Problem

Source: 2012cmo,problem5

Tags: modular arithmetic, inequalities, number theory proposed, number theory



Consider a square-free even integer $n$ and a prime $p$, such that 1) $(n,p)=1$; 2) $p\le 2\sqrt{n}$; 3) There exists an integer $k$ such that $p|n+k^2$. Prove that there exists pairwise distinct positive integers $a,b,c$ such that $n=ab+bc+ca$. Proposed by Hongbing Yu