Let V be a collection of points in the plane with the property that every point in V is a midpoint of a line segment joining two points of V. Prove that V must be an infinite set.

Discussion: This is a very elegant treatment of the problem using the extremal principle. The student assumes that the set is finite, which allows one to assert that there are a pair of points a, b at maximal distance and then is able to derive a contradiction.

Written response: Nice solution using the extremal priinciple.