Find all integer numbers $a,b,c$ such that $\frac{(a+b)(b+c)(c+a)}{2}+(a+b+c)^{3}=1-abc$.
Problem
Source: 6-th Hong Kong Mathematical Olympiad 2003
Tags: number theory unsolved, number theory
Source: 6-th Hong Kong Mathematical Olympiad 2003
Tags: number theory unsolved, number theory
Find all integer numbers $a,b,c$ such that $\frac{(a+b)(b+c)(c+a)}{2}+(a+b+c)^{3}=1-abc$.