We have
LHS = abc(\frac{a}{c} + \frac{a}{b}) + abc(\frac{b}{c} + \frac{b}{a})+ abc(\frac{c}{b} + \frac{c}{a})
= abc(\frac{a}{c} + \frac{a}{b} + \frac{b}{c} + \frac{b}{a} + \frac{c}{b} + \frac{c}{a})
= abc(\frac{a}{c} + \frac{c}{a} + \frac{b}{c} + \frac{c}{b} + \frac{a}{b} + \frac{b}{a})
as \frac{a}{c} + \frac{c}{a} \ge 2
\frac{b}{c} + \frac{c}{b} \ge 2
\frac{a}{b} + \frac{b}{a} \ge 2
The above 3 relations follow from am gm inequality
putting the value we get
LHS >= abc(2+2+2) or LHS >= 6abc
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