Problem-Solving and the standard curriculum in typical undergrad mathematics seems to be on different levels of difficulty. IN undergrad math, you learn new concepts and try some problems. However, although the problems in math contests require less concepts, they are actually harder than the problems solved in undergrad math if the playground is parallel. And it seems to me that math professors doesn't necessary know how to solve competition problems.I've heard that some of the competition problems on USAMO can be parallel difficulty of research math.
Is it true that some professors emphasize teaching and education in general while others do research? Because I really feel some math instructors teach brilliant classes but don't seem to do very well on those high level competitions such as USAMO or Putnam. My question is what really is mathematics. I heard it is about ideas. But what kind of ideas? Ideas that focus on problem-solving or ideas of developing new areas of mathematics or others? And are they fundamentally connected? Moreover, what is more important and what is better to pursue for different kinds of people? Maybe this is a silly question, but I ponder.