Question 100
Let $N$ be the set of all positive integers. Find the largest positive integer $k$ for which $N$ can be partitioned into subsets $A_1,A_2,\dots,A_k$ such that, for every integer $n\ge15$ and every $i\in\{1,2,\dots,k\}$, there are two distinct elements $a,b\in A_i$ with $a+b=n$.