MK-MO 2009 Problem 4

Let $a,b,c$ be positive real numbers for which $ab+bc+ca=\frac{1}{3}$ . Prove the inequality $$\frac{a}{a^2-bc+1}+\frac{b}{b^2-ca+1}+\frac{c}{c^2-ab+1}\ge\frac{1}{a+b+c}$$