I am a beginning graduate student in Mathematics. When I was in school, there were lots of math programs like summer programs to hone problem solving skills. But at the graduate level, I cannot really find any such program. I am afraid this lack of regular grilling might make my gray cells be less effective after a while.
I do understand that it is difficult to create a program for research level students because, well, they are not school kids you could give hints to, and check their progress; I mean, the idea seems a bit funny. Also, there are various math topics, a lot of these topics require sufficient theory that cannot be expected from all research students, who generally tend to be specialized in a particular sub-topic.
But all this lack of activity makes it very difficult for me to stay motivated. In school there was this continuous motivation to solve problems because we had olympiads and national entrance exams in mind, but that suddenly vanished after entering college. Looking back, I must say I am disappointed by the lack of continuity from school to undergraduate studies, which mainly dealt with reading theory and solving exercises. In school we used to look up to math summer camps but that is just not present today.
I sincerely believe that what I am doing now is a lot of reading, which I am not sure will help me eventually, because I feel my gray matter is not able to work as efficiently as it used to before. For example, I was looking at some number theory solutions I had written as a school student and it's very hard for me to fathom how I could think in those directions.
I do not really wish to become a yawning graduate student who has lost all problem solving skills. Will it be helpful to solve Olympiad problems once again? A quick internet search tells me that answers posted on this site clearly mention Olympiad problem solving to be different than open ended research, so it would probably not be that helpful.
I am desperately looking for some solution. I love research a lot but as you know, it is difficult to attack unsolved problems as they take time, patience and a lot of reading. But at the same time, if I do not exercise my problem solving skills, they may get lost.