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I was reading a note of Hojoo Lee on inequality which is written for International Math Olympiad (IMO) participants. Although he writes that “target readers are challenging high schools students and undergraduate students“, it appears to be quite advanced.

It occurred to me to ask, do these IMO problems contribute towards research work in math? Do these math notes/books give good overview for research work?

I am not interested in examples of Fields Medal winners who had previously participated in the IMO.

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    Olympiads are for competition, find or inspire talent, not to open new fields of research. – Greg Mar 14 '17 at 3:57
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    If you participated, that means you will be in demand by top universities. After you get in, which shouldn't be a problem, you get better resources ($ or other liked people), and hence you will be in a better position to make significant contributions. You basically have the Matthew Effect. – Prof. Santa Claus Mar 14 '17 at 4:52
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    Several academic colleagues I know did these types of competitions. They demonstrate interest, motivation, out-of-box thinking and many other qualities useful for research. In addition, of course, you need also long-term thinking to be successful, but some essential aspects of personality can be seen already in competitions. – Captain Emacs Mar 14 '17 at 9:16
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    Closely related: math.stackexchange.com/questions/4846/… – Sarastro Mar 14 '17 at 11:29
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    The first part of @Greg's comment supports a positive answer to the first question (interpreted broadly): yes, IMO problems contribute to math research because they inspire smart young people to pursue mathematics. – Kimball Mar 14 '17 at 21:10
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I think of Olympiad problems more as "parlour tricks". They're really difficult, and it's super-impressive if someone's good at them, but the skills are very different to the skills you need in research. As a big example of a difference: the Olympiad rewards quick accurate leaps of reasoning, because you're under such time pressure. Research rewards long-term grit and persistence through blind alleys and repeated failure.

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    In the very next sentence he acknowledges, that they are indeed "impressive". He does not belittle those skills in any way. This is a good answer. – student_of_mathematics Mar 14 '17 at 14:18
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    This is the same circumstance with programming competitions in which individuals aim to solve a puzzle in the quickest time or with the fewest lines of code. It's an impressive skill and a clear exercise of intelligence, but not very adaptable to, say, engineering software in a large-scale product. – 8protons Mar 14 '17 at 17:11
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    @Spork Something can't both be 'really difficult' and a 'parlour trick' - this is incongruous. No, but figuring out a trick and using it all in short time can be really difficult. – Joshua Taylor Mar 14 '17 at 17:50
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    Something can't both be 'really difficult' and a 'parlour trick'.Counterexample. Counterexample. Counterexample. – JeffE Mar 14 '17 at 21:22
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    @JeffE touché, and thank you for the entertainment. I'm surprised by how many people disagree with my understanding of the term 'parlor trick' (which, I thought, are simpler things than 'tricks' in general). Point taken. – Spork Mar 15 '17 at 10:41
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Disclaimer: I participated in the International Math Olympiad, and have a PhD in operations research, which is essentially a type of applied math.

There is some overlap between math olympiads and research math. However, as others have noted, mathematics is a very broad field, which includes subfields such as: algebraic topology, theoretical computer science, combinatorics, control theory, optimization, statistics/machine learning. The amount of overlap with math olympiads depends very much on what subfield of "research math" you are referring to. Math olympiads have more overlap with say combinatorics, and less with say control theory.

In training for math olympiads, I learned how to try special cases to get intuition about how a problem works, and how to simplify a problem step by step, and how to write a logical and complete proof. I also learned how to persevere, and to enjoy the challenge of tackling problems which are difficult, and also when to give up when sometimes I am just stuck. I think that these are skills that also are required in research math.

Therefore, I think that there is positive but imperfect correlation between performance at math olympiads and performance in research math. If you don't do very well at the IMO, you can still be a successful pure math professor; and if you get a perfect score at the IMO, that does not mean you are guaranteed to have a successful research math career.

Finally, the math olympiad is an artificial competition, in that the problems in the olympiad can all be solved in a fairly short time with a relatively small set of tricks. On the other hand, in the real world, research math is much more open ended, you need to find and define your own research problems, and oftentimes the problems cannot be solved!

I would use the analogy that olympiad math is like an RPG such as The Legend of Zelda: Breath of the Wild, whereas research math is like real life, extremely messy and open ended.

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    I want to upvote every paragraph individually and the second (non-disclaimer) paragraph twice! – Greg Martin Mar 14 '17 at 19:56
  • How do you train for the IMO? – theonlygusti Mar 15 '17 at 8:11
  • +1 for pointing out that the training is part of participation – Michael Klein Mar 15 '17 at 13:48
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    @theonlygusti I could write a whole essay about how to train for the IMO. I recommend that you refer to the site artofproblemsolving.com – I Like to Code Mar 15 '17 at 23:54
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I'll go against the other answers and say experience with math Olympiads helps a student to become a better researcher, although in a limited way. Math Olympiads give you a larger "bag of tricks", with which you can solve faster the easy-medium problems that you may encounter during your research.

Moreover, you arrive at university with a larger math background and understanding, struggle less with the material and are probably more likely to retain what you see in the lectures. For instance, separating the nontrivial ideas from the tedious details is much easier when you already have a lot of experience in solving problems and writing proofs.

And in maths, everything you know can suddenly become useful in another field. It's useful to have already seen something. As a numerical analyst, I have on occasion used ideas from other fields in my research: combinatorics, algebra, inequalities...

That said, Olympiads tend to produce "problem solvers" rather than "theory builders", and some students burn out after doing maths for so many years (but it's a very small minority) or lose focus in the lectures at the undergraduate level because they find them not challenging.

Disclaimer1: I have been an IMO contestant twice, and now I am heavily involved in the organization of the Italian Math Olympiad.

