Let m be a positive integer, and let $\mathbb{T}$ denote the set of all subsets of $\{1,\ 2,\ \cdots,\ m\}$. Call a subset $\mathbb{S}$ of $\mathbb{T}$ ${\delta}-good$ if for all $s_1,\ s_2\in\mathbb{S}$, $s_1\ne s_2$, $\mid\Delta(s_1,\ s_2)\mid\ge{\delta}m$, where $\Delta$ denotes symmetric difference (the symmetric difference of two sets is the set of elements that is in exactly one of the two sets). Find the largest possible integer $s$ such that there exists an integer $m$ and a $\frac{1024}{2047}-good$ set of size $s$.

Compute the number of possible words $w = w_1w_2\cdots w_{100}$ satisfying:

• $w$ has exactly $50$ $A$’s and $50$ $B$’s (and no other letters).
• For $i = 1,\ 2,\ \cdots,\ 100$, the number of $A$’s among $w_1$, $w_2$, $\cdots$, $w_i$ is at most the number of $B$’s among $w_1$, $w_2$, $\cdots$ , $w_i$.
• For all $i = 44,\ 45,\ \cdots ,\ 57$, if $w_i$ is an $B$, then $w_{i+1}$ must be an $B$
  

Solve the recursion $$a_n=\sum^{n-1}_{k=0}a_{k}a_{n-k-1}=a_0a_{n-1}+a_1a_{n-2}+\cdots+a_{n-1}a_0$$

where $a_0=a_1=1$.

Find the smallest square which can cover $n$ congruent equilateral triangles so that these triangles do not overlap.

Find the last $4$ digits of $2018^{2019^{2020}}$.

Compute $$\lim_{x\to 4}\frac{3-\sqrt{x+5}}{x-4}$$

Is the $y=\frac{1}{x}$ a continuous function?

Show that $$\lim_{x\to 0}\ \frac{x}{\sin{x}}=1$$

Let $0 < x < \frac{\pi}{2}$. Show that $\sin x < x <\tan x$.

Show the following sequence is convergent:

$$\frac{1}{1^2},\ \frac{1}{2^2},\ \frac{1}{3^2},\ \cdots,\ \frac{1}{n^2},\ \cdots$$

Show that the limit of $f(n)=\left(1+\frac{1}{n}\right)^n$ exits when $n$ becomes infinitely large.

Show that $$\lim_{x\to 0}\frac{e^x-1}{x}=1$$

Find the value of 

$$\lim_{x\to\infty}\frac{\sin{x}}{x}$$

Compute the derivative of $f(x)=x^n$.

Show that $$\frac{d}{dx} e^x = e^x$$

Given $\frac{d}{dx} e^x = e^x$, find the value of $\frac{d}{dx} \ln x$.

Find the derivative of function $y=\sin{x}$.

Find the derivative of $\arcsin{x}$.

Let $f(x)$ be an odd function which is differentiable over $(-\infty, +\infty)$. Show that $f'(x)$ is even.

Compute the limit of the power series below as a rational function in $x$:

$$1\cdot 2 + (2\cdot 3)x + (3\cdot 4)x^2 + (4\cdot 5)x^3 + (5\cdot 6)x^4+\cdots,\qquad (|x| < 1)$$

Compute $$1-\frac{1\times 2}{2}+\frac{2\times 3}{2^2}-\frac{3\times 4}{2^3}+\frac{4\times 5}{2^4}-\cdots$$

Construct one polynomial $f(x)$ with real coefficients and with all of the following properties:

• it is an even function
• $f(2)=f(-2)=0$
• $f(x) > 0$ when $-2 < x < 2$, and
• the maximum of $f(x)$ is achieved at $x=\pm 1$.

Find the coordinates of the center of mass of the $\frac{1}{4}$ disc defined by

$$\{(x, y) | x\ge 0, y\ge 0, x^2 + y^2 \le 1\}$$

assuming the density is uniform.

Consider the ellipse $x^2+\frac{y^2}{4}=1$. What is the area of the smallest diamond shape with two vertices on the $x$-axis and two vertices on the $y$-axis that contains this ellipse?

Compute $$I=\int \frac{x\cos{x}-\sin{x}}{x^2 + \sin^2{x}} dx$$