ISL 1999 C4

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

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