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Sum of distances power k is constant on unit circle

Source: 2019 China TST Day 1 Q3

March 5, 2019
algebra

Problem Statement

Find all positive integer nn, such that there exists nn points P1,…,PnP_1,\ldots,P_n on the unit circle , satisfying the condition that for any point MM on the unit circle, ∑i=1nMPik\sum_{i=1}^n MP_i^k is a fixed value for \\a) k=2018k=2018 \\b) k=2019k=2019.
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