Belorusija 2012

Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_1,\dots ,A_k$ such that for all integers $n\ge 15$ and all $i\in \{1,2,\dots ,k\}$ there exist two distinct elements of $A_i$ whose sum is $n$.