Question 51
Let $ABC$ be an equilateral triangle, and let $P$ be some point in its circumcircle. Determine, with reasons, all the positive integers $n$ such that the sum
$$
S_n P = |PA|^n + |PB|^n + |PC|^n
$$
is independent of the choice of the point $P$.
$$
S_n P = |PA|^n + |PB|^n + |PC|^n
$$
is independent of the choice of the point $P$.