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Inequality over integers

Source: Saint Petersburg MO 2020 Grade 10 Problem 4

May 7, 2020
inequalitiesnumber theory

Problem Statement

Let mm be a given positive integer. Prove that there exists a positive integer kk such that it holds 1≤1m+2m+3m+…+(k−1)mkm<2.1\leq \frac{1^m+2^m+3^m+\ldots +(k-1)^m}{k^m}<2.
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