It is sometimes easy to lose sight of the fact that math contests are meant to be fun and engaging, but not ultimate ends in and of themselves.
The main value of math contests is not to be found in the honors and awards (which are very nice, of course, but somewhat beside the point). The main value of math contests is to be found in wonderful mathematics it can draw students into discovering. Not only do they discover mathematics, but they can also discover the joys of working hard towards shared goals with a community of kindred spirits.
Math contests are a little bit like mountain climbing. The views from the top are nice, but even spectacular views do not justify the extreme efforts it can take to get there. What can justify those mountain-climbing efforts are the joys of the journey, especially if shared along with others, and the beautiful things you can learn about the natural world and the power within yourselves as you help one another to discover hidden potentials and problem solving abilities within you that you did not know existed.
Just as you should "stop and smell the flowers" as you climb the mountain, so too should you remember to "savor the problems" you encounter in math competitions. By construction, those contest problems have already been solved, but reflecting on them after the contests are over may inspire you to create new and fresh ones of your own devising--or you may discover exciting new ways to solve old problems.
A recently revived discussion on the Art of Problem Solving discussion forums reminded me of a remarkable essay written three years ago by a student on his contest experiences, which has resonated with many readers. The essay is very moving, thoughtful, and beautifully written. Both the essay (linked in the first post on the thread) and the discussion which followed it are readings that I would highly recommend to students, parents, and math teachers and coaches.
I would also recommend reading an inspiring speech given by our alumna speaker at Math Prize for Girls last fall, which also addresses these issues, and gives some excellent advice well worth bearing in mind for making the most of your math contest adventures.
You may also want to explore the links to the authors' blogs, which discuss some of the wonderful mathematics they have been inspired to learn and share with others.
There is more recommended reading on these topics here. Some of our older alumni may also find useful advice and insights in a similar vein here.