Problem of the month (elementary group theory): Let G be a finite group with n elements and let m be a positive integer. If m and n are relatively prime, then every element of G has a unique mth root. That is, for each g in G, prove there is a unique h with hm = g.
Problem of the month (elementary geometry): Find a polytope (such as a tetrahedron) in 3 dimensions with an odd number of vertices, edges, and faces or prove one does not exist. For example, two tetrahedrons stuck together (along a triangle) is a polytope with 5 vertices, 9 edges, but 6 faces, so is not such an example. (Thanks Mike at Noisebridge!)
Problem of the month (polynomials): Given a positive integer n, prove that the nth cyclotomic polynomial has linear x term -µ(n), where µ is the mobius function on the positive integers. (Thanks Lionel Levine!)