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Trigonometry, arithmetic means and the greek alphabet.

Source: Baltic Way 1998

January 11, 2011
trigonometryinequalities proposedinequalities

Problem Statement

Let the numbers α,β\alpha ,\beta satisfy 0<α<β<π20<\alpha <\beta <\frac{\pi}{2} and let γ\gamma and δ\delta be the numbers defined by the conditions:
(i) 0<γ<π2(\text{i})\ 0<\gamma<\frac{\pi}{2}, and tan⁡γ\tan\gamma is the arithmetic mean of tan⁡α\tan\alpha and tan⁡β\tan\beta;
(ii) 0<δ<π2(\text{ii})\ 0<\delta<\frac{\pi}{2}, and 1cos⁡δ\frac{1}{\cos\delta} is the arithmetic mean of 1cos⁡α\frac{1}{\cos\alpha} and 1cos⁡β\frac{1}{\cos\beta}.
Prove that γ<δ\gamma <\delta .
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