Is it worth it to get better at contest math? I have never done well in math competitions, and am now past the point at which I can participate in them. I am asking if it is worth it to go back and practice such types of problems until I gain some level of sufficient level of mastery, or if time is better spent trying to learn more advanced topics and specializing in an area of research. It seems a little odd that I should be struggling on problems that people many years younger than me can easily solve. Is this something I should worry about? Is it possible to be a successful mathematician and not be good at contest math, and are there any examples you know of? What score should a decent mathematician be able to get on the Putnam exam?
I have heard many people express the sentiment that the two types of thinking required for research and contest math are different. But still, if a mathematician couldn't solve any problems on say, the AMC 12, this would be somewhat alarming. I'm just trying to gauge the threshold at which a lack of skills in contest math will not impede research ability.
 A: I wouldn't worry about it.. I didn't do well (by the standards of this site) in math contests in high school. I never dreamed I could even compete at the national level, and was not even sure I'd be going into math. Today I'm a mathematician, and I never made any conscious effort to become better at contest math. If I look at the Putnam exam nowadays, I can do a few of the "easier" questions and also a couple related to my current research area if they're on it. Actually sometimes I'll see a B-6,A-6 etc which is totally obvious to me because I think about similar things.. then I look at a supposedly easy one involving combinatorics or matrices or other things I don't know anything about, and I am not any more able to do them today than 20 years ago when I took the exam. 
I really don't think any of this has impacted my research at all. If there are some contest-type skills you'll need later on, you can pick them up by doing lots of problems (not always of a contest type) and thinking in such areas. I think even if you go into a problem-oriented area like combinatorics it would make more sense to work through Stanley's Enumerative Combinatorics book for example than contest problems, because the former is more organized and sytematically goes through a subject in a way conducive to becoming a researcher in the area. 
A: If someone is very good at contest math this is a good indication that they will be good at math research, but the reverse is just not true. So I understand that you are unhappy about not having some kind of confirmation that you are good, but continuing to practice Putnam exercices will not change that. You will also not get this confirmation by asking strangers on the internet.
Note that success at competitions indicates proficiency at certain type of math skills (and strategy skills), but also quite importantly, it indicates


*

*interest in math

*perseverance

*a certain kind of stress resilience
These qualities can also be indicated by non-contest and even non-math activities.
I suggest to find your strengths and interests and direct your energies there. Obsessing over comparisons with other people will impede your research ability.
A: No, it is completly useless. The fact that they are timed, require no advanced mathematics, often solutions are ad-hoc/brute-force-ish, are good indicators.
Its relevance for research is comparable to that of beeing able to recite the digits of Pi.
A: This is too long for a comment. So I posted it as an answer. Quoted from Tao's advice on mathematics competition: "But mathematical competitions are very different activities from mathematical learning or mathematical research; don’t expect the problems you get in, say, graduate study, to have the same cut-and-dried, neat flavour that an Olympiad problem does.(While individual steps in the solution might be able to be finished off quickly by someone with Olympiad training, the majority of the solution is likely to require instead the much more patient and lengthy process of reading the literature, applying known techniques, trying model problems or special cases, looking for counterexamples, and so forth.)"
A: I think you are not alone in this. I myself have felt this embararssment (that having a graduate degree in Mathematics, I can't solve some problems that says a 17 year old can). Call it rationalizing if you will (and I'll disagree if you do), I think contest math calls upon a totally different set of skills than advanced mathematics. As adjectives they may sound the same (breaking a problem down, recognizing patterns etc), they are in fact quite different and lack of one does not imply lack of the other. IMHO one of the key skills required in advanced mathematics requires one to be able to grasp abstract concepts, which may not be relevant to contest maths at all. I think my answer to your question is, of course it is possible to be a good mathematician without winning math olympiad. I cannot remember any examples off the top of my head but as they say, absence of evidence is not evidence of absence.
When I have earned myself that title (of a 'mathematician'), I'll come back and update this answer.
