<s>[INST] Minimum value of $\frac{x}{y^2}+\frac{y}{z^2}+\frac{z}{x^2}$ Let $x,y,z>0$ and $x+y+z=xyz$. What is the minimum value of $$\frac{x}{y^2}+\frac{y}{z^2}+\frac{z}{x^2}?$$
In the case when $x=y=z$, the equation $x+y+z=xyz$ translates to $3x=x^3$, or $x=\sqrt{3}$. If $A$ denotes the quantity that we want to minimize, then $A=\sqrt{3}$ as well. 
If we use the inequality of arithmetic and geometric means, we get
$$A\geq \frac{3}{\sqrt[3]{xyz}}=\frac{3}{\sqrt[3]{x+y+z}}.$$
 [/INST] The constraint allows the substitution $x=\tan A, y = \tan B, z = \tan C$ for some acute triangle $\triangle ABC$.  Further from rearrangement inequality we have:
$$\sum_{cyc} \frac{x}{y^2} \geqslant \sum_{cyc} \frac1x = \sum_{cyc} \cot A$$
Now $x \mapsto \cot x$ is convex for $x \in (0, \frac{\pi}2)$, so we may use Jensen to conclude
$$\sum_{cyc} \cot A \geqslant 3 \cot \frac{\pi}3$$
Equality is when $x=y=z$.
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