I'll be a sophomore in high school (U.S.) next year. So far, I've taken a few classes in math at school, I'll be taking Precalculus and College Algebra this upcoming year, and I might self-study and take the AP Calc AB/BC tests before I take the classes (so I can go ahead and take a couple of classes my junior year at the local university).
I've always had an interest in math and math-heavy subjects, but it's really only now that I'm really beginning to take those interests seriously. I want to get into a really good college, and those colleges apparently want to see that you have a passion in which you've done a good amount. I have pretty good grades overall and do a number of extracurriculars, and I think I want math to be what I excel in.
The question is, what is something substantial I can do? I know that math research is pretty much out of the question because of what I've seen and heard from undergraduates and graduates. I really don't want to take the AMC and get into math olympiad stuff (though if I need to, I can). Would something like the Intel Science Fair be viable?
What subjects could I go into to and do something? Is something like number theory, which doesn't require much outside of regular high school math to get started, a good idea? Is there something that's slightly outside of mathematics that I can get into (like cryptography)?
Any suggestions would be really helpful.