Problem 5
A set of 2n+1 points in the plane has no three collinear and no four concyclic. A circle divides the set if it passes through 3 of the points and has exactly n-1 points inside it. Prove that the number of circles dividing the set is even if and only if n is even.
Step 1 of 5: Order the circles through a fixed pair
Detailed analysis
Fix two points A and B. The other 2n-1 points determine the 2n-1 circles through A and B. Order these circles by the positions of their centers on the perpendicular bisector of AB, and call the third point of Ci the point Xi. Let L and R be the two half-planes bounded by AB.