IR-TST 2011 Problem 9

We have $n$ points in the plane such that they are not all collinear. We call a line $\ell$ a 'good' line if we can divide those $n$ points in two sets $A,B$ such that the sum of the distances of all points in $A$ to $\ell$ is equal to the sum of the distances of all points in $B$ to $\ell$ . Prove that there exist infinitely many points in the plane such that for each of them we have at least $n+1$ good lines passing through them.