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?

Matonya 2022-08-12 Answered
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?
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Answers (2)

Trevor Copeland
Answered 2022-08-13 Author has 21 answers
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.
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schlichs6d
Answered 2022-08-14 Author has 2 answers
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.
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