ELMO SL 2012 A2

Let $a,b,c$ be three positive real numbers such that $a \le b \le c$ and $a+b+c=1$. Prove that $$\frac{a+c}{\sqrt{a^2+c^2}}+\frac{b+c}{\sqrt{b^2+c^2}}+\frac{a+b}{\sqrt{a^2+b^2}} \le \frac{3\sqrt{6}(b+c)^2}{\sqrt{(a^2+b^2)(b^2+c^2)(c^2+a^2)}}.$$

Owen Goff.