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Is it possible to arrange these numbers, $1, 1, 2, 2, 3, 3, \cdots, 1986, 1986$ to form a sequence for such there is $1$ number between two $1$'s, $2$ numbers between two $2$'s, $\cdots$, $1986$ numbers between two 1986's?

Let $BD$ be a median in $\triangle{ABC}$. If $AB=\frac{4\sqrt{6}}{3}$, $\cos{B}=\dfrac{\sqrt{6}}{6}$, and $BD=\sqrt{5}$, find the length of $BC$ and the value of $\sin{A}$.


Let $n$ be a positive integer and function $S_1(n)$ return the square of the sum of $n$'s digits. Additionally, let $S_{k+1}(n)=S_1\left(S_k(n)\right)$, where $k$ is a positive integer. Find the value of $S_{1991}(2^{1990})$.

Let $m$ and $n$ be positive integers satisfying $1 < m < n$. Show that $(1+m)^n > (1+n)^m$.


Let $\{a_n\}$ be a geometric sequence whose initial term is $a_1$ and common ratio is $q$. Show that $$a_1\binom{n}{0}-a_2\binom{n}{1}+a_3\binom{n}{2}-a_4\binom{n}{3}+\cdots+(-1)^na_{n+1}\binom{n}{n}=a_1(1-q)^n$$

where $n$ is a positive integer.


Find the constant term in the expansion of $\left(\frac{x}{2}+\frac{1}{x}+\sqrt{2}\right)^5$.


A deck of poker has three different colors each of which contains $10$ cards numbered from $1$ to $10$, respectively. In addition, there are two jokers both of which are numbered as $0$. A card with number $k$ is valued as $2^k$ points. How many different ways are there to draw several cards from this deck so that their total value equals $2004$?


Let $\mathbb{A}=\{a_1,\ a_2,\ \cdots,\ a_{100}\}$ be a set containing $100$ real numbers, $\mathbb{B}=\{b_1,\ b_2,\ \cdots,\ b_{50}\}$ be a set containing $50$ real numbers, and $\mathcal{F}$ be a mapping from $\mathbb{A}$ to $\mathbb{B}$. Find the number of possible $\mathcal{F}$ if  $\mathcal{F}(a_1) \le \mathcal{F}(a_2)\le\cdots\mathcal{F}(a_1)$, and for every $b_i\in\mathbb{B}$, there exists an element $a_i\in\mathbb{A}$ such that the $\mathcal{F}(a_i)=b_i$.


How many ways are there to arrange $8$ girls and $25$ boys to sit around a table so that there are at least $2$ boys between any pair of girls? If a sitting plan can be simply rotated to match another one, these two are treated as the same.


There are $n \ge 6$ points on a circle, every two points are connected by a line segment. No three diagonals are concurrent. How many triangles are created by these sides and diagonals?


Compute $$\lim_{x\to 0}\frac{\int_0^x\sin(xt)^2dt}{x^5}$$


Compute

$$\lim_{x\to 0}\frac{\frac{x^2}{2}+1-\sqrt{1+x^2}}{(\cos{x}-e^{x^2})\sin(x^2)}$$


Calculate $$\lim_{n\to\infty}\frac{1}{n^2}\sum_{k=1}^{n}\left(k\sin\frac{k\pi}{n}\right)$$


Find all the inflection points of $$\left\{\begin{array}{rl} x &=\cot {t} \\ y&=\frac{\cos(2t)}{\sin{t}} \end{array}\right., t\in(0, \pi)$$

Let a differentiable function $f(x)$ satisfy $$f(x)\cos{x} + 2\int_0^xf(t)\sin{t}dt = x+1$$

Find $f(x)$.


Find the value of $\displaystyle\lim_{n\to\infty}\sin^2\left(\pi\sqrt{n^2+n}\right)$.


Let $f(x)$ be a twice differentiable continuous function, and $f(0)=f'(0)=0$, $f''(0)=6$. Find the value of $$\lim_{x\to 0}\frac{f\left(\sin^2{x}\right)}{x^4}$$

Find the value of $$I=\int\frac{e^{-\sin{x}}\sin(2x)}{(1-\sin{x})^2}dx$$


Compute the value of $$\lim_{x\to\pi}\frac{\ln(2+\cos{x})}{\left(3^{\sin{x}}-1\right)^2}$$


Compute the value of $$\lim_{n\to\infty}n^2\left(1-\cos\frac{\pi}{n}\right)$$


Let $$f(x)=\left\{\begin{array}{ll} \cos{x} &, x\in[-\frac{\pi}{2}, 0)\\e^x&,x\in[0,1] \end{array}\right.$$

Compute $\displaystyle\int_{-\frac{\pi}{2}}^{1}f(x)dx$.


The three segments marked with lengths are perpendicular to each other. Find the area of the outside square.