IMO 1999 Shortlist Question 24

C4 (GBR). Let $A$ be a set of $N$ residues $(\bmod N^2)$. Prove that there exists a set $B$ of $N$ residues $(\bmod N^2)$ such that the set $A+B=\{a+b\mid a\in A,b\in B\}$ contains at least half of all residues $(\bmod N^2)$.