IMO 2013, Day 2-P5

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Masum
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IMO 2013, Day 2-P5

Unread post by Masum » Mon Jul 29, 2013 1:13 pm

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 $x \in \mathbb Q_{>0}$.
One one thing is neutral in the universe, that is $0$.

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