BALTIC-WAY 2011 Problem 4

Let $a,b,c,d$ be non-negative reals such that $a+b+c+d=4$ . Prove the inequality $$\frac{a}{a^3+8}+\frac{b}{b^3+8}+\frac{c}{c^3+8}+\frac{d}{d^3+8}\le\frac{4}{9}$$