I will be starting as a freshman in August 2018 in an undergrad institution. My school does not have the best reputation for mathematics in my country. Although it has a tie-up with another nearby graduate institution, which is the well known to top 10 US graduate program in Mathematics. Some Faculties from it teach courses at my institution. As from there, I do have hopes of getting good LOR's. Because of fewer facilities, many students in my undergrad pursue their honors thesis at a different location. Few of them pursued outside of my country under AMS fellows.
Ph.D. placements at my school include Dartmouth ,Northwestern, Paris, UIUC etc.
I do have the option of taking a gap year and target two other schools A & B.
School A has recently started its undergrad program. Although it is fairly reputed in my country and abroad. Their placements include Chicago, Ohio State, Wisconsin etc.
School B has the best math undergrad program in my country. It has a stellar reputation in top 10 grad schools in the US and many students are pursuing PhDs in those schools.
I know Harvard or Princeton is beyond my reach, but do schools like Chicago or Berkeley care more about elite undergrad status. Should I take a gap year? Can stellar letters of recommendation and good honor's thesis outweigh undergrad reputation?