2. 25 boys and 25 girls sit around a table. Prove that it is always possible to find a person that sits next to two girls (one on each side of the person).
3. Prove that among any 16 distinct positive integers not exceeding 100, there are four different ones a,b,c,d such that a+b = c+d.
4. The Hamming distance between two equal-length bit strings is the number of positions in which they differ. For example, the distance between 0001 and 0101 is 1. Suppose that, for some integer n, there are K n-bit strings such that every pair has Hamming distance at least 101. Prove that there are K n+1-bit strings such that every pair has Hamming distance at least 102.
5. Given a finite sequence {x1,x2,…,xN}, shifting the sequence gives the sequence {x2,x3,…,xN,x1}. Prove that for any sequence of N real numbers, if the sum of the numbers is non-negative, then one can shift the sequence some number of times so that the resulting sequence has the following property: for each K ≤ N, the sum of the first K numbers in the resulting sequence is non-negative.
6. (just for fun, not for proof practice). Find a way to do the following card trick. In advance, Alice and Bob agree on some protocol. Then, the audience chooses 5 arbitrary cards from a standard deck of cards and gives the 5 cards to Alice. Alice then shows 4 of the cards, one at a time, to Bob. Bob then announces the identity of the 5th card (the one the audience gave to Alice that she did not show to Bob). Alice is not allowed to signal Bob in any manner, other than which 4 cards she reveals and the order in which she reveals them.