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Find extremes of \sum x_i

Source: Chinese National Competition 2001

January 28, 2005
inequalitiesinequalities proposed

Problem Statement

If nonnegative reals x1,x2,…,xnx_1, x_2, \ldots, x_n satisfy ∑i=1nxi2+2∑1≤k<j≤nkjxkxj=1 \sum_{i=1}^n x_i^2 + 2\sum_{1 \leq k < j \leq n} \sqrt{\frac{k}{j}}x_kx_j = 1 what are the minimum and maximum values of ∑i=1nxi\sum_{i=1}^n x_i?
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