Question: In a room where there are more than 50 people with ages between I and 100, show that
(a) Either two people have the same age or there are two people whose ages are consecutive integers.
(b) Either two people have the same age or one person's age is a multiple of another's.
(c) Some of the people shake hands. Show that at least two shook the same number of hands. ("No hands" is a possibility.)
(d) Some people shake hands. Show that among those who shook at least one hand, two people shook the same number of hands.