$(HUN 3)$ In the plane $4000$ points are given such that each line passes through at most $2$ of these points. Prove that there exist $1000$ disjoint quadrilaterals in the plane with vertices at these points.
Choose a direction such that any line of that slope passes through at most one of the points. Draw lines of that slope through each of the points - thus creating $3999$ strips. Starting from one side, use groups of four points on consecutive lines to form $1000$ quadrilaterals, clearly disjoint (but not necessarily all convex).