IMO Shortlist 2011 problem C4
Dodao/la:
arhiva23. lipnja 2013. Determine the greatest positive integer
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
that satisfies the following property: The set of positive integers can be partitioned into
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
subsets
![A_1, A_2, \ldots, A_k](/media/m/e/d/0/ed053ddcc89dfc2c2ad62711c11f857a.png)
such that for all integers
![n \geq 15](/media/m/7/2/8/728dc28cfcc66cd0dc1f4109604db893.png)
and all
![i \in \{1, 2, \ldots, k\}](/media/m/8/a/7/8a741b20481961fbee9b6905298c6980.png)
there exist two distinct elements of
![A_i](/media/m/5/f/0/5f0935569a883b13bb70b83ea33eee14.png)
whose sum is
![n.](/media/m/4/7/8/478d25bfe04800537ae6e85be9d11ea2.png)
Proposed by Igor Voronovich, Belarus
%V0
Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_1, A_2, \ldots, A_k$ such that for all integers $n \geq 15$ and all $i \in \{1, 2, \ldots, k\}$ there exist two distinct elements of $A_i$ whose sum is $n.$
Proposed by Igor Voronovich, Belarus
Izvor: Međunarodna matematička olimpijada, shortlist 2011