2022/022) Find all solutions in positive integers x, y, z of the equation $\frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 3$

We have by AM GM inequaity among $\frac{x}{y}$ ,$\frac{y}{z}$, $\frac{z}{x}$

we have 

$\frac{\frac{x}{y} + \frac{y}{z} + \frac{z}{x}}{3} >= \sqrt[2]{\frac{x}{y} * \frac{y}{z} * \frac{z}{x}}$

 or  $\frac{\frac{x}{y} + \frac{y}{z} + \frac{z}{x}}{3} >= \sqrt[2]{1}$

or  $\frac{x}{y} + \frac{y}{z} + \frac{z}{x}>=  3$

this is greater than 3 if all are not equal and is 3 if all are equal

or  $\frac{x}{y} = \frac{y}{z} = \frac{z}{x}$

or x = y = z (any integer)