More problems

Problem 1. Show that every graph contains two vertices of equal degree.

Hint: Use The Pigeon Hole Principle

Problem 2. Given six people, show that either three are mutual friends, or three are strangers to one another. (Assume that ``friendship'' is mutual; i.e., if you are my friend then I must be your friend.

Problem 3. (USAMO 1986) During a certain lecture, each of five mathematicians fell asleep exactly twice. For each pair of these mathematicians, there was some moment when both were sleeping simultaneously. Prove that, at some moment, some three of them were sleeping simultaneously.

Sketch of the proof. Suppose the opposite. Then we must have that every mathematician was sleeping simultaneously the first time with two mathematician and the next with the other two. Really, suppose that there is a mathematician A, who fell asleep simultaneously (say first time) with B, C and D. Then either two of the three, say B and C were sleeping simultaneously with A, or all three B, C and D were sleeping simultaneously. In both cases we will have a contradiction with the initial assumption.

We can now draw a graph corresponding to the problem. Its 5 vertices will correspond to the mathematicians. Fixing the first vertex, A, we join it with another vertex with an edge if and only if both corresponding mathematicians were sleeping simultaneously the first time. After that the same procedure for the already connected with A vertices. Two cases are possible: either we have a cycle of length 4, or a cycle of length 5. In both cases it is easy to reach a contradiction.