Prove that $\sum\limits_{cyc}\sqrt{a^2+10bc}\geq\frac{1}{2}\sum\limits_{cyc}\sqrt{22a(b+c)}$

Let $a$, $b$ and $c$ be non-negative numbers. Prove that: $$\sum\limits_{cyc}\sqrt{a^2+10bc}\geq\frac{1}{2}\sum\limits_{cyc}\sqrt{22a(b+c)}$$ I tried SOS, C-S, Holder, Mixing Variables and more, but without success.