I came from a top LAC with a decent, though small, math program. My professors say I can get into math PhD program in schools like UCLA, Columbia or Michigan (as I will start taking grad level courses in my sophomore. I'm also doing a decent REU for freshman summer). However, I want to have a better chance for Harvard, Princeton, MIT or Berkeley. I want to go to Math in Moscow (http://www.mccme.ru/mathinmoscow/index.php?page=10
) program and take 12 to 14 courses on a gap year. Then I want to do a domestic exchange to Harvard/ MIT (many people did that in my College, not very hard) with probably more than a dozen grad courses taken in Moscow so that I may secure some good rec letters together with some personal relations (and yet more grad courses) in those big school. The only problem is it is going to take one more year to finish my undergrad and I don't know how much it will actually increase my chance. Do you think it is advisable?