From Wikiversity

[edit] Contests

Each country typically has its own set of contests. For high school, these usually include one national Mathematics Olympiad and one or more qualifier contests intended to help select students to invite to the national Olympiad. The top six students of every country are then invited to participate in the International Mathematics Olympiad (IMO).

[edit] List of contests

In the United States:

Middle School

MathCounts - A national middle school math contest

AMC 8 - The American Mathematics Contest for students in 8th grade and below

High School

AMC 10 - The American Mathematics Contest for students in 10th grade and below

AMC 12 - The American Mathematics Contest for students in 12th grade and below

AIME - The American Invitational Mathematics Examination, qualified for through the AMC 10/12

USAMO - The United States of America Mathematics Olympiad, qualified through the AIME and the AMC 10/12

Undergraduate

Putnam - A contest based on high school mathematics as well as first year calculus and algebra

More Listings:

http://k12mathcontests.wikispaces.com/

http://homeschoolmathcontests.com/

[edit] Books

There are many problem solving books available for students. The list is broken up into three levels by difficulty. Most of these books can be purchased online.

Math Contest Books