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#### Pranav

##### Well-known member

- Nov 4, 2013

- 428

In $\Delta$ABC, $\displaystyle \cos \left( \frac{A}{2} \right) \cos \left( \frac{B}{2} \right) \cos \left( \frac{C}{2} \right)\leq \frac{1}{4}$, then greatest angle of triangle

A)lies in $\left(0,\frac{\pi}{2}\right)$

B)lies in $\left(\frac{2\pi}{3},\frac{5\pi}{6}\right)$

C)lies in $\left(\frac{5\pi}{6},\pi\right)$

D)lies in $\left(\frac{\pi}{2},\frac{2\pi}{3}\right)$

Attempt:

I haven't been able to proceed anywhere with this problem. I could only simplify the given inequality to

$$\sin A+\sin B+\sin C\leq 1$$

(The above can be proved by using $C=\pi-(A+B)$)

But I am not sure if the above helps.

Any help is appreciated. Thanks!