Determine the greatest positive integer
that satisfies the following property: The set of positive integers can be partitioned into
subsets
such that for all integers
and all
there exist two distinct elements of
whose sum is
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
Školjka