ELMO SL 2010 A6

For all positive real numbers $a,b,c$ , prove that $$\sqrt{\frac{a^4 + 2b^2c^2}{a^2+2bc}} + \sqrt{\frac{b^4+2c^2a^2}{b^2+2ca}} + \sqrt{\frac{c^4 + 2a^2b^2}{c^2 + 2ab}} \geq a + b + c.$$

In-Sung Na.