Number theory · Quadratic residues · Divisibility · Remainders · Pigeonhole
Let \(k\), \(m\) and \(n\) be natural numbers, none of which is divisible by \(5\). Prove that at least one of the three numbers
\(k^2 - m^2, \qquad m^2 - n^2, \qquad n^2 - k^2\)
is divisible by \(5\).
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), drzavno (national) round 2001, 1. letnik, category A, problem 1. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source