[Wayu]

Let a,b,c be positive real numbers.Prove that
$\sqrt{\frac{a^2+b^2}{a+b}+\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}}$$\geqslant$$\sqrt{\frac{2ab}{3a+b+2c}}$+$\sqrt{\frac{2bc}{3b +c+2a}}$+$\sqrt{\frac{2ca}{3c+a+2b}}$.