AMC 8
The American Mathematics Competition 8 (AMC 8) is a 40 minute, 25 question multiple choice exam taken by middle schoolers. The score is calculated as the number of correct answers and various distinctions exist for those that meet certain criteria and scores.
AMC 10
The American Mathematics Competition 10 (AMC 10) is a 75 minute, 25-question multiple choice test offered by the Mathematical Association of America. It has 25 questions, with correct answers worth 6 points, blank answers worth 1.5 points, and 0 points for incorrect answers (discouraging guessing). The perfect score in this contest is 150, and contestants generally need to score around a 110 (top 2.5-5%) to participate in the AIME.
AMC 12
The American Mathematics Competition 12 (AMC 12) is a 25-question multiple-choice exam identical in format to the AMC 10. However, it does cover topics that are traditionally not covered by the tenth grade, such as trigonometry and logarithms. The top 5-10% participate in the AIME, and participants score between 90 to 100 to advance.
AIME
The American Invitational Mathematics Examination is a 3 hour, 15-question exam where all answers are integers between 0 and 999. The questions are usually far more difficult than the AMC 10 or 12, and participants who score well on the exam can qualify for the USA Mathematical (Junior) Olympiad.