I am a high school student interested in thinking about math. I don't know a lot of high-powered math (I only know up to calculus), instead I focus on discrete topics related to math Olympiads (combinatorics, number theory, geometry etc). Olympiad problems typically take 2-3 hours to solve. I want to start thinking about interesting problems over extended periods of time.
So I am wondering where I can find a bank of problems that satisfy the following criteria: they are simply stated, related to discrete topics (not graduate level math please), and are difficult enough that they cannot be solved in a day, but not as difficult as full fledged research problems. I am not talking about open problems nessecarily; I don't want to think about something like the Collatz conjecture, since that is too difficult as its been open for a long time.
I am sorry if I'm not being clear, but I dont know what more specifics I can give. Maybe someone can help me narrow down what I'm actually asking?
You could go to different university course pages and look for their undergraduate discrete math course. They are generally accessible to the public.
For exmaple:
http://web.stanford.edu/class/cs103/
http://people.cs.uchicago.edu/~simon/TEACH/DISCRETE/
http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-spring-2010/index.htm
And there are plenty more, have fun!