AIME stands for American Invitational Math Exam.
You start with AMC 8 when you are an 8th grader or below. If you are good or older than 8th grade, you write AMC 10 or /and AMC 12. Anybody can write those, you just need to register in time and pay. If your scores are in top 1% of all those writing AMC 10 or top 5% of those writing AMC 12, you are INVITED to take the AIME. Costs stop at this point. Your school gets the exam that is administered over a three hour periods. The highest scoring kids (generally around 500 or so from the entire USA) are then invited to write USAMO (United States of America Math Olympiad). The top scorers go onto special summer program and then six of them are selected to represent the USA in the International Math Olympiad (IMO).
Hope S will one day make USAMO!
