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IMO ShortList 1999, combinatorics problem 4

Source: IMO ShortList 1999, combinatorics problem 4

November 14, 2004
modular arithmeticcombinatoricscountingAdditive combinatoricsAdditive Number TheoryIMO ShortlistHi

Problem Statement

Let AA be a set of NN residues (modN2)\pmod{N^{2}}. Prove that there exists a set BB of of NN residues (modN2)\pmod{N^{2}} such that A+B={a+b∣a∈A,b∈B}A + B = \{a+b|a \in A, b \in B\} contains at least half of all the residues (modN2)\pmod{N^{2}}.
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