Problem 2
Suppose 997 points are given in the plane. Every segment joining two points has its midpoint coloured red. Prove that there are at least 1991 red points. Find a special case with exactly 1991 red points.
Step 1 of 4: Order the points by height
Detailed analysis
Rotate the coordinate axes if necessary; only finitely many directions are forbidden. Label the points so their y-coordinates are strictly increasing.