Let $a$ and $b$ be two positive real numbers satisfying $(a-b)^2=4(ab)^3$. Find the minimal value of $\frac{1}{a}+\frac{a}{b}$.

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

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

Show that $\ln x < \sqrt{x}$ holds for all positive $x$.

Show that $1-\cos{x} < x^2$ holds for all $x > 0$.

Show that $$\lim_{n\to\infty}\int_0^1 x^n(1-x)^n dx = 0$$

$\textbf{Shatter the Ball}$

You are in a $100$-story building with two identical bowling balls. You want to find the lowest floor at which the ball will shatter when dropped to the ground. What is the minimum number of drops you need in order to find the answer?

A circle of radius $2$, center on the origin, is drawn on a grid of points with integer coordinates. Let $n$ be the grid points that lie within or on the circle. What is the smallest amount of radius needs to increase by for there to be $(2n-5)$ grid points within or on the circle?

Prove that, if $|\alpha| < 2\sqrt{2}$, then there is no value of $x$ for which $$x^2-\alpha|x| + 2 < 0\qquad\qquad(*)$$

Find the solution set of (*)  for $\alpha=3$.

For $\alpha > 2\sqrt{2}$, then the sum of the lengths of the intervals in which $x$ satisfies (*) is denoted by $S$. Find $S$ in terns of $\alpha$ and deduce that $S < 2\alpha$.

Find the minimal value of $4^m + 4^n$ if $m+n=3$.

Let $a$, $b$, $c$, $x$, $y$, and $z$ be positive numbers. Show that $$\sqrt{a^2+x^2} + \sqrt{b^2 + y^2}+\sqrt{c^2+z^2} \ge \sqrt{(a+b+c)^2 + (x+y+z)^2}$$

Let positive numbers $a$, $b$ and $c$ satisfy $a+b+c=8$. Find the minimal value of $\sqrt{a^2+1}+\sqrt{b^2+4}+\sqrt{c^2+9}$.

Let $a, b, c \in\mathbf{R}^+$ and $\frac{a^2}{1+a^2}+\frac{b^2}{1+b^2}+\frac{c^2}{1+c^2}=1$, show that $$abc\le\frac{\sqrt{2}}{4}$$