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.