<s>[INST] Inequality $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq a + b+ c$ when $abc = 1$ 
If $a,b,c > 0$ are such that $abc=1$, then
  $$ \frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq a + b+ c. $$

I would be pleased if you give me a hint. Thanks in advance.
 [/INST] $$a+b+c =$$
$$\{ \mbox{multiply by 1: maybe } abc\mbox{, maybe }\sqrt{abc}, ... \}$$ 

  $$= \frac{a}{\sqrt[3]{abc}}+\frac{b}{\sqrt[3]{abc}}+\frac{c}{\sqrt[3]{abc}}=$$

$$ 
=\sqrt[3]{\frac{a}{b} \cdot \frac{a}{b} \cdot \frac{b}{c}}
+\sqrt[3]{\frac{b}{c} \cdot \frac{b}{c} \cdot \frac{c}{a}}
+\sqrt[3]{\frac{c}{a} \cdot \frac{c}{a} \cdot \frac{a}{b}}
\le
$$
$$
\{\mbox{ GM}\le\mbox{AM }\} 
$$
$$
\le \frac{\frac{a}{b}+\frac{a}{b}+\frac{b}{c}}{3} 
+ \frac{\frac{b}{c}+\frac{b}{c}+\frac{c}{a}}{3}
+ \frac{\frac{c}{a}+\frac{c}{a}+\frac{a}{b}}{3} = 
$$
$$
=\frac{a}{b}+\frac{b}{c}+\frac{c}{a}.
$$
</s>