JBMO-SL 2014 A8

Let $x, y$ and $z$ be positive real numbers such that $x y z=1$. Prove the inequality

$$\frac{1}{x(a y+b)}+\frac{1}{y(a z+b)}+\frac{1}{z(a x+b)} \geq 3 \text {, if: }$$

a) $a=0$ and $b=1$; b) $a=1$ and $b=0$; c) $a+b=1$ for $a, b>0$

When does the equality hold true?