Post problems/thoughts on WOOT forum if you want to talk about them :O (Random sidenote: I like the removal of ratings for class boards, because IMO, the purpose of this class is mainly for people to learn, and asking dumb questions should be encouraged.) If you want to talk with people in real life, that's harder.

Note: Terence tao explains this better than I do: http://terrytao.wordpress.com/career-advice/ and in particular http://terrytao.wordpress.com/career-advice/ask-yourself-dumb-questions-%E2%80%93-and-answer-them/

There are two ways of looking at a problem: (oversimplification, but will suffice for the moment)

1. Looking at it by itself
2. Looking at it as a piece of a larger picture

In general, 1 will have you solve the problem more quickly; 2 will have you learn much more though. (And learning intuition is largely doing 2.)

Note that I'm not saying 2 is better, I'm just saying 2 is better for learning.

The difference between them: 1. when you solve a problem, you can react in different ways.

You can be like "OK AND NOW WE GO TO THE NEXT PROBLEM."

Except then it's a good question to ask yourself why you're treating learning as a list of problems you have to solve. (Those are actual contests; while this can be a good idea on contests and practice contests, if you do everything like a contest then a lot of your learning has to come from post-contest reflection)

or you can be like hmm what generalizations of this problem are there? could other methods have worked? how much is each constraint needed, how much can i modify it? how did I think of which method to use? why did the method work in this specific problem? what other general types will the method work in? what sort of general result is this problem? is this a problem a case of a more general framework? could this seemingly not-so-motivated method actually be a more motivated method disguised? how could other people have thought of those other solutions, and are those other solutions disguised (btw, if you're reading IMO compendium, IMO compendium solutions are always other solutions disguised)? where could I find this problem naturally?

2. How do you approach a problem?

You can take a very solution-focused route. But there's a lot of understanding that you'll miss.

People sometimes are like "WOW WHY DID I TRY THIS SOLUTION METHOD IT OBVIOUSLY WON'T WORK SDG:NSL:GNSDK:GNSDGNSDFJISDLNGDSLKGJDSFSFDS."

Sidenote: Induction is not a solution method. If your answer to "what have I tried" is "Induction", unless its a very clever induction, you actually haven't really tried anything. On the other hand, if you figure out a clever way to make induction work, then you've done something. If you realized that the problem has some fundamental structure that makes induction very natural, you've done something. If you randomly decided to use induction because the problem had a big number in it, you haven't done anything.

But basically, there are... 3 levels of solution methods in some sense.

level 1: A method that you think will solve the problem.
level 2: A method that will help you understand the problem or a method that you think will teach you something.
level 3: A method that you tried because you were like "I have no clue what to do, so lets randomly try using pigeonhole on a typical Euclidean geometry problem!"

In general, people will make one of two mistakes:

One mistake is to have a lot of your methods be level 3 methods. If you're stuck, you don't randomly try a method for the sake of randomly trying a method. In that case, try to understand the problem; There are ways of doing this: small/specific cases, easier problems, general considerations of what methods will work.

(And now to go into a somewhat-rant about the "general considerations of what methods will work.") I think I first consciously realized this existed (probably subconsciously knew before that) that these sort of considerations are useful when I read some of the polymath project work and read something sounding like "we know the answer can be as small as log n, so these type of methods that will prove an lower bound of sqrt n won't work." And these type of general considerations are really powerful (I would go as far as to say these type of general considerations are probably the largest improvement based on a single concept that one can make. Because one of the things I dislike about most olympiads (read: all non-russian olympiads) is that for almost every problem, I will be able to figure out in half an hour or less the general path of a solution though general consierations and just need to figure out details.)

The other mistake is to never use level 2 methods and try only methods that you are fairly sure will solve the problem. Not only does this method break down for harder problems, its also harder to learn with this method.

For example, if you never try to use algebraic (not really right word, but w/e) ideas in combinatorics in problems that are more based in elementary combinatorics, you will end up missing a lot of harder problems because you have no familiarity with probabilistic method, linear algebra, etc. Likewise, if all you try is algebraic ideas, you won't learn any elementary combinatorics.

As another example, if after you learn algebraic number theory, you never try to use basic algebraic number theory on number theory problems, you'll miss a lot of easier solutions on hard problems. Likewise, if all you do is true to use basic ANT on number theory problems, you'll miss a lot of easier solutions on hard problems.

A basic summary: (Note: Go read the entire thing if you actually want to learn anything; this basic summary doesn't actually say anything useful, its just a reminder of what you read) Think as part of the bigger picture in learning, Try to connect problems after you solve them, and try methods that you think you'll learn from.