9. [25] Let $A_{1} A_{2} \ldots A_{k}$ be a regular $k$-gon inscribed in a circle of radius 1 , and let $P$ be a point lying on or inside the circumcircle. Find the maximum possible value of $\left(P A_{1}\right)\left(P A_{2}\right) \cdots\left(P A_{k}\right)$.
Solution: Place the vertices at the $k$ th roots of unity, $1, \omega, \ldots, \omega^{k-1}$, and place $P$ at some complex number $p$. Then
$$
\begin{aligned}
\left(\left(P A_{1}\right)\left(P A_{2}\right) \cdots\left(P A_{k}\right)\right)^{2} & =\prod_{i=0}^{k-1}\left|p-\omega^{i}\right|^{2} \\
& =\left|p^{k}-1\right|^{2},
\end{aligned}
$$
since $x^{k}-1=(x-1)(x-\omega) \cdots\left(x-\omega^{k-1}\right)$. This is maximized when $p^{k}$ is as far as possible from 1, which occurs when $p^{k}=-1$. Therefore, the maximum possible value of $\left(P A_{1}\right)\left(P A_{2}\right) \cdots\left(P A_{k}\right)$ is 2 .