I did math competition when I was in high school and the prize I won opened the door to a top university for me. I would say wrestling with a problem with limited tools (no calculus, no advanced algebra etc), finding a solution as elegant as possible is a game that I enjoyed at that time and still enjoy to this date (I'm close to 30 now, still challenging myself with IMO problems from time to time). But that's it. A game is a game and whatever you do inside the game world, no matter how fancy it is, is not practical in the real world. I didn't feel any privilege compared to fellow students without that experience after entering the university.

Getting into the "higher math" world was really painful for me actually. Seeing how some "modern" techniques (which has already existed for ~400 yrs, not literally modern at all) have solved the problem I struggled with in such an ultimate way made me feel overwhelmed. I felt like a track athlete, gloating over how fast I was running, realizing how modern transportation has revoluted - people no longer move long distances with human power. The quality a good driver needs is attention - on the road and the car simultaneously - not a pair of sporty legs.

The transfer from high school math olympics towards "higher math" requires not only practically a major upgrade in knowledge and toolsets, but also some shift of thinking paradigm - the task is not looking for an elegant and ingenious shortcut to a particular problem, but a highway that is general and inspiring to a fully new field (like how equations opened the door to linear algebra and everything subsequent). I pushed myself to embrace the transition, but it didn't seem to work. I managed to pass the exams and obtained a higher degree, with some expertise in a particular field of application. Still, I always have the feeling that my mathematic understanding is like house built on sand and lack a solid foundation. There is some sort of chasm that I failed to break through...

I agree with the part about mental skills, partially. Experience with math competition improved my concentration and persistence. I found piano practice more contributing in this means.