) + \left(x^{3} + \frac{1}{y^{3}}\right) + \cdots + \left(x^{2001} + \frac{1}{y^{2001}}\right) \).

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>.

Determine the largest value of $S$ such that any finite collection of small squares with a total area $S$ can always be placed inside a unit square $T$ in such a way that no two of the small squares share an interior point.

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>.

There exist $r$ unique nonnegative integers $n_1 > n_2 > \cdots > n_r$ and $r$ unique integers $a_k$ ($1\le k\le r$) with each $a_k$ either $1$ or $- 1$ such that \[a_13^{n_1} + a_23^{n_2} + \cdots + a_r3^{n_r} = 1025.\]
Find $n_1 + n_2 + \cdots + n_r$.

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>.

The region shown is bounded by the arcs of circles having a radius 8 units, with each arc having a central angle measure of 45 degrees and intersecting at points of tangency. Determine the area of this region, which can be expressed in the form $a\sqrt{b} + c\pi$ square units, where $\sqrt{b}$ is a radical in simplest form. What is the value of $a + b + c$?

Answer: $-22$


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

Given the ranges $-6 \leq x \leq -2$ and $0 \leq y \leq 4$, determine the largest possible value of the expression $\frac{x+y}{x}$.

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 the convex quadrilateral $ABCD$, the sum of $AB+BD+DC$ is at most 2, and the area of the quadrilateral is $1/2$. What can be the length of diagonal $AC$?

Answer: $\sqrt{2}$


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

The numbers \(1, 2, 3, \ldots, 400\) are written on 400 cards. Two players, \(A\) and \(B\), play the following game:

1. In the first step, \(A\) takes 200 cards for themselves.
2. \(B\) then takes 100 cards from both the remaining 200 cards and the 200 cards that \(A\) has, totaling 200 cards for themselves, and leaves the remaining 200 cards for \(A\).
3. In the next step, \(A\) takes 100 cards from both players' cards, totaling 200 cards for themselves, leaving the remaining 200 cards for \(B\).

This process continues until \(B\) completes the 200th step. At the end, the sums of the numbers on the cards held by \(A\) and \(B\) are calculated as \(C_A\) and \(C_B\) respectively. Then, player \(A\) pays player \(B\) the difference \(C_B - C_A\).

What is the maximum difference that \(B\) can achieve if both players play optimally?

Answer: $20000$


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

Given a tetrahedron $ABCD$, each edge is colored red or blue with equal probability. What is the probability that point $A$ can reach point $B$ through red edges within the tetrahedron $ABCD$?

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


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

The function \(f(x) = 5x^2 - 15x - 2\) has a minimum value when x is negative.

Answer: $-13.25$


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>.

Fill each cell in the given grid with a number from 1 to 4 so that no number repeats within any row or column. Each "L" shaped block spans two rows and two columns. The numbers inside the circles on the line indicate the sum of the numbers in the two adjacent cells (as shown in the provided example, where the third row, from left to right, is 2, 3, 1). What is the four-digit number formed by the two numbers in the bottom row of the given grid, in left to right order?

Answer: $2143$


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

In circle $O$, $\overline{EB}$ is a diameter and the line $\overline{DC}$ is parallel to $\overline{EB}$. The line $\overline{AB}$ intersects the circle again at point $F$ such that $\overline{AB}$ is parallel to $\overline{ED}$. If angles $AFB$ and $ABF$ are in the ratio 3:2, find the degree measure of angle $BCD$.

Answer: $72$


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

How many four-digit positive integers are multiples of 7?

Answer: $1285$


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

A typical triangle is defined as one in which each angle is at least \(30^{\circ}\).

A triangle is identified using the following procedure. The perimeter of a circle is divided into 180 equal arcs, and 3 distinct points are randomly chosen from these division points. (Any three points are chosen with equal probability.) What is the probability that the selected points form a typical triangle?

What is the probability of selecting a typical triangle if the vertices can be chosen arbitrarily on the perimeter of the circle?

Answer: $0.25$


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

In a blackboard, it's written the following expression $ 1-2-2^2-2^3-2^4-2^5-2^6-2^7-2^8-2^9-2^{10}$ We put parenthesis by different ways and then we calculate the result. For example: $ 1-2-\left(2^2-2^3\right)-2^4-\left(2^5-2^6-2^7\right)-2^8-\left( 2^9-2^{10}\right)= 403$  and $ 1-\left(2-2^2 \left(-2^3-2^4 \right)-\left(2^5-2^6-2^7\right)\right)- \left(2^8- 2^9 \right)-2^{10}= -933$ How many different results can we obtain?

Answer: $1024$


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

A circle can be inscribed in a trapezoid with bases of lengths 3 and 5, and a circle can also be circumscribed about it. Calculate the area of the pentagon formed by the radii of the inscribed circle that are perpendicular to the legs of the trapezoid, its smaller base, and the corresponding segments of the legs.

Answer: $\frac{3 \sqrt{15}}{2}$


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

Calculate the value for the expression $\sqrt{25\sqrt{15\sqrt{9}}}$.

Answer: $5\sqrt{15}$


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

Let \(a\), \(b\), \(c\) be real numbers such that \(9a^2 + 4b^2 + 25c^2 = 1\). Find the maximum value of
\[
10a + 3b + 5c.
\]

Answer: $\sqrt{134}$


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

The hour and minute hands of a clock move continuously and at constant speeds. A moment of time $X$ is called interesting if there exists such a moment $Y$ (the moments $X$ and $Y$ do not necessarily have to be different), so that the hour hand at moment $Y$ will be where the minute hand is at moment $X$, and the minute hand at moment $Y$ will be where the hour hand is at moment $X$. How many interesting moments will there be from 00:01 to 12:01?

Answer: $143$


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

Given the arithmetic sequence {an} satisfies a3 − a2 = 3, a2 + a4 = 14.
(I) Find the general term formula for {an};
(II) Let Sn be the sum of the first n terms of the geometric sequence {bn}. If b2 = a2, b4 = a6, find S7.

Answer: $-86$


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

In triangle $XYZ$, points $X'$, $Y'$, and $Z'$ are on the sides $YZ$, $ZX$, and $XY$, respectively. Given that lines $XX'$, $YY'$, and $ZZ'$ are concurrent at point $P$, and that $\frac{XP}{PX'}+\frac{YP}{PY'}+\frac{ZP}{PZ'}=100$, find the product $\frac{XP}{PX'}\cdot \frac{YP}{PY'}\cdot \frac{ZP}{PZ'}$.

Answer: $102$


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

Given the cubic function $f(x) = x^3 - 9x^2 + 20x - 4$, and the arithmetic sequence $\{a_n\}$ with $a_5 = 3$, find the value of $f(a_1) + f(a_2) + ... + f(a_9)$.

Answer: $18$