Let $S$ be a set of points in a plane $P$, having the following property: for any point $X in P$ there is at least one point $M in S$ so that the distance $|XM|$ is rational. Find the minimum cardinal of such a set.
First, let’s notice there is such a set, by taking $S=P$. Also if $S$ is a line in $P$. Someone claimed there is finite set $S$ having the required property. Does anyone have an idea how to solve this?
UPDATE
A related question here.