CN-TST 2019 Problem 3

Find all positive integer $n$ , such that there exists $n$ points $P_1,\ldots,P_n$ on the unit circle , satisfying the condition that for any point $M$ on the unit circle, $\sum_{i=1}^n MP_i^k$ is a fixed value for a) $k=2018$ b) $k=2019$ .