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\sum_{k=1}^n\frac{1}{\sin{k}\sin{k+1}}

Source: 2-nd Taiwanese Mathematical Olympiad 1993

January 13, 2007
trigonometryinductionirrational numberalgebra unsolvedalgebra

Problem Statement

Let mm be equal to 11 or 22 and n<10799n<10799 be a positive integer. Determine all such nn for which ∑k=1n1sin⁡ksin⁡(k+1)=msin⁡nsin⁡21\sum_{k=1}^{n}\frac{1}{\sin{k}\sin{(k+1)}}=m\frac{\sin{n}}{\sin^{2}{1}}.
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