KZ-MO 2013 Problem 3

Consider the following sequence : $a_1=1 ; a_n=\frac{a_[{\frac{n}{2}]}}{2}+\frac{a_[{\frac{n}{3}]}}{3}+\ldots+\frac{a_[{\frac{n}{n}]}}{n}$ . Prove that $a_{2n}< 2*a_{n } (\forall n\in\mathbb{N})$