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IMO ShortList 1999, algebra problem 4

Source: IMO ShortList 1999, algebra problem 4

November 14, 2004

algebraColoringpartitionRamsey TheoryIMO Shortlistcombinatorics

Problem Statement

Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that, for any two integers

x,y

taken from two different subsets, the number

x^2-xy+y^2

belongs to the third subset.