Prove that $\frac{x^5-x^2}{x^5+y^2+z^2}+\frac{y^5-y^2}{x^2+y^5+z^2}+\frac{z^5-z^2}{x^2+y^2+z^5}≥0 $. Given $x, y, z $ are 3 positive reals such that $xyz≥1$. Prove that
$$\frac{x^5-x^2}{x^5+y^2+z^2}+\frac{y^5-y^2}{x^2+y^5+z^2}+\frac{z^5-z^2}{x^2+y^2+z^5}≥0.$$
This question is so complicated. I failed many times to get the proof. Can anyone help me please? Thank you.

This is problem №3 from IMO 2005. Here you can find its solution.
