Problem

Source: V International Festival of Young Mathematicians Sozopol 2014, Theme for 10-12 grade

Tags: number theory, Sequence



Let $c>1$ be a real constant. For the sequence $a_1,a_2,...$ we have: $a_1=1$, $a_2=2$, $a_{mn}=a_m a_n$, and $a_{m+n}\leq c(a_m+a_n)$. Prove that $a_n=n$.