Problem Solving

One of my hobbies is swapping competition-type math problems with friends. Here are some of my favorites.

• Fix a > 0. Compute the sum over all nonnegative integers k of 1/(k² + a²).
• Recall that, if A and B are commuting matrices over C, then they have a common eigenvector. Is this still true if [A,B] = AB-BA has rank one?
• Let f(x) be a function on R such that |f^(k)(x)| ≤ 1 for all real x and all nonnegative k. Show that, if additionally f'(0) = 1, then f(x) = sin(x).
• Let G be a graph with even clique number. Prove that G can be partitioned into two subgraphs which have the same clique number (as each other). (2007 IMO Problem 3)
• Let f be a discrete harmonic function on Zⁿ, i.e., suppose that at each point f(x) is the average of f(y) over the 2n neighbors y of x. If f is bounded, must it be constant?