MK-MO 2008 Problem 2

Positive numbers $a$ , $b$ , $c$ are such that $\left(a + b\right)\left(b + c\right)\left(c + a\right) = 8$ . Prove the inequality $$\frac {a + b + c}{3}\ge\sqrt [27]{\frac {a^3 + b^3 + c^3}{3}}$$