I did MathCounts back in the '80s, and only 5 students from my GT class were invited to participate. We could only field a team of 4, so it was just the 4 of us plus one alternate. As I recall, it was the math teacher's best judgment as to which of us would do best in the competition, although he may have given us some quiz questions to see who could think most creatively.
I think I still have my competition Tshirt somewhere and I'll admit it would give me some thrill to have my daughter participate (and, given that there were almost NO girls competing back then, maybe we'd be the first mother-daughter MathCounts mathletes!)
How does it work now?