The question is from the multiple choice problem:
A circular region is divided by 5 radii into sectors, as shown above. Twenty-one points are chosen in the circular region, none of which is on any of the 5 radii. Which of the following statements must be true?
I. Some sector contains at least 5 of the points
II. Some sector contains at most 3 of the points.
III. Some pair of adjacent sectors contains a total of at least 9 of the points.
It is not hard to show that I is true while II is not. Intuitively III is also true. But here are my questions:
How to prove that III is true? (Can Pigeonhole principle be used here?)
What topics/theorems/principles/ is statement III in this problem related to in combinatorics?