Problem

Source: Iran TST 2013-First exam-2nd day-P5

Tags: modular arithmetic, algebra, polynomial, quadratics, Vieta, number theory proposed, number theory



Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$? Proposed by Mahan Malihi