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Inequality with sin θ

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September 6, 2010
inequalitiestrigonometrymodular arithmeticinequalities unsolved

Problem Statement

Let θ1,θ2,⋯ ,θn\theta_1,\theta_2,\cdots,\theta_n be nn real numbers such that sin⁡θ1+sin⁡θ2+⋯+sin⁡θn=0\sin \theta_1+\sin \theta_2+\cdots+\sin \theta_n=0. Prove that ∣sin⁡θ1+2sin⁡θ2+⋯+nsin⁡θn∣≤[n24]|\sin \theta_1+2 \sin \theta_2+\cdots +n \sin \theta_n| \leq \left[ \frac{n^2}{4} \right]
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