TITLE: Induction: Prove that it is possible to seat people in a circle so that everyone sits beside a friend
QUESTION [7 upvotes]: Use induction to prove the following:
If each person in a group of $n$ people is a friend of at least half the people in the group, then prove that it is possible to seat them in a circle so that every one sits next to a friend of his/hers.
No idea how to solve this problem.

REPLY [6 votes]: Base Case: The statement is true for $2$ people (and less than two doesn't count as a group). Because each must be friends with the other.
Induction Hypothesis: Let the statement be true for $n$ people.
Inductive Step: If true for $n \ge 2$ people and a new one arrives, consider separately $n$ even or odd. 
If $n = 2k$, then the new arrival makes a group of $2k + 1$, and by the premise of the problem, everyone in this new group must be friends with at least $k + 1$ of the group. So the new arrival has $k + 1$ friends already seated and at most $k - 1$ people separating them so he must have two adjacent friends he can be placed between.
If $n = 2k + 1$ his arrival makes $2k + 2$ and he must have $k + 1$ friends in that group. Again they are already seated with at most $k$ people separating them so there must be two adjacent that he can be seated between.
So, in either case the statement is true for $n + 1$, and is therefore true for all  $n \ge 2$.
The key to the solution is that with the arrival of a new person he has in any case a minimum of $k + 1$ friends among $2k$ or $2k + 1$ people already seated.