3+ [Mutual Position of Two Circles]

In a regular $n$-gon ( $n \geq 3$ ), the midpoints of all sides and diagonals are marked.

What is the maximum number of marked points that can lie on one circle?
Let's consider two cases.

1) $n=2 m$. The diagonals and sides of a regular $2 m$-gon have $m$ different lengths. Therefore, the marked points lie on $m-1$ concentric circles (with $n$ points on each) or at the common center of these circles. Since different circles have no more than two common points, a circle not belonging to this family of concentric circles contains no more than $1+2(m-1)=2 m-1=n-1$ marked points.
2) $n=2 m+1$. The diagonals and sides of a regular $(2 m+1)$-gon have $m$ different lengths. Therefore, the marked points lie on $m$ concentric circles (with $n$ points on each). A circle not belonging to this family of concentric circles contains no more than $2 m=n-1$ marked points.

In both cases, the maximum number of marked points lying on one circle is $n$.

## Answer

p points.