Problem

Source: 69 Polish MO 2018 Second Round - Problem 6

Tags: algebra, analysis, Poland, Sequence



Let $k$ be a positive integer and $a_1, a_2, ...$ be a sequence of terms from set $\{ 0, 1, ..., k \}$. Let $b_n = \sqrt[n] {a_1^n + a_2^n + ... + a_n^n}$ for all positive integers $n$. Prove, that if in sequence $b_1, b_2, b_3, ...$ are infinitely many integers, then all terms of this series are integers.