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Sequence of sums of k-th powers

Source: Cono Sur 2023 P6

August 8, 2023
algebra

Problem Statement

Let x1,x2,…,xnx_1, x_2, \ldots, x_n be positive reals; for any positive integer kk, let Sk=x1k+x2k+…+xnkS_k=x_1^k+x_2^k+\ldots+x_n^k.
(a) Given that S1<S2S_1<S_2, show that S1,S2,S3,…S_1, S_2, S_3, \ldots is strictly increasing.
(b) Prove that there exists a positive integer nn and positive reals x1,x2,…,xnx_1, x_2, \ldots, x_n, such that S1>S2S_1>S_2 and S1,S2,S3,…S_1, S_2, S_3, \ldots is not strictly decreasing.
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