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IMO Shortlist 2012, Number Theory 4

Source: IMO Shortlist 2012, Number Theory 4

July 29, 2013
modular arithmeticnumber theoryequationIMO Shortlist

Problem Statement

An integer aa is called friendly if the equation (m2+n)(n2+m)=a(m−n)3(m^2+n)(n^2+m)=a(m-n)^3 has a solution over the positive integers. a) Prove that there are at least 500500 friendly integers in the set {1,2,…,2012}\{ 1,2,\ldots ,2012\}. b) Decide whether a=2a=2 is friendly.
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