TITLE: What aspects of math olympiads do you find still useful in your math research?
QUESTION [24 upvotes]: I was rereading the book Littlewood's Miscellany and this passage struck me:

It used to be said that the discipline in 'manipulative skill' bore 
  later fruit in original work. I should deny this almost absolutely - such 
  skill is very short-winded. My actual experience has been that after a 
  few years nothing remained to show for it all except the knack, which has 
  lasted, of throwing off a set of (modern) Tripos questions both suitable 
  and with the silly little touch of distinction we still feel is called for; 
  this never bothers me as it does my juniors. (I said 'almost' absolutely; 
  there could be rare exceptions. If Herman had been put on to some 
  of the more elusive elementary inequalities at the right moment I can 
  imagine his anticipating some of the latest and slickest proofs, perhaps 
  even making new discoveries.)

I would like to ask a question to former math olympiad students who now are actively involved in math research. Do you find the training for olympiads useful in later research career as a mathematician?

REPLY [8 votes]: I'd point out a trivial thing: often, you just need to sit down and do a calculation, or a case-by-case analysis, etc. I mean something that can't be directly fed to Mathematica, requiring higher-level reasoning, still technical enough so that you need to crunch it with pen and paper. 
Olympiad kids are trained to concentrate on such tasks and do them quickly and cleanly. Or course, any research mathematician, knowing the problem boils down to a computation, will be eventually able to do it. But they might spend more time, get distracted, make a mistake, spend even more time because of that, etc. Even more importantly, you often don't know in advance whether the outcome of your computation will solve the problem at hand. The quicker and more confident you are at such things, the more you can try.