Advantages of Mathematics competition/olympiad students in Mathematical Research Everyone in this community I think would be familiar with International Mathematical Olympiad, which is an International Mathematics Competition held for high school students, with many countries participating from around the world. 
What's interesting to note is that many of the IMO participants have gone to win the Fields Medal. Notable personalities include Terence Tao (2006), Ngo Bao Chau (2010), Grigori Perelman (2006), etc.
I would like to know: What advantages does an IMO student possess over a 'normal' student in terms of mathematical research? Does the IMO competition help the student in becoming a good research mathematician or doesn't it? 
 A: I would say that olympiads build some, but far from all, of the skills needed to excel at mathematical research. I'd compare it to running 100 meters versus playing soccer. Usain Bolt is probably a better soccer player than the vast majority of the population, because he could outsprint anyone and because he's generally in fantastic shape. But that doesn't mean he's going to be able to play on a professional team.
Being a successful researcher requires

*

*the ability to learn new fields of mathematics, and develop ways of thinking about them that others haven't.

*the discipline to spend months or years returning to a problem and trying new angles on it.

*(or at least is strongly aided by) the ability to communicate and "sell" one's results, in writing and in talks.

*the ability to write good definitions, that will be useful and cover the boundary cases correctly.

*the ability to form an intelligent guess as to which unproven statements are true and which are false.

*the ability to hold a complex argument in one's head and play with it.

*(or at least is strongly aided by) the ability to find clever technical arguments.

I would say that olympiads are very helpful in developing the last skill, somewhat helpful in developing the fifth and sixth, and not at all in developing the first four.
I definitely, at some points in my research, find myself needing lemmatta which would be fair to put on an IMO or a Putnam exam. And when that I happens I feel myself relaxing, because I know I can do that. But I also spend a lot of my time trying to learn how to think about a subject, or figuring out what to prove, or trying to figure out how broadly a phenomenon holds. And those are not skills which I found olympiad training helpful in.
In case someone wants to know my Olympiad credentials to evaluate this advice, I was the first alternate to the US team in 1998 and, during my senior year of high school, I regularly came in somewhere in the top 10 spots in national (USA) contests.
A: Well the education system in India doesn't provide room for problem solving skill development.Most of my friends and students who have done well at Regional Math Olympiads and INMO take up coaching from elsewhere.They usually start early say by 7th-8th grade.When I reached my high school level I found I that I was just nothing in mathematics.Frankly speaking I am not that good in maths .But sometimes I even amazed myself by solving some Olympiad problems quite quickly!!!.I found that i succeded because the problem was solvable with just jugglery of some very elementary concept.But I often failed with other intricate problems that demanded observing a trick or pattern.
   Now,I feel I should have taken this rigorous approach.At least it does good to your skills and improves thinking.Being a Computer Science student(well I gave up the idea of taking maths further coz I thought I am no good :( )I think that rigor would have been a great asset in my career.
  Well as far as IMO,Putnam etc.. and Fields Medal is concerned I would say that inventing something is a matter of chance but the better prepared guy holds the advantage.
A: Keep in mind that what you have here is a correlation, not a causation. While doubtlessly Olympiad training would help develop some skills necessary for research, I think it is likely that many of the strongest mathematicians participate in these competitions when they are in high school and go on to do research later.
A: I think tricks can be helpful to a tiny extent. I am not sure if short problem solving skills are helpful. I think combinatorics/number theory problems are the most useful practice for doing simple research (in probably combinatorics).
I think the courage/tenacity to solve a problem you are not sure you can solve is what you should develop from olympiad training. I sometimes am discouraged in my dream to do research to mathematics. Sometimes, my self esteem is shaken when I see so many people at my school better/faster/smarter than me at both doing mathematics and learning it (and taking tests and doing psets). But whenever I am discouraged, I am reminded of the hard work I went through during the olympiad years, and I become calmer and more focused and more determined and less afraid of working towards a goal despite the odds. (the last sentence is an exaggeration, but now that I've said, I can see it becoming true)
A: I can say from personal experience that the bulk of people who train for the IMO tend not to become any more exceptional at research than any other person/s who take up the subject - with intent to become a researcher - at university and beyond.
Some of the top-performing students at the IMO - including a good number of Gold and Silver medalists (from the US atleast) - I have known: and I can't say that they became any more exceptional at research than any other non-IMO participants and / or top-scorers.
Basically, the competition tends to make participants into very sharp-minded and 'clever' problem solvers (which, perhaps, has some advantages in some contexts in research); but as far as giving you a -significant- advantage, it really doesn't do much as far as I've seen.
Rather, follow the advice of a well-known mathematician, John Milnor -- think carefully, think deeply, and work patiently and diligently at whatever problem you're working on.
I think you might find that proves the best approach to research, regardless of academic specialization.
cheers
A: I think the high score on the IMO helped get them into a great school where they had great teacher, and that is what really helped them to do good research and also, along with the skills and exercise they did to do well on the exam.  I mean how many Field medals went to student of poorly ranked colleges/universities.   I think it more of the added benfits that result because they got into the best universities and the high IMO score helped their entrance. 
A: Training for competitions will help you solve competition problems - that's all. These are not the sort of problems that one typically struggles with later as a professional mathematician - for many different reasons. First, and foremost, the problems that one typically faces at research level are not problems carefully crafted so that they may be solved in certain time limits. Indeed, for problems encountered "in the wild", one often does not have any inkling whether or not they are true. So often one works simultaneously looking for counterexamples and proofs. Often solutions require discovering fundamentally new techniques - as opposed to competition problems - which typically may be solved by employing variations of methods from a standard toolbox of "tricks". Moreover, there is no artificial time limit constraint on solving problems in the wild. Some research level problems require years of work and immense persistence (e.g. Wiles proof of FLT). Those are typically not skills that can be measured by competitions. While competitions might be used to encourage students, they should never be used to discourage them. 
There is a great diversity among mathematicians. Some are prolific problem solvers (e.g. Erdos) and others are grand theory builders (e.g. Grothendieck). Most are somewhere between these extremes. All can make significant, surprising contributions to mathematics. History is a good teacher here. One can learn from the masters not only from their mathematics, but also from the way that they learned their mathematics. You will find much interesting advice in the (auto-)biographies of eminent mathematicians. Time spent perusing such may prove much more rewarding later in your career than time spent learning yet another competition trick. Strive to aim for a proper balance of specialization and generalization in your studies.
A: Students who manage to make IMO are much smarter than the average math major. People will disagree with me, but anyone who's gone through undergad with IMO medalists will know that I have a point. To make it to the top levels of any competitive intellectual pursuit, competitors must already have a very high level of base intelligence that's transferrable to many other domains. Every IMO participant I know is insanely smart in general, not just in math competitions.
I don't buy these two common arguments:


