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Nice geometric inequality of IMO LongList 1977

Source:

January 11, 2011
inequalitiesgeometrytrigonometryarea of a triangleHeron's formulageometry proposed

Problem Statement

Prove that for every triangle the following inequality holds: ab+bc+ca4S≥cot⁡π6.\frac{ab+bc+ca}{4S} \geq \cot \frac{\pi}{6}. where a,b,ca, b, c are lengths of the sides and SS is the area of the triangle.
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