IMO Shortlist 2010 problem A7
Dodao/la:
arhiva23. lipnja 2013. Let
![a_1, a_2, a_3, \ldots](/media/m/3/3/2/332a89ed59afef28333231596fbc0c80.png)
be a sequence of positive real numbers, and
![s](/media/m/9/0/8/908014cbadb69e42261a56b450a375b9.png)
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.](/media/m/d/6/4/d640a2c1ccffcae66208eaf89fcc2a15.png)
Prove there exist positive integers
![\ell \leq s](/media/m/1/8/6/1862096a554e06d8b8e3d6357d927c0c.png)
and
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
, such that
![a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.](/media/m/2/f/2/2f2a2dda4a6c3d74651ddee3e53a76ba.png)
Proposed by Morteza Saghafiyan, Iran
%V0
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
Izvor: Međunarodna matematička olimpijada, shortlist 2010