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\sum \sum ij \cos ( \alpha_i - \alpha_j ) \geq 0

Source: Romanian IMO Team Selection Test TST 1987, problem 9

September 25, 2005
trigonometryinequalities proposedinequalities

Problem Statement

Prove that for all real numbers α1,α2,…,αn\alpha_1,\alpha_2,\ldots,\alpha_n we have ∑i=1n∑j=1nijcos⁡(αi−αj)≥0. \sum_{i=1}^n \sum_{j=1}^n ij \cos {(\alpha_i - \alpha_j )} \geq 0. Octavian Stanasila
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