Irish Mathematical Olympiad

Let $S$ denote the set of the 2016 given points. Consider the collection of perpendicular bisectors of pairs of distinct points in $S$. This is a finite collection of lines, so we can pick a point $P$ not on any one of these lines, and also not in $S$.

For $r \ge 0$, let $N(r)$ be the number of points in $S$ whose distance from $P$ is at most $r$. By construction, the circle $C(r)$ of radius $r$ about $P$ contains at most one point of $S$ for each $r$, so $N(r)$ increases from 0 for small $r$ to 2016 for very big $r$, and each increase occurs as a jump of 1. Thus, there are numbers
$$0 = r_0 < r_1 < r_2 < \dots < r_{2016} < r_{2017} = \infty$$
such that if $0 \le n \le 2016$, then $N(r) = n$ exactly when $r_n \le r < r_{n+1}$. The circle $C(r)$ has the desired properties if we pick $r_{666} < r < r_{667}$.