Problem 1090

3. A segment connecting two points which divides the given circle into two arcs, one of which contains exactly $n$ points in its interior, we will call a good segment. Good segments determine one or more closed polygonal lines that we will call stars. Let us compute the number of stars. Note first that $\operatorname{gcd}(n+1,2 n-1)=\operatorname{gcd}(n+1,3)$. (i) Suppose that $3 \nmid n+1$. Then the good segments form a single star. Among any $n$ points, two will be adjacent vertices of the star. On the other hand, we can select $n-1$ alternate points going along the star, and in this case no two points lie on a good segment. Hence $N=n$. (ii) If $3 \mid n+1$, we obtain three stars of $\left[\frac{2 n-1}{3}\right]$ vertices. If more than $\left[\frac{2 n-1}{6}\right]=\frac{n-2}{3}$ points are chosen on any of the stars, then two of them will be connected with a good segment. On the other hand, we can select $\frac{n-2}{3}$ alternate points on each star, which adds up to $n-2$ points in total, no two of which lie on a good segment. Hence $N=n-1$. To sum up, $N=n$ for $3 \nmid 2 n-1$ and $N=n-1$ for $3 \mid 2 n-1$.