Summary: Pigeonhole Principle
October 15, 2003
If N objects are placed into k boxes, then there is at least one box
containing at least N/k objects.
1. Show that among any n + 1 positive integers not exceeding 2n there
must be an integer that divides one of the other integers.
2. Assume that in a group of six people, each pair of individuals consists
of two friends or two enemies. Show that there are either three mutual
friends or three mutual enemies in the group.
3. (Problem 12) Prove that for any real number and any positive integer
n there exist integers p, q with 1 q n such that
| - p/q| <
4. Prove that every sequence of (m - 1)(n - 1) + 1 distinct real numbers
has either an increasing subsequence with m terms or a decreasing
subsequence with n terms.
2 Previous Problems