Trig Revolution 2020

By sine rule

$\huge\ \sum { \frac { \sqrt { \sin { A } } }{ \sqrt { \sin { B } } +\sqrt { \sin { C } } -\sqrt { \sin { A } } } =\sum { \frac { \sqrt { a } }{ \sqrt { b } +\sqrt { c } -\sqrt { a } } } }$ .

Now, let

$\large\ {x=\sqrt { b } +\sqrt { c } -\sqrt { a } \\ y=\sqrt { c } +\sqrt { a } -\sqrt { b } \\ z=\sqrt { a } +\sqrt { b } -\sqrt { c }}$ .

Substituting it the inequality to be solved becomes

$\huge\ \sum { \frac { y+z }{ 2x } }$ , whose minimum value by AM-GM is $3$ .

Thus,,

$\huge\ 2020\sum { \frac { \sqrt { \sin { A } } }{ \sqrt { \sin { B } } +\sqrt { \sin { C } } -\sqrt { \sin { A } } } } \ge \left( 2020\times 3 \right) =\boxed { 6060 }$