Problem 1082

Erdős Problem #1082 [status: falsifiable; formalized: yes]. Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no three on a line. Does $A$ determine at least $\lfloor n/2\rfloor$ distinct distances? In fact, must there exist a single point from which there are at least $\lfloor n/2\rfloor$ distinct distances? Prize: no. Tags: distances, geometry.