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Inequality with cot alpha, cot beta, cot gamma

Source: Moroccan MO 11th grade 5th exam 2011

April 24, 2011
inequalitiesgeometryperimetertrigonometryinequalities proposed

Problem Statement

Let α,β,γ\alpha , \beta ,\gamma be the angles of a triangle ABCABC of perimeter 2p 2p and RR is the radius of its circumscribed circle. (a)(a) Prove that cot⁡2α+cot⁡2β+cot⁡2γ≥3(9⋅R2p2−1).\cot^{2}\alpha +\cot^{2}\beta+\cot^{2}\gamma\geq 3\left(9\cdot \frac{R^{2}}{p^{2}} - 1\right). (b)(b) When do we have equality?
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