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For $x_i\ge0,i=1,2,...,n$ satisfying

$\sum_{i=1}^nx_i^2+2\sum_{1\le k <j\le n}\sqrt{\frac{k}{j}}x_kx_j=1$,

find the maximum and minimum of $\sum_{i=1}^n x_i.$

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1 Answer

Actually the minimum is pretty trivial:

$(\sum x_i)^2\ge \sum_{i=1}^{n}x_{i}^{2}+2\sum_{1\le k <j\le n}\sqrt{\frac{k}{j}}x_{k}x_{j}=1 $. For equality can we take $x_1=1$ and $x_i=0$ for $i>1$.

The maximum is much interesting.

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