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Sum of differences of fourth and fifth powers

Source: China TST 1999, problem 1

May 22, 2005
inequalitiessmoothingChina TST

Problem Statement

For non-negative real numbers x1,x2,…,xnx_1, x_2, \ldots, x_n which satisfy x1+x2+⋯+xn=1x_1 + x_2 + \cdots + x_n = 1, find the largest possible value of ∑j=1n(xj4−xj5)\sum_{j = 1}^{n} (x_j^{4} - x_j^{5}).
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