Question 42
Let $n$ be a positive integer and $N=n^{2021}$. There are $2021$ concentric circles of pairwise distinct radii centered at $O$, and $N$ equally spaced rays starting at $O$. Some of the $2021N$ intersections of the circles and rays are painted red; the others are unpainted.
It is known that, however one intersection point from each circle is chosen, there is an angle $\theta$ such that rotating all the chosen points together through $\theta$ about $O$ takes them to red points. Prove that the smallest possible number of red points is $2021n^{2020}$.
It is known that, however one intersection point from each circle is chosen, there is an angle $\theta$ such that rotating all the chosen points together through $\theta$ about $O$ takes them to red points. Prove that the smallest possible number of red points is $2021n^{2020}$.