IMO 2013 Problem 5

Let $\mathbb Q_{>0}$ be the set of all positive rational numbers. Let $f:\mathbb Q_{>0}\to\mathbb R$ be a function satisfying the following three conditions:

(i) for all $x,y\in\mathbb Q_{>0}$ , we have $f(x)f(y)\geq f(xy)$ ; (ii) for all $x,y\in\mathbb Q_{>0}$ , we have $f(x+y)\geq f(x)+f(y)$ ; (iii) there exists a rational number $a>1$ such that $f(a)=a$ .

Prove that $f(x)=x$ for all $x\in\mathbb Q_{>0}$ .

Proposed by Bulgaria