1. Three squares adjoin each other as in the figure. Find the sum of the angles A, B, and C.

2. The Fibonacci sequence 1,1,2,3,5,8,13,21,34,55,... is defined by setting the first two terms equal to 1, and thereafter by letting each term be the sum of the previous two. In other words, a_n+2 (term number n+2) is defined by a_n+2 = a_n+1 + a_n for n = 1,2,3,... Prove that if n is divisible by m, then the nth term of the sequence is divisible by the mth term. For example, 8 is divisible by 4, and the 8th term (21) is divisible by the 4th term (3).

3. Points A, B, and C move counterclockwise around three coplanar circles. Each point moves with the same constant angular velocity with respect to the center of its circle. How does the centroid of triangle ABC move?

4. For a class with two or more students, show that at least two students have the same number of friends in the class. Assume that you cannot be your own friend. Also assume that if I am your friend, then you are my friend (and vice versa).

5. Write an essay of 500 to 700 words (with bibliography) on:

(a) An unusual application of mathematics, or

(b) Recent developments on Fermat's Last Theorem.