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/(k*.^{2}+ a^{2}) - 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*| ≤ 1 for all real^{(k)}(x)*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**Z**^{n}, 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?