Regular n-gon and constant sum

[ins-]

It is given a regular $n$-gon $A_1...A_n$. On its
a) incircle
b) circumcircle
is chosen a random point $P$.
For which natural numbers $m$ $(m \geq 2)$ and $n$ $(n \geq 3)$ the following expression is a constant:
$X = PA_{1}^{m} + ... + PA_{n}^m$

[Dimoniada]

ins- в сообщении #431554 писал(а):

It is given a regular $n$-gon ${A_1}...{A_n}$ that inscribed in cicrcle $R$. On its
c) concentric circle $r$
is chosen a random point $P$.
Than for $k < n$, $k\in N$
$X = PA_{1}^{2k} + ... + PA_{n}^{2k} = n{(R^2+r^2)^k} \cdot\sum\limits_{l=0, even}^k (\frac {Rr} {R^2+r^2})^k\cdot {\frac {k!} {(k-l)!((l/2)!)^2}}$.

For odd powers $m<n$ there exist sequence of plus and minus that $X = -PA_{1}^{m} + PA_{2}^{m}... + (\mp)PA_{n}^{m} = 0$.

[ins-]

Thank you very much.
How did you got this result?
Is it very well known fact?

[Dimoniada]

(Оффтоп)

I'm never seen that result, to my mind it's hard problem for schoolchildren.
P.S.: Во втором утверждении пропустила, $n$ как и $m$ должно быть нечётным, иначе "не получится".

[ins-]

(Оффтоп)

It is not intended for schoolchilren if it is not TST or last round of NOM. What i like/dislike is the problem is not pure geometric. I think i saw similar Vietnamese proposal - the particular case (n=3, m=4) for IMO. I was just curious to see what more skilled mathematicians know about this. When I smoke I wonder such things