Question Number 169271 by Shrinava last updated on 27/Apr/22
Answered by mr W last updated on 27/Apr/22
$${a}={r}\:\mathrm{cot}\:\frac{{B}}{\mathrm{2}}+{r}\:\mathrm{cot}\:\frac{{C}}{\mathrm{2}} \\ $$$$\frac{{a}}{{r}}=\mathrm{cot}\:\frac{{B}}{\mathrm{2}}+\mathrm{cot}\:\frac{{C}}{\mathrm{2}}\geqslant\mathrm{2}\left(\mathrm{cot}\:\frac{{B}}{\mathrm{2}}\:\mathrm{cot}\:\frac{{C}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$$\left(\frac{{a}}{{r}}\right)^{\mathrm{6}} \geqslant\mathrm{2}^{\mathrm{6}} ×\left(\mathrm{cot}\:\frac{{B}}{\mathrm{2}}\:\mathrm{cot}\:\frac{{C}}{\mathrm{2}}\right)^{\mathrm{3}} \\ $$$$\Sigma\left(\frac{{a}}{{r}}\right)^{\mathrm{6}} \geqslant\mathrm{2}^{\mathrm{6}} ×\Sigma\left(\mathrm{cot}\:\frac{{B}}{\mathrm{2}}\:\mathrm{cot}\:\frac{{C}}{\mathrm{2}}\right)^{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\geqslant\mathrm{2}^{\mathrm{6}} ×\mathrm{3}×\left(\mathrm{cot}\:\frac{{A}}{\mathrm{2}}\:\mathrm{cot}\:\frac{{B}}{\mathrm{2}}\:\mathrm{cot}\:\frac{{C}}{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\geqslant\mathrm{2}^{\mathrm{6}} ×\mathrm{3}×\mathrm{3}^{\mathrm{3}} =\mathrm{5184}\:\checkmark \\ $$
Commented by Shrinava last updated on 27/Apr/22
$$\mathrm{Thankyou}\:\mathrm{dear}\:\mathrm{sir}\:\mathrm{perfect} \\ $$