What aspects of math olympiads do you find still useful in your math research? 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? 
 A: 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.
A: My personal experience and view is that there are certain olympiad type problems that, if you can solve them, demonstrate genuinely useful skills applicable in research. 
Often in research you encounter similar questions perhaps as sub-problems to your main problem or in the search for special cases or counterexamples. Unless there is an obvious approach by existing theory progress usually then involves simply making an educated guess and trying to prove it. The ability to "feel" ones way to a good guess in reasonable time say by raw instinct, clever use of heuristics or analogy is vital in this case. Similar skills are needed I believe in solving many Olympiad Questions, especially under time pressure.
For example the "Pentagon Game" discussed on Matt Baker's Math Blog involves a pentagon with integers at the vertices and a rule to evolve those. You have to prove that the game ends in a finite time - the solution involves finding an positive integer invariant that always decreases. Finding this invariant quickly is non-trivial and requires the ability to guess some good options and/or exclude many which won't work. There is no standard theory to fall back on you which is often the case for genuine research problems.  
(See this question for a discussion of Olympiad Questions with connections to real mathematics, many of which fit the above criteria and this paper for more mathematical developments from the "Pentagon Game".)
More generally concerning the question of whether problem solving skills are important to learn and practise for budding mathematicians you might be interested in John Hammersley's view which was certainly an outlier amongst mathematicians at the time - he believed that manipulative skills, problem solving skills were much more important than abstraction and theory which often did not help in solving real world problems.- see his article "On the enfeeblement of mathematical skills by "Modern Mathematics" and by similar soft intellectual trash in schools and universities" 
A: It's my belief that a large part of mathematical research, perhaps more than we would like to admit, comes down to finding clever elementary arguments.  This is particularly true in my own area (combinatorics and related fields) but it is true of many other fields as well.  Of course there's usually some machinery to be mastered, but at the end of the day, you're trying to come up with something new, and it's not so often that you're building some gigantic new machine out of whole cloth. Typically you're taking various known ideas and trying to figure out how to adapt them and put them together in a new way to prove something new.  When the pieces of the puzzle finally fall into place, I find the experience to be not unlike the process of solving an Olympiad problem.  The Olympiad training is useful for building a sense of confidence that something nontrivial can emerge with a bit of persistence and cleverness.  I find that some of my colleagues without this kind of problem-solving background will sometimes give up too quickly, because they are at a loss as to how to proceed when their usual box of tools doesn't apply.
A second way in which I find Olympiad training useful is that when I am confronted with a new and difficult problem that seems too hard to tackle directly, I can often find a way to invent a toy version of the problem whose solution may give some insight.  Experience with Olympiad problems has given me a sense of what a "bite-sized" problem looks like—subtle enough to be nontrivial, but simple enough to be tractable.  Interestingly, I often find that some of my colleagues who are better than I am at solving Olympiad problems can often solve my bite-sized problems when I can't; at the same time, I often seem to be better than those same colleagues at coming up with the bite-sized problems in the first place.  This may partly explain why some Olympiad stars don't become good mathematical researchers.  Research requires several skills, and those who only know how to solve tractable problems and don't know how to formulate them in the first place may not do so well at research.  But I think that experience with Olympiad problems can help with the formulation process as well.
