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Trigonometric inequality - ILL 1966

Source:

September 25, 2010

inequalitiestrigonometrygeometric inequalityIMO ShortlistIMO Longlist

Problem Statement

Prove the inequality

\tan \frac{\pi \sin x}{4\sin \alpha} + \tan \frac{\pi \cos x}{4\cos \alpha} >1

for any

x, \alpha

with

0 \leq x \leq \frac{\pi }{2}

and

\frac{\pi}{6} < \alpha < \frac{\pi}{3}.