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55

Part of 1988 IMO Longlists

Problems(1)

IMO LongList 1988 Trignometric Inequality

Source: IMO LongList 1988, Mongolia 5, Problem 55 of ILL

11/3/2005
Suppose αi>0,βi>0\alpha_i > 0, \beta_i > 0 for 1≤i≤n,n>11 \leq i \leq n, n > 1 and that ∑i=1nαi=∑i=1nβi=π. \sum^n_{i=1} \alpha_i = \sum^n_{i=1} \beta_i = \pi. Prove that ∑i=1ncos⁡(βi)sin⁡(αi)≤∑i=1ncot⁡(αi). \sum^n_{i=1} \frac{\cos(\beta_i)}{\sin(\alpha_i)} \leq \sum^n_{i=1} \cot(\alpha_i).
inequalitiestrigonometryinductionrearrangement inequalityalgebra unsolvedalgebra