@BiffBrown : I am aware of the page.
Prior knowledge: A few and some are naturally ready, though this is more common at very top privates (the usual, you know who they are) and elite publics (Ones with renowned STEM programs like Tech, but I am thinking mainly Berkeley and Michigan) that are more likely to attract more folks at the level of the 2 Goldwater Scholars recently generated by the math department (specifically people who at least made it to IMO or IPO camps). Some students come from privates and charter high schools (usually outside of the south) with renowned math or physics programs. Sobuggme schools naturally have a much bigger Putnam culture (they may be the schools that send a substantial amount of their physics and math undergraduates to doctoral programs in those areas).