Blarg I got jumped by a whole bunch of people in writing this, but I don't think I'm just restating anything here...

Re above about proof writing: there's no reason you can't work on solving and writing at the same time. Write out solutions to the problems you think you've solved while practicing, and later find someone who knows their stuff to check what you did.
I think you can replace Starcraft and math with anything seriously competitive. In all of the competition-like things I've ever pushed myself to improve in, the same things apply: learn the fundamentals before memorizing advanced tricks, spend a lot of time playing/solving, and actually use that time correctly - take a hard look at every game/problem you finish to figure out what you need to improve. I'm sure there's more too.

Specifically applying the three things I mentioned above to math (but again, probably more can be said):

1. Fundamentals before advanced stuff. As an example, learning inequalities. Just learning what AM-GM/Cauchy are and how to solve some simple problems that use them is a bad idea (see the 2nd paragraph of this post for what the US IMO team leader thinks of that). You should know AM-GM and Cauchy of course, but you can't stop there. There are basic algebraic manipulations you have to know how to do, which you generally pick up by doing enough problems to have encountered them all. In general, thinking that you couldn't solve a problem (like USAMO 1 this year for instance) because you didn't know some high-powered result and that you should start practicing everything from Rearrangement to Popoviciu is a terrible, terrible idea.

2. Every time one of these threads about improving comes up, the answer is the same: do lots of problems. The sooner you realize there is no way around this the better. Books like ACOPS, PSS, the 10x subject books will all help with this, but the Contests section here suffices if you know where to look. There are a lot of past topics here about what books/problems/olympiads are suited for various levels.

3. Learning everything you can from the problems you do. This is really important and probably most of what pythag meant when he said "learn how to learn". You can easily do hundreds of problems or every past USAMO or whatever and make less progress than someone who just does a few dozen problems the right way. Your time spent on a problem should not end once you solve it or give up. If you get stuck on a problem, make sure you've worked on the problem for a long time before you even think about looking at a solution. For olympiad problems, getting no progress in the past hour is a good rule of thumb to start with. If you do get the problem, read other solutions anyway.

Now that you've worked hard on the problem and read the solutions, ask yourself some questions. Were there any approaches you tried that neither you nor any other solutions ended up using? If so, try to understand what about that method is fundamentally unsuited to the problem. On the flip side, were there any approaches you tried, discarded because you thought they didn't work, and yet other solutions successfully used them? If so, figure out which part of the solution you got stuck on enough to make you think you weren't going anywhere. How would you change your thinking to never miss such a step in the future? Were there any approaches you didn't think of at all? Stop reading the solution there, look up some things about that approach if you're not familiar with it, and then go back and spend awhile trying to apply it to the problem yourself. (This is a much better way to learn weird tricks like Vieta jumping or Combinatorial Nullstellensatz then just looking them up after having heard of the name.)

One important thing to keep in mind while you're doing this is there is a lot more to improving than just adding to your list of known theorems or what tricks work where, because once you hit a problem that thwarts every piece of your knowledge bank you have to think of something new. So you should also be trying to build a strong intuition about the results and tricks you know or even things much vaguer than that. How fast/slow is this sequence/function allowed to grow? Do these conditions on in the problem not seem to care about anything except 's residue modulo something? Does this number theory problem ever actually use the addition operation? There are some results, like the majorization condition in Muirhead or the Chinese Remainder Theorem, that are really just formalizing a piece of intuition you could easily pick up on your own from experience.

Along these lines, a problem you couldn't solve might have an incredibly long and intimidating solution, but that solution might just be the result of writing out all of the details of an idea that you could come up with in less than five minutes. For that reason, for the scarier-looking solutions, or really any solutions, you should try to understand the steps and motivations until you can state all the main ideas in a couple sentences and basically know how to solve it from that. There's a good chance this will be a lot easier to do than it looks, and it will be of tremendous help when you run into a problem that utilizes similar ideas.


You might notice that in all of this I'm suggesting that there's no reason to believe someone who claims to have done thousands of problems and know every arcane theorem and trick out there is going to be successful, and that's exactly what I intend to say. It's always discouraging to see people say that they're planning to do every problem in PSS or every IMO SL, because it sounds like they're more intent on being able to say they've done that than actually doing the problems as thoroughly as they should. Reminds me of this xkcd.


(EDIT: Took a second read and realized I implied some things I didn't mean in places.)