<s>[INST] Prove that $ \frac{x}{yz} + \frac{y}{xz} + \frac{z}{xy} \geq \frac{1}{x} + \frac{1}{y} + \frac{1}{z} $ Prove that $ \frac{x}{yz} + \frac{y}{xz} + \frac{z}{xy}  \geq  \frac{1}{x} + \frac{1}{y} + \frac{1}{z} $ for $x,y,z \in \Bbb{R}$ and $xyz > 0 $.
I know that i can use the axioms of the real numbers, but i can't finde an usefull equivalent transformation 
 [/INST] By C-S
$$\sum_{cyc}\frac{x}{yz}=\sum_{cyc}\frac{x^2}{xyz}\geq\frac{(x+y+z)^2}{3xyz}\geq\frac{3(xy+xz+yz)}{3xyz}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$
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