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Sequence of roots

Source: 69 Polish MO 2018 Second Round - Problem 6

April 28, 2018
algebraanalysisPolandSequence

Problem Statement

Let kk be a positive integer and a1,a2,...a_1, a_2, ... be a sequence of terms from set {0,1,...,k}\{ 0, 1, ..., k \}. Let bn=a1n+a2n+...+annnb_n = \sqrt[n] {a_1^n + a_2^n + ... + a_n^n} for all positive integers nn. Prove, that if in sequence b1,b2,b3,...b_1, b_2, b_3, ... are infinitely many integers, then all terms of this series are integers.
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