TUYMAADA 2012 S7

Prove that for any real numbers $a,b,c$ satisfying $abc = 1$ the following inequality holds $$\dfrac{1} {2a^2+b^2+3}+\dfrac {1} {2b^2+c^2+3}+\dfrac{1} {2c^2+a^2+3}\leq \dfrac {1} {2}.$$

Proposed by V. Aksenov