I'm sure that some of you have noticed that the kids who're really good at competition math tend to be better than pretty much everyone else at basically every subject in school.   This also doesn't seem to work in reverse - the kids who're really good at history or biology or whatever are not necessarily very good at math.
I've also noticed that there are people who spend large amounts of time studying for school math tests only to fail. Four years ago I believed that this was because the other kids were just "dumb." Two years ago I believed that what kept them from becoming smart was their mindset.  I don't think that I was wrong back then, but only now have I really honed in on why I get so much better than them despite putting in the same amount of time. Math is hard because you can't just grind. You have to sort.
I think that these two things have the same cause - the brain of a successful math contestant isn't just stronger, but also better organized.
The two main things that newer contestants struggle with are making stupid mistakes ("sillying") and speed. In this case, I think the standard advice of "practice makes perfect" is probably the furthest you could get from helpful.
I actually think that speed is a pretty terrible thing to focus on if you're trying to improve for math competitions. Of course, a little bit of speed practice in the weeks before the AMCs doesn't hurt, but I think that it doesn't really matter whether you do it or not. My advice is to just do problems, learn new concepts, and most importantly, sort out what's happening in your head.
I find that a significant portion of people really good at math competitions do this by writing somehow. So write handouts, write notes, or whatever. Just write about what you're learning in some way. Write outlines of the solutions to medium-hard AMC and AIME problems and keep track of them somewhere. It's totally fine if what you write is complete garbage at first. I definitely did, and I can't deny that seeing the material that I wrote two or three years ago is incredibly embarrassing.  
I notice that one of two things happen:
When the latter happens, you'll tend to not know for a while. I panicked after the AMC 10A because I legitimately feared not making AIME. I can't deny that I still feel that very real fear. It's not nonexistent or far-fetched, it's piercing. But it's not the most important thing out there - just prepare a little, do some practice tests before the real one rolls around, and you'll be fine on test day.
So the idea isn't to practice speed. The idea - and I can't believe nobody has ever said this before - is to sort out "What the heck is happening in my head?" Speed will come naturally with that. It's like cleaning up your room. Of course you'll find things faster with a neat room. The way to find things in your room faster isn't to practice finding stuff in your room, or giving yourself a lower time limit to find things in your room. It's to clean the darn thing, for crying out loud.
But there's a difference between cleaning your room and sorting your brain. Your room gets dirtier the more you use it. Your brain gets cleaner and more efficient the more you use it. And best the way to use and sort your brain is to write. And you might end up giving back to the math community that gave you so much in the process, which is certainly a very pleasant byproduct.
Sort. Write. Polish. That's really it.
 Also applicable to the kids who do competitive physics or computer science, though to a much lesser degree.
 And if they're not better than someone else at some subject, it's because the other person specializes in that subject, though from my experience this basically never happens.
 Specifically, "I'm studying because I need to not fail a test" is a terrible way to study. I find that at least when you're in school, doing something to please a bureaucracy tends to make you suck at it. As another example, see the kids who dedicate their entire lives to getting accepted into a good college - even if they do succeed, they tend to realize sooner or later that their entire high school career was pretty much pointless. MIT even explicitly states that the best way to get into college is to not do things for college.
 I also strongly believe I'll have the same feeling about the stuff I'm currently writing in one or two years, including this essay. And I think the only difference between me now and me back then is that only now will I have the foresight to know that I'll cringe when I see this two years later.
 If you don't find anything you wrote two or three years ago embarrassing at all, that's a huge red flag for stagnation.
 At least compared to how much you know and how much you score on harder contests.
 This is what happened to me - in fact, I thought I would be on the decline because of low MATHCOUNTS/AMC scores compared to my teammates. But this year, at least, has proven that completely wrong.