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 4 of 4: Count the red points and attain equality
Detailed analysis
There are therefore at least 1991 red points. For P_i=(0,2i), every midpoint is (0,i+j), with 3<=i+j<=1993; all values in this interval occur, giving exactly 1991 red points.