<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] Multiply through by $xyz$ and you get $$x^2 + y^2 + z^2 \geq xy + yz + zx.$$ Now note that $$(x-y)^2 + (y-z)^2 + (z-x)^2 \geq 0$$ with equality only if $x = y = z$.
</s>