$\frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 10 \to {(\frac{x}{y} + \frac{y}{z} + \frac{z}{x})^2} = {10^2}$
اتحاد مربع سهجملهای $ \to \frac{{{x^2}}}{{{y^2}}} + \frac{{{y^2}}}{{{z^2}}} + \frac{{{z^2}}}{{{x^2}}} + 2(\frac{x}{y}.\frac{y}{z} + \frac{x}{y}.\frac{z}{x} + \frac{y}{z} + \frac{z}{x}) = 100$
$ \to \frac{{{x^2}}}{{{y^2}}} + \frac{{{y^2}}}{{{z^2}}} + \frac{{{z^2}}}{{{x^2}}} + 2(\frac{x}{z} + \frac{z}{y} + \frac{y}{x}) = 100$
$ \to \frac{{{x^2}}}{{{y^2}}} + \frac{{{y^2}}}{{{z^2}}} + \frac{{{z^2}}}{{{x^2}}} = 100 - 22 = 78$