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
$$A\geq \frac{3}{\sqrt[3]{xyz}}=\frac{3}{\sqrt[3]{x+y+z}}.$$

Let $x_1 = 1/x, y_1 = 1/y, z_1 = 1/z$. Then $x_1y_1 + x_1z_1 + y_1z_1 = 1$.
$$
  \sum_{cyc} \frac{x}{y^2} \geq \sum_{cyc} \frac1x = x_1 + y_1 + z_1
$$
Due to rearrangement inequality $(x_1 + y_1 + z_1)^2 \geq 3(x_1y_1 + x_1z_1 + y_1z_1) = 3$. So
$$
x_1 + y_1 + z_1 \geq \sqrt{3}
$$
