1. I have a bunch of identical pairs of gloves jumbled together in a box. Suppose I take 5 gloves out at random. The pigeonhole principle guarantees that I will get at least one Left handed glove and at least one Right handed glove.

Is the preceding statement true or false ...

1. In a tournament with $\mathbf{n}$ teams, if every team plays each of the other teams exactly once, and every team wins at least once, then at least two of the teams will end up with the same win-loss record. Explain why.

There is a crazy mailman who does not like to put more than $\mathbf{x}$ letters in a mailbox. If there are $\mathbf{b}$ mailboxes and $\mathbf{L}$ letters, is it possible for him to put all the letters in the boxes without exceeding $\mathbf{x}$ letters in any box ...