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 5 of 5: Finish the parity conclusion
Detailed analysis
Therefore G is even exactly when n is even, as required.