The American Mathematics Competitions (AMC) is dedicated to the goal of strengthening the mathematical capabilities of America’s youth by identifying, recognizing and rewarding excellence in mathematics through a series of national contests. The entry examinations are the AMC 8, AMC 10, and AMC 12 and are designed for students in grades 6 through 12. The AMC contests are intended for everyone from the average student at a typical school who enjoys mathematics to the very best student at the most special school. More information can be found at http://amc.maa.org/.
The Putnam Competition is held annually for undergraduate college students. Scholarships and cash prizes are awarded to high scorers. It is considered to be the most prestigious collegiate level math competition in the world. Learn more at http://math.scu.edu/putnam/index.html.
The United States of America Mathematical Olympiad (USAMO) is a high school mathematics competition. This competition serves as the final round of the AMC. Find more information at http://amc.maa.org/usamo/usamo.shtml.