I suppose this is the place for my questions as much as any place is:
I'm a math student coming on my 3rd year of undergrad, and I am working as a counselor at a Summer math camp. The camp is for 12-15 year old kids. About 40 are taken from across Texas each year. The kids take 38 hours of math each week for three weeks, by the end of it completing algebra 1, geometry, algebra 2, or pre-calculus. Some kids will finish the class they had initially been placed in before the three weeks are up, and they might finish two or three different courses in those three weeks.
The point is that these kids are bright, and by bright I mean bright. A handful of them will most likely become mathematicians themselves. I actually did the camp when I was 15, and it paved the way for me to enter university two years early, as well as made me want to pursue math.
What I want to do is give these kids a problem of the day or week, depending on what I can come up with. These should be problems the higher-performing kids will be able to do, but not easily. They should take a good amount of thought, but still be doable for very gifted 12-15 year olds. I'd rather it wasn't highly technical, i.e. that they don't have to use some calculus method or something to solve it; I also want them to seem like challenge questions, rather than just more homework. Moreover, I'd rather it were fairly tough to Google.
I am, however, not an educator, so I was hoping some instructors with a history of dealing with precocious and intelligent students could offer some advice. Thanks in advance.