Olympiad Problems

SMO 2011:
Find all integers $z \in \mathbb{Z}$, such that $$2^{z}+2=r^2,\quad\quad\quad$$ where $r \in \mathbb{Q}$ is a rational number.
SMO 2012:
Let $a,b,c >0$ be real numbers with $abc=1$. Prove $$1+ab+bc+ca \ge \min\left\{\frac{(a+b)^2}{ab},\frac{(b+c)^2}{bc},\frac{(c+a)^2}{ca}\right\}.\quad\quad\quad$$ When does equality occur?
Swiss TST 2013:
Let $P$ denote a polynomial with integer coefficients, such that for all $u,v \in \mathbb{N}$ the following holds: $$u^{2^{2013}}-v^{2^{2013}}|P(u)-P(v).\quad\quad\quad$$ Prove that there is a polynomial $Q$ with integer coefficients such that $P(x)=Q(x^{2^{2013}})$ holds.
Swiss TST 2013:
Find all natural numbers $n$ which satisfy the following equality: $$\sum_{d|n}d^4=n^4+n^3+n^2+n+1.\quad\quad\quad$$
MEMO 2013:
Let $x,y,z,w \in \mathbb{R} \backslash \{0\}$ such that $x+y\neq 0, z+w \neq 0$, and $xy+zw \ge 0$. Prove the inequality $$\left( \frac{x+y}{z+w}+\frac{z+w}{x+y} \right)^{-1}+\frac{1}{2} \ge \left( \frac{x}{z}+\frac{z}{x} \right)^{-1}+ \left( \frac{y}{w}+\frac{w}{y} \right)^{-1}.\quad\quad\quad$$
SMO 2017:
Let $x,y,z \in \mathbb{R}^+_0$ satisfying $xy+yz+zx=1$. Prove the following inequality $$\frac{4}{x+y+z}\le (x+y)(\sqrt{3}z+1).\quad\quad\quad$$