This is my response to a topic on AOPS. I thought I would share my views on problem-solving. Personally, I feel rather weak in problem-solving, and it's great to see many young people talented in that area. And now, my response to the question, "How do you learn from unsolved problems?"
"I am going to offer some personal experience from both contest math and academic math.
I did competition math for three years. I competed in the AMC 10, AMC 12, Cuyamaca College Field Day, and the UCSD Honors Math Contest. Did I get awards? No, only participation. But I learned something from each competition - problems are tough, and even tougher with time constraints. I was so glad to be around people that shared a love for mathematics and problem solving, and I am grateful to my Algebra II teacher for showing me this side of math.
Now, let's talk about problems I wasn't to solve. I thought about these problems for a long time, years after the contests. I found that it was best to simplify the problem, if possible. That is, solve a problem similar to the one posed. Most of the time I wasn't able to solve it entirely, but I gained insight. These problems I sat on the back-burner for a while.
Let's get to college. Fresh out of high school, my first math class was Calculus III. This was a MIM class in our math program - for Math-Intensive Majors. I wrestled with so many problems in that class and had fun doing so (probably I was the only one having fun, but who am I to say so!). Perhaps one problem I remember the most was from a take-home quiz. It was my first one, assigned the last week of August, and was meant to refresh our skills of integration techniques in Calculus II.
But I learned something: be adventurous. Make wacky substitutions, solve similar problems, and try to come up with the solution.
Now let's jump to the present: two more semesters of undergrad to go (including one semester of student teaching at a local school), and I'm learning how to really write mathematics. One my greatest experiences was this past summer, when a professor and I went through the book Concrete Mathematics as an independent study. Those are TOUGH problems. I wasn't able to solve some, but I got a really good start on them. I compiled the most interesting (and perhaps easier) problems into a paper, which I may post at the request of the forum members.
As my real analysis professor puts it, I have to "wrestle" with the problems. Wrestling with a problem that you may not solve teaches you something - either you find the solution eventually or you get a bit closer to a solution. Another thing I learned from these problems I wasn't able to solve is that mathematics is hard work - extremely hard work. Sure, I don't have to chug out papers analyzing a work by Mark Twain or compile a project on the success of inclusion in Texas classrooms, but I have fun tackling tough problems.
So what did I really learn? I learned what mathematics really is."