9 Macmaster students and 9 Ryerson students are standing in a circle. Prove that some student is standing between 2 Ryerson students. Hint: Number the circle positions 1 to 18, then consider the even and odd-numbered positions. Is the result still valid if there were 8 students from each campus? Explain your answer.

25 people are at a party. Every pair of partiers either know each other or do not know each other. Knowing yourself does not count. Every partier knows at least one other partier. (a) Prove that there are two partiers who know the same number of people at the party. (b) Now suppose it is possible that some partier knows nobody at the party. Is it still the case that there are two partiers who know the same number of people at the party? Explain your answer.