International Mathematical Olympiad 2010 Problem 6

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers. Suppose that for some positive integer $s$, we have $$a _ {n} = \max \left\{a _ {k} + a _ {n - k} \mid 1 \leq k \leq n - 1 \right\}$$ for all $n > s$. Prove that there exist positive integers $\ell$ and $N$, with $\ell \leq s$ and such that $a_{n} = a_{\ell} + a_{n - \ell}$ for all $n \geq N$.