Putnam 1978 A5

Let $0 < x_i < \pi$ for $i=1,2,\ldots, n$ and set $$x= \frac{ x_1 +x_2 + \ldots+ x_n }{n}.$$ Prove that $$\prod_{i=1}^{n} \frac{ \sin x_i }{x_i } \leq \left( \frac{ \sin x}{x}\right)^{n}.$$