Problem-Solving and other things in mathematics relations? 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.
 A: Well, you have quite a number of things confused. Here are some points.
1) Contest problems are challenging, but to succeed at them you have to study and practice them. This is because the contests use their own repertoire of tricks and techniques. Although they are good practice in thinking and problem solving, there is no correlation between success in contests and research. Most likely many profs have realised that they have nothing left to learn from the contests. 
2) Research involves a mixture of study, creativity, analysis, problem solving, and judgment (which questions to tackle, which to avoid, when to work harder, when to relax). This makes it hard to pin down and describe exactly.
3)(and this is where everyone is going to get mad at me) Colleges and universitys do not teach any kind of real thinking, in fact there are hostile towards it and students who try to think for themselves are punished in a number of subtle ways.
4)If you want to know what mathematics is about you have to study it.
A: Those are just what I thought.
Yes, mathematics is about ideas, specially about numbers and calculations. 
The difference between professors who does (A) research, or (B) education, or (C) over look the contests, are: professor-A makes a complex problem simple so that it can be solved; professor-B makes a complex concept simple so that student understand; and professor-C makes a simple question complicated so that to pick up the best student from the mass.
As far as I know, the complicated questions are made from one or many very simple facts or smaller questions. How? Just like the parents to hide the fake eggs in the lawn for kits to find out. No difference.
In fact, it is easy to make simple into complex but hard in the other way. Most professors and the students in the competition are doing the hard work. The difference here is, for professors, there is none knows the solution, for the students, the solution is known to exist, just how fast they can find it.
Hope it helps you. The professors do not teach how to compete for a good reason, right? 
