IMO Shortlist 2013 problem A3


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21. rujna 2014.
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Let \mathbb{Q}_{>0} be the set of positive rational numbers. Let f : \mathbb{Q}_{>0} \to \mathbb{R} be a function satisfying the conditions 
f(x) f(y) \geq f(xy)    \quad \text{and} \quad  f(x + y) \geq f(x) + f(y)
for all x, y \in \mathbb{Q}_{>0}. Given that f(a) = a for some rational a > 1, prove that f(x) = x for all x \in \mathbb{Q}_{>0}.
Izvor: Bulgaria