Question 34
In the usual notation for a triangle $ABC$, prove that
$$
2 \sin \frac{A}{2} \le \frac{a}{\sqrt{bc}},
$$
with equality iff $c = b$. Deduce, or prove otherwise, that
$$
8 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \le 1,
$$
and
$$
\sin^2 \frac{A}{2} + \sin^2 \frac{B}{2} + \sin^2 \frac{C}{2} \le \frac{a^3 + b^3 + c^3}{4abc},
$$
with equality in both cases iff $ABC$ is equilateral.
$$
2 \sin \frac{A}{2} \le \frac{a}{\sqrt{bc}},
$$
with equality iff $c = b$. Deduce, or prove otherwise, that
$$
8 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \le 1,
$$
and
$$
\sin^2 \frac{A}{2} + \sin^2 \frac{B}{2} + \sin^2 \frac{C}{2} \le \frac{a^3 + b^3 + c^3}{4abc},
$$
with equality in both cases iff $ABC$ is equilateral.