<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 $$ a=\frac{x}{y} $$ $$ b=\frac{y}{z} $$ $$ c=\frac{z}{x} $$
Then $$abc=1$$
$$ a^2+b^2+c^2=a+b+c$$
which implies
$$ (a-0.5)^2+(b-0.5)^2+(c-0.5)^2=0.75$$
I was struck up here, i would appreciate if any one helps me here
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