[poper]

$\frac{A}{C}+\frac{B}{A}+\frac{C}{B}+\frac{B}{C}+\frac{A}{B}+\frac{C}{A}=\frac{A+B}{C}+\frac{B+C}{A}+\frac{A+C}{B}$
$=\frac{1-C}{C}+\frac{1-A}{A}+\frac{1-B}{B}$
$=\frac{1}{A}+\frac{1}{B}+\frac{1}{C}-3$
$=9-3=6$