[supernewton]

They're the feeder contests in the U.S. that determine who gets to represent the U.S. in the IMO.

AMC = multiple choice test, open to all grade school students.

AIME = open response test, all answers are numerical, open to students who score high enough on AMC only.

USAMO = USA Math Olympiad. IMO-style proof problems. Open only to top N scorers on AMC and AIME.