I think you have a chance at CalTech and the other highly-rejective colleges, both on your own merit with likely a small bump for being female from a rural area/state. With the standard caveat that of course they are reaches for everyone.
You could also look at Harvey Mudd which is part of the Claremont/Pomona consortium and where you can cross-register at any of the 5 consortium schools and they all share a campus and a library etc.
Brown has a very well-regarded applied math program if applied math is of interest.