# 2015-10 Product of sine functions

Let $$w_1,w_2,\ldots,w_n$$ be positive real numbers such that $$\sum_{i=1}^n w_i=1$$. Prove that if $$x_1,x_2,\ldots,x_n\in [0,\pi]$$, then $\sin \left(\prod_{i=1}^n x_i^{w_i} \right) \ge \prod_{i=1}^n (\sin x_i)^{w_i}.$

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# 2012-7 Product of Sine

Let X be the set of all postive real numbers c such that  $\frac{\prod_{k=1}^{n-1} \sin\left( \frac{k \pi}{2n}\right)}{c^n}$  converges as n goes to infinity. Find the infimum of X.

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Evaluate the sum $\sum_{n=1}^{\infty} \frac{n \sin n}{1+n^2}.$
(UPDATED: 2011.2.18) I have fixed a typo in the formula. Initially the following formula $\sum_{n=1}^{\infty} \frac{\sin n}{1+n^2}$ was posted but it does not seem to have a closed form answer. I’m sincerely sorry!