Problem

Source: Indian Postal Coaching 2007 set 3 p1

Tags: Sum, geometry, constant, circumcircle, square



Let $P$ be a point on the circumcircle of a square $ABCD$. Find all integers $n > 0$ such that the sum $$S_n(P) = |PA|^n + |PB|^n + |PC|^n + |PD|^n$$is constant with respect to the point $P$.