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The term "olympiad" is used generically to refer to a math contest in which students are asked not to compute numerical answers, but to give proofs of specified statements. (Example: "Prove that 2003 is not the sum of two squares of integers.") The most famous example is the International Mathematical Olympiad; most countries that participate at the IMO have national olympiads as part of their team selection process. Some areas have additional olympiads at the regional or local level.
The jump from short answers to olympiads is a tough one. Here are some tips for students making this transition.
by Kiran S. Kedlaya (kedlaya'at'math.mit.edu)
The Art of Problem Solving hosts an online Forum for mathematics students to collaborate and develop a community of peers.
The MAA has many books on Problem Solving available through the MAA Book Store.
Exercise your mind daily with an AMC problem, posted daily on MAA Minute Math.
Check out this list of websites that features fun math.