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 2 of 4: Select two families of midpoints
Detailed analysis
The numbers m_i and n_i are y-coordinates of midpoints of actual segments, so distinct values correspond to distinct red points.