59th Ukrainian National Mathematical Olympiad

Due to inequality of arithmetic and geometric means applied several times: $$\begin{gathered} \left(x^4+\frac{1}{y^4}\right)+\left(y^4+\frac{1}{z^4}\right)+\left(z^4+\frac{1}{x^4}\right)\ge 2\cdot \left(\frac{x^2}{y^2}+\frac{y^2}{z^2}+\frac{z^2}{x^2}\right)= \\ =\left(\frac{x^2}{y^2}+\frac{y^2}{z^2}\right)+\left(\frac{y^2}{z^2}+\frac{z^2}{x^2}\right)+\left(\frac{z^2}{x^2}+\frac{x^2}{y^2}\right)\ge 2\cdot \left(\frac{x}{z}+\frac{y}{x}+\frac{z}{y}\right), \end{gathered}$$ this completes the proof.