Specialized math instruction is available during high school years in US. However, it costs a lot of money unless you qualify for financial aid. A case in point:

http://www.exeter.edu/documents/COI/COI2015-16.pdf#page=55

Tuition is $36,430 for day students. It is free for families making less than $75,000.

UM, my DD's school is K-12 and I don't think they offer anything beyond AP Calculus BC but they send a significant number of students to HPYS, MIT, Caltech, and other highly selective colleges each year. I get the impression that these kids do a lot more on their own (or orchestrated by their parents...). They do independent research, apply for patents, start their own software company/non-profit, etc. You know, the usual college rat race.