problem: a,b,c>0 and $abc=1$ prove that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq a+b+c$
my attempt:
$LHS=\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=(a+b+c) \frac{\frac{a}{b}+\frac{b}{c}+\frac{c}{a}}{a+b+c}$
From weighted AM-GM;
$LHS\geq (a+b+c) ((\frac{1}{b})^a(\frac{1}{c})^b(\frac{1}{a})^c)^{\frac{1}{a+b+c}}$
and we have :
$ ((\frac{1}{b})^a(\frac{1}{c})^b(\frac{1}{a})^c)^{\frac{1}{a+b+c}}\geq 1$
because: if $a\leq b \leq c $ :then $((\frac{1}{b})^a(\frac{1}{c})^b(\frac{1}{a})^c)\geq ((\frac{1}{b})^a(\frac{1}{c})^a(\frac{1}{a})^a)=(\frac{1}{abc})^a=1$
and if $c\leq a \leq b $ :then $((\frac{1}{b})^a(\frac{1}{c})^b(\frac{1}{a})^c)\geq ((\frac{1}{b})^c(\frac{1}{c})^c(\frac{1}{a})^c)=(\frac{1}{abc})^c=1$.and the same for others case
so finally:
$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=(a+b+c) \frac{\frac{a}{b}+\frac{b}{c}+\frac{c}{a}}{a+b+c}\geq a+b+c.$
question:
-does my attempt is true?