Chapter 6.3, Problem 25E

In a room where there are more than 50 people with ages between 1 and 100, show the following:

1. Either two people have the same age or there are two people whose ages are consecutive integers.

2. Either two people have the same age or one person’s age is a multiple of another’s.

3. Some of the people shake hands. Show that at least two shook the same number of hands.( “No hands” is a possibility.)

4. Some people shake hands. Show that among those who shook at least one hand, two people shook the same number of hands.