59th Ukrainian National Mathematical Olympiad

The inequality can be rewritten as follows: $$\begin{array}{rl}& (x^4+\frac{1}{z^4}+z^4+\frac{1}{y^4})+(y^4+\frac{1}{t^4}+t^4+\frac{1}{z^4})+(z^4+\frac{1}{x^4}+x^4+\frac{1}{t^4})+(t^4+\frac{1}{y^4}+y^4+\frac{1}{x^4})\\ & \ge 4\cdot \left(\frac{x}{y}+\frac{y}{z}+\frac{z}{t}+\frac{t}{x}\right).\end{array}$$

$(x^4+\frac{1}{z^4})+(z^4+\frac{1}{y^4})\ge 2\sqrt{x^4\cdot \frac{1}{z^4}}+2\sqrt{z^4\cdot \frac{1}{y^4}}=\frac{2x^2}{z^2}+\frac{2z^2}{y^2}\ge 4\sqrt{\frac{x^2}{z^2}\cdot \frac{z^2}{y^2}}=\frac{4x}{yz},$ that finishes the proof.