2005 USAMO Problems/Problem 5
Let be an integer greater than 1. Suppose points are given in the plane, no three of which are collinear. Suppose of the given points are colored blue and the other colored red. A line in the plane is called a balancing line if it passes through one blue and one red point and, for each side of the line, the number of blue points on that side is equal to the number of red points on the same side.
Prove that there exist at least two balancing lines.