58th Ukrainian National Mathematical Olympiad

From known identity $$x^3+y^3+z^3-3xyz=(x+y+z)\cdot \frac{(x-y{)}^2+(y-z{)}^2+(z-x{)}^2}{2}.$$

From inequality between $n$ arithmetic mean and root-means-square we have, that $$\begin{array}{rl}(x-y{)}^2+(y-z{)}^2+(z-x{)}^2& \ge \frac{1}{3}(|x-y|+|y-z|+|z-x|{)}^2\ge \\ & \ge \frac{1}{3}(|x-y|+|y-z|+|x-z|{)}^2=\frac{4}{3}(x-z{)}^2.\end{array}$$

From this follows the inequality needed.