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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

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