IL-GILLIS 2022 Problem 6

Let $x,y,z$ be non-negative real numbers. Prove that:

$$\sqrt{(2x+y)(2x+z)}+\sqrt{(2y+x)(2y+z)}+\sqrt{(2z+x)(2z+y)}\geq$$ $$\geq \sqrt{(x+2y)(x+2z)}+\sqrt{(y+2x)(y+2z)}+\sqrt{(z+2x)(z+2y)}.$$