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2010 IMO Problems/Problem 6

Revision as of 11:14, 20 July 2010 by Bugi (talk | contribs) (Created page with '== Problem == Let <math>a_1, a_2, a_3, \ldots</math> be a sequence of positive real numbers, and <math>s</math> be a positive integer, such that <cmath>a_n = \max \{ a_k + a_{n-…')
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Problem

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 \[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\]

Author: Morteza Saghafiyan, Iran

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