I totally agree with Kriston when she says that those problems are very relative to age. It is much easier to work with younger child who is resisting something (simply doing it with them will be a remedy) then with a teenager.

As for Math Circles - there are different ones. Some are free, some you pay for, some are competition prep some are extended problems. The ones I encountered in Europe were focusing on extended math concepts. If you look back in my posts, we have actually managed to organize one at the college in our town.
I think that the math circle in Boston area is different, but the ones that I am familiar with are geared towards high school and advanced middle school kids. Post intermediate Algebra and Geometry.