JBMO 2012 Problem 1

Let $a, b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove that

$$\frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{c}{a}+\frac{a}{c}+6 \geq 2 \sqrt{2}\left(\sqrt{\frac{1-a}{a}}+\sqrt{\frac{1-b}{b}}+\sqrt{\frac{1-c}{c}}\right)$$

When does equality hold?