BALTIC-WAY 2008 Problem 2

Prove that if the real numbers $a, b$ and $c$ satisfy $a^{2}+b^{2}+c^{2}=3$ then

$$\frac{a^{2}}{2+b+c^{2}}+\frac{b^{2}}{2+c+a^{2}}+\frac{c^{2}}{2+a+b^{2}} \geq \frac{(a+b+c)^{2}}{12} .$$

When does equality hold?