Problem

Source: ELMO 2003 Problem 3

Tags: invariant, number theory unsolved, number theory



Let $k$ be a positive integer for which the equation \[ 2ab+2bc+2ca-a^2-b^2-c^2 = k \] has some solution in positive integers $a,b,c$. Prove that the equation has a solution for which $a$, $b$ and $c$ are the sides of a possibly degenerate triangle.