I feel a need to further P.J.'s comments on math contests, because I think he indadvertedly overlooked the most important part of math competition, and that is practice. Math contests are not sports. It matters not if you win or lose, what matters is that you try your best to do as well as you can. for most students, and teams hopefully, this does NOT involve endless drill and practice, nor does it have anything to do with memorizing endless formulas. Granted some teams do this, and they are often "winners" but they have missed the entire point of having a math team.
Math team is a place where students who like math and are interested in math, and may be talented in math, have a chance to gather together and work on interesting problems together. It is a chance to share creative insights and to learn about things that are not usually taught in the classroom. Math team practice is where I can ask the problems that are too hard to ask in class , and, yes, if no one can solve the particular problem, then take it home and work on it. The message I hope most coaches send is that we are here to learn math, to work on interesting problems and to have fun doing it. The contest itself is the carrot, but by no means is it why we spend so much time doing this. Out hope is that some of our mathletes will find that math is not about grades, nor is it about winning. It is about working on interesting problems with others who also think it is fun to work on interesting problems.
One of my favorite stories about math team, one I have told countless times, and one that convinced me that math contests are wonderful things, involves a young student named Eric Winfree. It was clear that Eric was brilliant but definitely not fast. He could solve the problems but often not within the artificial constraints set up by contests. One week he failed to make the Chicago area ARML team and did not qualify for theAIME. I talked to him on Friday of that week, and expected to find him dejected. Instead, he was cheerful, even robust. I conversed with him and he said I got all of the problems this week. I asked him what he meant, and he said that he had correctly solved all of the ARML tryout problems and all of the AMC problems by Thursday night, and three of them were really cool. Let me show you what I learned.
Eric is now a working scientist, mathematician and CalTech. . He attributes math contests as one of the many important parts of his education. I always tried to get our math team students to understand Eric's attitude about math contests. I hope most math team coaches see it this way, as that is why I spent considerable time and effort during my career promoting math contests, and coaching teams that did well but didn't win very often. I count my time spent with this as some of the most important successful time I spent doing anything, because many students were drawn into the math fold in part because of math contests.