. A regular polygon $\mathcal{P}$ with $n$ vertices is inscribed in a disk $D$. Determine the points of $D$ for which the product of the distances to the vertices of $\mathcal{P}$ is maximal.
. In the complex plane, by similarity, we bring the vertices of the polygon to the $n$-th roots of unity $e^{2 i k \pi / n}$, where $0 \leqslant k \leqslant n-1$. The product of the distances from the point with affix $z$ to the vertices of $\mathcal{P}$ is

$$
\prod_{k=0}^{n-1}\left|z-e^{2 i k \pi / n}\right|=\left|z^{n}-1\right|
$$

The maximum is obtained when $z^{n}=-1$, i.e., $z=e^{(2 k+1) i \pi / n}$, and its value is 2. (If the original disk has radius $R$, the value of the maximum is $2 R^{n}$ ).