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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.

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  • $\begingroup$ If you can manage to attend a summer math program, that can be a great experience. If there's a good math circle near you that could also be good. On your own, number theory is a great subject that you can learn by yourself just by reading an introductory number theory textbook (and working on problems, etc). Another good book to try reading is Calculus by Spivak, but there are many options. Whatever you're intellectually most excited about, do a lot of that. $\endgroup$ – littleO Jun 9 '17 at 5:58
  • $\begingroup$ I think math contests are generally the way that high school students show that they are interested in applying themselves in math. However, if you also enjoy other STEM fields you might see if there are any engineering type groups you can join. Like one of those robotics clubs. You may (or may not, just depending on the group) be able to do some interesting math and either way it'll look good on your application (and you might just have some fun, too). $\endgroup$ – user137731 Jun 9 '17 at 5:59
  • $\begingroup$ I don't really have a good answer for you, but I'd like to say that it's unfortunate that universities in the U.S. care so much about extracurricular stuff. How many millions of kids are forced to waste time pretending to have a passion for things they really don't? Anyway, at the high-school level, programming would certainly seem to be an area where you're more likely than in math to be able to do things that are creative and useful. $\endgroup$ – user49640 Jun 9 '17 at 9:57
  • $\begingroup$ Perhaps helpful: math.stackexchange.com/questions/1714966/… $\endgroup$ – Ethan Bolker Jul 11 '18 at 17:30
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You can get it to proofs, elementary number theory, and maybe some abstract algebra. Check out How to Prove It: A Structured Approach and then you could probably get through Elements of Number Theory by John Stillwell and Elements of Algebra: Geometry, Numbers, Equations by John Stillwell. The last two books are a pair and will introduce you to some topics in abstract algebra and elementary number theory. (This is all assuming you know at least algebra 2) If you can do that then after ap calculus you might even be able to go read a real analysis text like baby rudin but one step at a time.

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