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Prove this inequality holds for all triangle

Source: 2009 Jozsef Wildt International Math Competition

April 27, 2020
inequalitiesgeometrytrigonometry

Problem Statement

Prove that for all triangle △ABC\triangle ABC holds the following inequality ∑cyc(1−3tan⁡A2+3tan⁡A2)(1−3tan⁡B2+3tan⁡B2)≥3\sum \limits_{cyc} \left (1-\sqrt{\sqrt{3}\tan \frac{A}{2}}+\sqrt{3}\tan \frac{A}{2}\right )\left (1-\sqrt{\sqrt{3}\tan \frac{B}{2}}+\sqrt{3}\tan \frac{B}{2}\right )\geq 3
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