Question 139
A set of points in the plane is called good if the distance between any two points in it is at most $1$. Let $f(n, d)$ be the largest positive integer such that in any good set of $3n$ points, there is a circle of diameter $d$, which contains at least $f(n, d)$ points. Prove that there exists a positive real $\epsilon$, such that for all $d \in (1 - \epsilon, 1)$, the value of $f(n, d)$ does not depend on $d$ and find that value as a function of $n$.
(Kristiyan Vasilev, Konstantin Garov)
(Kristiyan Vasilev, Konstantin Garov)