Question 64
Let $S$ be a finite planar set no three points of which are collinear, and let $D(S, r) = \{\{x, y\} : x, y \in S, \text{dist}(x, y) = r\}$, where $r$ is a positive real number, and $\text{dist}(x, y)$ is the Euclidean distance between the points $x$ and $y$. Show that
$$
\sum_{r \gt 0} |D(S, r)|^2 \le 3|S|^2(|S| - 1)/4.
$$
$$
\sum_{r \gt 0} |D(S, r)|^2 \le 3|S|^2(|S| - 1)/4.
$$