*

*"IMO students have the same mathematical abilities as any other math student."


This simply isn't true. It's no coincidence that more than half of the participants of the US team get their PhD's, and essentially every US IMO participant who chooses to pursue graduate school ends up getting into a top program and suceeding. Do some Google searching for past participant's names if you don't believe me. It's also not a coincidence that a very significant percentage of IMO medalists end up taking graduate math courses their freshman year of college. That's much more impressive than the average undergraduate math major. 
Fields medals and the like are indicators of extreme outliers, and should not be the main factor used when guaging the general research ability of a population.


*

*"IMO students only train on learning competition tricks."


IMO performance doesn't happen in a vacuum. You have to consider the kinds of kids who win IMO: Smart, motivated, and passionate about math. Many IMO medalists will learn far ahead in high school, and many will learn enough math to conduct legitimate research during high school or early undergrad. This gives them a rather significant head start.
A: First of all, before I comment I would like to state that I am merely a high school student and I apologise if I give people an impression that I am an outsider(I am only a 15 year old and I doubt if my views will be taken seriously).
Participation in mathematical competitions, in my opinion, is not an end in itself.I personally believe that these competitions introduce a student to the rigors of mathematics much before others get a feel for it.Let's take this aspect: An IMO contestant has to attack 6 problems in 9 hours over 2 days.Imagine it for yourself.Someone gaining that experience at an early age ensures a smooth transition to "Real Mathematics".Automatically, the tenacity to attack a problem for a sustained period of time is gained.That is bound to help later on.But I also believe that someone who hasn't participated in these competitions has an equal chance of carving out a good career.
A: Regarding Bill Dubuque's comment about problem solvers versus theory builders there's a nice paper called "Birds and Frogs" by Freeman Dyson about this topic. 
A: Greatest creative mathematician of $21$st century Grothendieck said that

Mathematics is not a competitive sport

Watch  :Interview video of  the Abel prize Winner John Tate
video timing :$44:00-44:10$
He say that  I suppose I’m a theory builder or maybe a
conjecture maker. I’m not a conjecture prover very
much, but I don’t know. It’s true that I’m not good
at solving problems. For example, I would never be
good in the Math Olympiad. There speed counts
and I am certainly not a speedy worker. That’s one
pleasant thing in mathematics: It doesn’t matter
how long it takes if the end result is a good theorem. Speed is an advantage, but it is not essential
On this Articles
"Dr. Gelfand, who often said," You have to be fast only to catch fleas , "sought to teach not only the rules of math, but also the beauty and exactness of the field.
You have to be fast only to catch fleas  mean you have to be fast only to pass the university/college/GRE  exam
For more details read my post
A: IMO participants are way higher in mathematical problem solving capability than their peers.
The question to ask is
1) How many of the non IMO students, made it as a conscious decision?. Like, i dont want to participate in IMO because it is on technique. How many did go that route?. Honestly lot of the guys who are good in maths, would have tried IMO, and national level olympaids, would not have succeeded (nothing wrong it), or made a defensive decision that it can affect their school grades, or their parents made that decision for them. I think we can reasonably assume that the students who decided not to pursue IMO because, it didn't fit the principles will be extremely less.  This is the category where, they can qualify in the national olympiads if they wanted to , but didnt.
2) Some of the folks may have developed interest in maths in their undergrad years, and so would not have prepared or given IMO level exams.
It is outright silly to say that Maths is for the long run and to dismiss the skill levels of IMO folks. Looks to me like sour grapes.
A: My experience with doing some competition math in the US (MathCounts / AIME level, not IMO level) is that competition math exposes students to a lot more math and a lot more problem solving than regular US public school math. The traditional math courses a middle school or high school student takes in the US (through public schools) is something like a basic introduction to pre-algebra, a little bit of linear algebra, the very basics of geometry, and the basics of calculus. I've talked with friends who also went through the US education system and they took a similar set of courses.
All the courses are very "computational" in thinking and we did not prove any of the non-trivial results we used on the homework sets, which were again mainly computational. In fact my undergrad had a mandatory course for all STEM majors called "Concepts of Mathematics" which introduced some basic math tools like basic set theory, induction, functions, number theory, and an overview of the reals/rationals that I believe US highschool students typically get zero education for. USAMO / IMO problems are good to get students to start thinking abstractly and in ways they are never exposed to in, for example quantifying over a set, thinking about induction and "chain" arguments, reasoning with proof by contradiction, thinking of functions as maps between two sets instead of just as curves $y = f(x)$ that can be graphed, etc. The problems also require much more thinking and persistence compared to school work, and building up this persistence is very important in my view. Most highschool students are not used to thinking about one math problem for hours or days, juggling thoughts and intuitive feelings about problems.
