Good luck self studying for those -- I tried. I made AIME but had no chance there.
There's no way a significant number of generic public high schools start caring about the AMC. It will just be helicopter parents, wealthy private schools and the odd self motivated student.
You can study for the AMC–I used to (disclaimer: I knew a lot people who did so as well). However, doing well on the AMC usually means you need to actually understand what you're doing; it's fairly difficult to cheese your way through.
