CMO 2012 Problem 5

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