## Task 1 - 160931

A regular n-gon inscribed in the unit circle has the property that it maps onto itself when rotated by $180^{\circ}$ about the center of the unit circle. On the periphery of the unit circle, any point $P$ is given.

Determine, under these conditions, from the given number $n$ the sum $s$ of the squares of the distances from the point $P$ to all points of the n-gon!
Due to the point symmetry, $n=2 m$ is even.

Let the vertices of the $n$-gon be numbered in a mathematically positive direction as $P_{1}, P_{2}, \ldots, P_{2 m}$, such that $P_{k}$ and $P_{m+k}$ are opposite to each other. These form a diameter of the unit circle, and it holds that $\left|P_{k} P_{m+k}\right|^{2}=2^{2}=4$.

By Thales' theorem, every triangle $P_{k} P_{k+m} P$ is right-angled at $P$ (or $P$ coincides with one of the endpoints of the considered diameter).

In any case, by Pythagoras (or directly by substitution),

$$
\left|P P_{k}\right|^{2}+\left|P P_{k+m}\right|^{2}=\left|P_{k} P_{k+m}\right|^{2}=4
$$

Thus, for the desired sum $s$, the value is the number of these diameters, which is $4 m=2 n$, since each vertex of the $n$-gon is involved in exactly one diameter of the unit circle, and exactly two vertices of the $n$-gon lie on each such diameter.