[unsolved] Gooood but more difficult!



  • Let x1,x2,,xnx_1,x_2,\cdots,x_n (n2)(n\geq2) be a non-decreasing monotonous sequence of positive numbers such that x1,x22,,xnnx_1,\frac{x_2}{2},\cdots,\frac{x_n}{n} is a non-increasing monotonous sequence .Prove that
    i=1nxin(i=1nxi)1nn+12n!n \frac{\sum_{i=1}^{n} x_i }{n\left (\prod_{i=1}^{n}x_i \right )^{\frac{1}{n}}}\le \frac{n+1}{2\sqrt[n]{n!}}


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