IMO 2010 Problem 6

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers, and $s$ be a positive integer, such that $$a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.$$ Prove there exist positive integers $\ell \leq s$ and $N$ , such that $$a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.$$

Proposed by Morteza Saghafiyan, Iran