Find all the real values of $x$ that satistify: $$\sqrt{3x^2 + 1} - 2\sqrt{x} + x - 1=0$$

Find all the real values of $x$ that satistify: $$\sqrt{3x^2 + 1} - 2\sqrt{x} - x + 1=0$$

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 $S < 2\alpha$.

Which number is larger: $5^{4321}$ or $4^{5321}$?

Show that $2013^2 +2013^2\times 2014^2 + 2014^2$ is a perfect sqare.

Compute $\sqrt[32]{259\times 23\times 11 +9}$.

Let $f(x)$ be a second degree function satisfying $f(-2)=0$ and $2x \lt f(x) \le\frac{x^2+4}{2}$. Find the value of $f(10)$.

Solve $x^{x^{88}} - 88=0$.