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Difference between revisions of "2013 IMO Problems/Problem 5"
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Prove that <math>f(x)=x</math> for all <math>x\in\mathbb Q_{>0}</math>.
 
Prove that <math>f(x)=x</math> for all <math>x\in\mathbb Q_{>0}</math>.
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Proposed by Bulgaria
 

Revision as of 12:50, 21 June 2018

Problem

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 $x\in\mathbb Q_{>0}$.

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