Clarifications: The cut must go through the center of the rectangle One of the points should be on the cut itself 3. Of course, the problem carries the pigeon hole spirit if you haven't sensed that already 4. For part (b), the maximum refers to the maximum number of point on one perimeter or the other.

Problem 1: Rectangle

Use Pigeonhole Principle. Please explain all parts. Thank you

Eight different points are placed on the perimeter of a rectangle.

(a) Show that we can cut the rectangle into two pieces of equal area, such that one of them will have at least 5 points on its perimeter.

(b) We drop the requirement that the two pieces must have the same area, but maintain that both areas must be non-zero (so not a trivial cut). What is the maximum number of points that we can guarantee on a given perimeter if: you have the freedom to choose the cut for any cut

(c) How does your answer to (b) changes if the rectangle is replaced by a circle?