Here is an elementary geometry problem. I have two different solutions for it but they both use advanced mathematics (calculus at the very least, Riemann surfaces for the most elegant solution, will post my solutions later). The interesting thing is, although the problem is really elementary, no one around here seems to be able to produce an elementary solution.
Let C1 and C2 be two disjoint circles in the plane, C2 interior to C1 (with distinct centres, or else the problem is trivial). Let p0 be a point of C1. Draw a tangent to C2 passing by p0 and call p1 its other intersection with C1. Draw a tangent to C2 passing by p1 and call p2 (p2 different from p0) its other intersection with C1. Repeat the process to obtain p3, ..., pn. Assume pn=p0. Now let q0 be another point of C1 and construct q1, ..., qn similarly. Prove that qn=q0.
By the way, if you want to reply and want to be 100% sure to reach me, send your reply by e-mail; this group has too much stuff for me to read everything.