doturitip9

2022-07-07

Tricky inequality no avail to AM-GM
Let $a,b,c$ be 3 distinct positive real numbers such that abc = 1. Prove that

I tried AM-GM in many different ways, but it doesn't work since one of the terms on the LHS inevitably becomes negative. Any help is greatly appreciated.

Jordan Mcpherson

Expert

By AM-GM
$\sum _{cyc}\frac{{a}^{3}}{\left(a-b\right)\left(a-c\right)}=-\sum _{cyc}\frac{{a}^{3}}{\left(a-b\right)\left(c-a\right)}=-\sum _{cyc}\frac{{a}^{3}\left(b-c\right)}{\prod _{cyc}\left(a-b\right)}=a+b+c\ge 3$

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