Let $\mathbb Q_{>0}$ be the set of all rational numbers greater than zero. Let $f: \mathbb Q_{>0} \to \mathbb R$ be a function satisfying the following conditions:
(i) $f(x)f(y) \geq f(xy)$ for all $x, y \in \mathbb Q_{>0}$,
(ii) $f(x+y) \geq f(x) + f(y)$ for all $x, y \in \mathbb Q_{>0}$,
(iii) There exists a rational number $a> 1$ such that $f (a) = a$.
Show that $f(x) = x$ for all $x \in \mathbb Q_{>0}$.
IMO 2013, Day 2-P5
One one thing is neutral in the universe, that is $0$.