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Part of 1975 IMO Shortlist

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Inequality on trigonometric circle the arcs - ISL 1975

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9/21/2010
Consider on the first quadrant of the trigonometric circle the arcs AM1=x1,AM2=x2,AM3=x3,…,AMv=xvAM_1 = x_1,AM_2 = x_2,AM_3 = x_3, \ldots , AM_v = x_v , such that x1<x2<x3<⋯<xvx_1 < x_2 < x_3 < \cdots < x_v. Prove that ∑i=0v−1sin⁡2xi−∑i=0v−1sin⁡(xi−xi+1)<π2+∑i=0v−1sin⁡(xi+xi+1)\sum_{i=0}^{v-1} \sin 2x_i - \sum_{i=0}^{v-1} \sin (x_i- x_{i+1}) < \frac{\pi}{2} + \sum_{i=0}^{v-1} \sin (x_i + x_{i+1})
trigonometryTrigonometric inequalityTrigonometric IdentitiesInequalityoptimizationIMO Shortlist