Participating in math competitions can be a terrific experience for gifted students. Many mathematically gifted students have been accelerated in math; a student with a couple of years of high school math can be a huge asset to a middle school math team, even if he/she has never been in a math class with any of his/her teammates. It adds a social element to doing math (some of the problem solving is done by teams rather by individuals), and, for some gifted students who are not also gifted athletes, this may be the only opportunity to be on a TEAM.
An excellent author for appropriate books is George Lenchner.
Note that there are books with Problem of the Week formats (NOT math workbooks!).
This is a good one:
The handbooks from past years can be useful for preparing to compete, and contain some interesting problems.
There are many resources for problems from past ARML contests:
There are books containing problems from past exams.
This is the "big time!" So, naturally, there are a host of available resources for students competing at the very highest levels. Here are links to a sample of international math olympiad problem books.
2001
2001
2002
2003
2005
Multiyear problem sets:
I think two of the best books for high school competition prep, and for deeper understanding of mathematical topics than high school classes typically provide are these:
The Art of Problem Solving Volume 1
and
The Art of Problem Solving Voume 2.
The Art of Problem Solving Volume 1
and
The Art of Problem Solving Voume 2.
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