Question 27
C4 (FIN) Determine whether or not there exist two disjoint infinite sets $\mathcal A$ and $\mathcal B$ of points in the plane satisfying the following conditions:
(i) No three points in $\mathcal A\cup\mathcal B$ are collinear, and the distance between any two points in $\mathcal A\cup\mathcal B$ is at least $1$.
(ii) There is a point of $\mathcal A$ in any triangle whose vertices are in $\mathcal B$, and there is a point of $\mathcal B$ in any triangle whose vertices are in $\mathcal A$.
(i) No three points in $\mathcal A\cup\mathcal B$ are collinear, and the distance between any two points in $\mathcal A\cup\mathcal B$ is at least $1$.
(ii) There is a point of $\mathcal A$ in any triangle whose vertices are in $\mathcal B$, and there is a point of $\mathcal B$ in any triangle whose vertices are in $\mathcal A$.