New posts in Muirhead Inequality

Prove that $ a^2+b^2+c^2 \le a^3 +b^3 +c^3 $

A hard inequality indian olympiad problem

How can we not use Muirhead's Inequality for proving the following inequality?

Prove this inequality: $\frac{b+c}{\sqrt{a}}+\frac{c+a}{\sqrt{b}}+\frac{a+b}{\sqrt{c}}\geq \sqrt{a}+\sqrt{b}+\sqrt{c}+3$