The most difficult mathematics competition for students is the International Mathematical Olympiad (IMO), or "The World Championship" for high school students. It is held annually, but hosted by different countries (e.g. Kazakhstan in 2010).
The first IMO was held in 1959 in Romania (with seven countries participating), and has expanded to include more than 100 countries from five continents. The contest is spread over two days, with each day involving three problems which competing students have 4 1/2 hours to solve.
To document the 2006 competition in Slovenia and the amazing students representing the United States, the Mathematical Association of America (MAA) produced the film Hard Problems. This competition involved 90 countries and 498 students.
Key items in the film are "the dedication to mathematics shown by the coaches and team members, as well as the importance of the support of the student's parents." The film Hard Problems also provides a glimpse of the preparation and pressure faced by students participating in the IMO.
Though the film Hard Problems can be purchased from the MAA, it is being broadcast by public television stations at different times in October. Check your local listings for the specific dates and times in your area....then encourage your students and other mathematics teachers to watch.
If you can't watch it, you might want to try your hand at solving the problems. For about the last 15 years of contests, the problems are accessible at the IMO website.