Disclaimer2: all of this is anecdotal (but so are all the other answers I have read up to now). I don't know if there is any rigorous statistical investigation on that.

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    I took part in the B(ritish)MO but didn't make the IMO. I don't think the insights gained from the BMO really helped with undergraduate mathematics, let alone research (which I didn't do), but of course the level I was at, my total preparation and retrospective would be best expressed in hours rather than days. But having the sort of interests that lead to me trying the BMO in the first place certainly did make undergrad easier. To some quite large extent, participating in these competitions is a symptom of being good at mathematics, not a cause of it! – Steve Jessop Mar 14 '17 at 14:10
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    @SteveJessop 1. I agree with you on the cause/effect relationship, and this is certainly one of the reasons why quantifying the influence of math Olympiads is difficult. 2. At least in the way we structure the competitions in Italy, there is a big step in the amount of study and theory needed to compete at the national and international level. I believe that the preparation that may mostly affect positively your future research career is the one for international competitions. – Federico Poloni Mar 14 '17 at 15:00
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    @SteveJessop I'm not sure what your basis is for saying that insights from the Olympiad wouldn't help with mathematics research, since you didn't do that. Research in any field is very different from studying it at undergraduate level, so "it didn't help me as an undergrad" doesn't let you conclude "it wouldn't help a researcher, either." – David Richerby Mar 16 '17 at 12:00
  • @DavidRicherby I think there is a greater burden of proof on those claiming that Olympiads help with research rather than being symptopm-not-cause as Steve Jessop suggests. You're welcome to email me if you want particular discussion but some indication of my POV is mccaughan.org.uk/g/personal/maths.html (Disclaimer: I did some pre-BMO competitions but did not put myself forward for the selection competition to BMO, let alone IMO) – Yemon Choi Mar 16 '17 at 15:11
  • @YemonChoi This whole answer is a justification of the claim that Olympiads help with research! Steve made a claim to the contrary with, IMO (*baddum-tsh!*), very poor justification so I called him out on it. – David Richerby Mar 16 '17 at 15:27
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I think that there is only a weak correlation between capable researchers and Math Olympiad champions. In fact, some of those "math savant" types make poor researchers since they cannot frame their mathematics in formal and well constructed ways. At my university, we had a few students score in the 40s and 50s on the Putnam exam. I think two of these six or so students went on to a graduate degree in math. And it took one of them three years after graduating to finally get around to doing so.

Also, the problems on exams like the Putnam exam and the Math Olympiad exams are already established mathematics. It would, in my experience, require large leaps to produce publishable research from such questions.

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    I don't know the Putnam exam, but Math Olympiads already require you to write in formal and well constructed ways. I speak from experience only on a national level (in Germany), but it was always necessary to formulate correct proofs and I can hardly imagine it would be different at the IMO. – dirkk Mar 14 '17 at 11:39
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    "I think that there is only a weak correlation between capable researchers and Math Olympiad champions." Over which population? Do you mean that an IMO participant isn't much more likely to become a good mathematician than a random guy on the street. – JiK Mar 14 '17 at 12:48
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    Successful Math Olympiad competitors are no more "math savants" than trained researchers: their success is the result of a long, thorough, and deep training process. The whole "special math genius" trope is harmful both to society at large (where it puts people off of mathematics) and within mathematics (where it fools us into valuing apparent genius over hard work). – Greg Martin Mar 14 '17 at 19:54
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    The correlation between IMO success and research success is indeed less than complete, but by calling it "weak" you are understating it a lot. Everyone from the German IMO team in my years (2004-2006) is involved in research (maths or CS) today. Same holds for the majority of the top 20 from IMO 2004 (just one example). This is not to say that the rest have failed at research -- they might have chosen a career that suits them better. – darij grinberg Mar 15 '17 at 3:53
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    At every International Congress of Mathematicians since 1990, at least one of the Fields Medalists was previously a medalist at the IMO. Seven of them got perfect scores. (Fewer than 200 people have received a perfect score through 1995, the most recent year relevant to the Fields Medal.) – A. Rex Mar 15 '17 at 6:27
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There are very different areas of mathematics, some are more theory-oriented, some are more problem-oriented. Theory-oriented areas (like e.g. algebraic geometry) are built from bottom to top, while problem-oriented areas (like combinatorics, discrete optimization) offer you a bunch of methods that are suitable for solving problems and you need to cleverly combine them. The spirit of the latter areas is much nearer to the "Maths olympiad feeling" while the former areas require different skills.

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Math competitions serve more as platforms for students to join the mathematical community than as actual tools for research.

Many of my mathematically inclined friends and I would probably not have developed an interest in math until college (or not at all, if some other subject got to us first). The impetus to compete in middle and high school led us to develop curiosity in the field, eventually leading us to pursue more advanced mathematics and enter the research world.

However, to directly answer your question, math competitions themselves are not directly used for research purposes.

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do these IMO problems contribute towards research work in math?

No, not the problems themselves. IMO problems must be unambiguously solvable in a few hours. They are not research questions. However, you can easily make an argument that the IMO process helps mathematical research by encouraging young talent (see some of the other answers).

Do these math notes/books give good overview for research work?

No, if you mean an overview of current research. Research is heavily sub-specialized. IMO must be accessible to not yet super-specialized high school students. However, in a very general sense, IMO questions and research both involve a fair bit of writing proofs; perhaps IMO can be considered an "overview" for research in that sense.

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There is some IMO-problem which was influencing research-math: https://terrytao.wordpress.com/2009/07/20/imo-2009-q6-as-a-mini-polymath-project/

In short it is used for testing

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