Let
![a,b,c](/media/m/3/6/4/36454fdb50fc50f021324b33a6b513e3.png)
be positive real numbers such that
![a+b+c=1](/media/m/d/b/8/db883e2d12c15d7b713305ba8c309cb3.png)
. Prove that
![\frac{a}{b} + \frac{a}{c} + \frac{c}{b} + \frac{c}{a} + \frac{b}{c} + \frac{b}{a} + 6
\geq 2\sqrt{2} \cdot \left( \sqrt{\frac{1-a}{a}}+\sqrt{\frac{1-b}{b}}+\sqrt{\frac{1-c}{c}} \right)](/media/m/f/5/2/f5221e438687c943a031617d8049a02e.png)
When does equality hold?
%V0
Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that $$
\frac{a}{b} + \frac{a}{c} + \frac{c}{b} + \frac{c}{a} + \frac{b}{c} + \frac{b}{a} + 6
\geq 2\sqrt{2} \cdot \left( \sqrt{\frac{1-a}{a}}+\sqrt{\frac{1-b}{b}}+\sqrt{\frac{1-c}{c}} \right)
$$ When does equality hold?