4. On a circle, 100 points are marked, painted either red or blue. Some of the points are connected by segments, with each segment having one blue end and one red end. It is known that no two red points belong to the same number of segments. What is the maximum possible number of red points?
Answer: 50.

Solution: Take 50 red and 50 blue points. The first red point is not connected to any other, the second is connected to one blue, ..., the 50th is connected to 49 blues. Obviously, there cannot be more than 50 red points, because if there are 51 or more, then there are no more than 49 blues, hence the number of connection options is no more than 50, i.e., (by the pigeonhole principle) some two red points will belong to the same number of segments.