I came across the following problem:
Let $X$ be a set of $1003$ distinct points on the plane s. t. no three of them are colinear and no four of them lie on the same circle. We say that a pair of points $a,b\in X$ is good if there exists a closed ball $B$ s. t. $B\cap X=\{a,b\}$. Let $S$ be the set of all good pairs. Show that $2003\leq|S|\leq 3003$.
Second inequality is quite easy - we observe that $S$ induces a planar graph with each connected component having at least $3$ vertices and use inequality $e\leq 3v-6$.
I have a problem with the first inequality. I can show that $1003\leq |S|$ (just by showing that every point of $X$ is contained in at least $2$ good pairs, by considering cases when this point is in $int(conv(X))$ or $\partial(conv(X))$, but I was unable to get anything significantly better than that.
Any help will be appreciated.