RU-ALLRUSSIAN 2009 Problem 95

Let $a$ , $b$ , $c$ be three real numbers satisfying that $$\left\{\begin{array}{c c c} \left(a+b\right)\left(b+c\right)\left(c+a\right)&=&abc \left(a^3+b^3\right)\left(b^3+c^3\right)\left(c^3+a^3\right)&=&a^3b^3c^3\end{array}\right.$$ Prove that $abc=0$ .