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Describe the numbers

Source: Romanian IMO TST 2006, day 2, problem 3

April 22, 2006
modular arithmeticinductionnumber theory proposednumber theory

Problem Statement

For which pairs of positive integers (m,n)(m,n) there exists a set AA such that for all positive integers x,yx,y, if ∣x−y∣=m|x-y|=m, then at least one of the numbers x,yx,y belongs to the set AA, and if ∣x−y∣=n|x-y|=n, then at least one of the numbers x,yx,y does not belong to the set AA?
Adapted by Dan Schwarz from A.M.M.
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