socrates wrote: Let $ABC$ be an equilateral triangle, and let $P$ be some point in its circumcircle. Determine all positive integers $n$, for which the value of the sum $S_n (P) = |PA|^n + |PB|^n + |PC|^n$ is independent of the choice of point $P$. Very nice @socrates sir. Is it your creation ?

It's from Mediterranean Mathematics Olympiad 2017.

If we set $D$ - midpoint of arc $BC$ then $S_n(D)=(\frac{2\sqrt{3}a}{3})^n+2*(\frac{\sqrt{3}a}{3})^n$ where $a$ - length of $AB$ $S_n(A)=2a^n$ So we need $2a^n=(\frac{2\sqrt{3}a}{3})^n+2*(\frac{\sqrt{3}a}{3})^n$ or $2=(\frac{\sqrt{3}}{3})^n(2+2^n)$ or $(\sqrt{3})^n =2^{n-1}+1$ Solutions are $n=2,4$ So we need to check only $n=2$ and $4$

$S_4(P)=18R^4$ $S_2(P)=6R^2$ https://artofproblemsolving.com/community/c6h133115p753947