Track / Stage 5 / 129 of 400 #729 of 2000

4. In $\triangle A B C$, prove that: $\frac{1}{\sin \frac{A}{2}}+\frac{1}{\sin \frac{B}{2}}+\frac{1}{\sin \frac{C}{2}} \geqslant 6$.

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

I solved it I didn't Skip

Official solution

4. $y=\frac{1}{\sin x}$ is a convex function in $\left(0, \frac{\pi}{2}\right)$, so the left side $\geqslant \frac{3}{\sin \frac{A+B+C}{6}}=6$.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.