<s>[INST] Prove That $x=y=z$ If $x, y,z \in \mathbb{R}$,
and if
$$ \left ( \frac{x}{y} \right )^2+\left ( \frac{y}{z} \right )^2+\left ( \frac{z}{x} \right )^2=\left ( \frac{x}{y} \right )+\left ( \frac{y}{z} \right )+\left ( \frac{z}{x} \right ) $$
Prove that $$x=y=z$$
 [/INST] Let $a=\frac{x}{y}, b=\frac{y}{z}, c=\frac{z}{x}$, and let
$$
G=\sqrt[3]{abc}, A = \frac{a+b+c}{3}, Q=\sqrt{\frac{a^2+b^2+c^2}{3}}
$$
Then the given condition is $Q^2=A$, but by the power mean inequality
$$
Q^2\ge Q \ge A \ge G = 1
$$
with equality in each case only if $a=b=c=1$, i.e. if $x=y=z$.
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