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I sometimes see people arguing against concentrating too many resources in high school maths competition (such as IMO) training. Their reasons they give are usually the following:

  1. Competitions are a very skewed representation of the subject. They only cover small areas like number theory, combinatorics, "ancient" parts of modern algebra, and the obsolete Planar Euclidean geometry. Major areas (analysis, modern algebra, and modern geometry) are not covered at all.
  2. Nobody disagrees that 16th-century tricks (NOTE: I mean tricks, not big theorems like Pythogras) of Euclidean geometry are not of much use to any professional mathematicians today. Some even doubt if it should be part of the high-school curriculum at all. In general, competition stuff are irrelevant to the research of most of the professional mathematicians in analysis, algebra and geometry (and of course, more irrelevant for other people).

EDIT: the competition stuff might be the application of an advanced theory, but never part of the advanced theory itself.

  1. Although students are likely to learn some knowledge beyond high-school if they manage to get to IMO level, in general, for most students who are training for competitions, the contests tend to limit them within the scope of high-school contents. This, again, creates a biased picture of the subject in their mind.
  2. Some people do not like the fact that maths is made into a "sport" - the long hours of training are laborious, but does not necessarily lead to deep thoughts.

All those reasons (especially points 1 and 2) appear to me that they are implying the following thing (although this is rarely explicitly said) - students who are good at maths would better not pursue their interests through competitions; instead, they can learn higher level university-level maths in advance.

It appears that in some countries, intensive competition training is unnecessary, as the resources used in contests is more than sufficient to allow promising students to learn undergrad math in advance (if less attention is paid to contests).

I know it is slightly opinion-based, but there doesn't seem to be similar discussions before, so should students learn higher-level university-level maths in advance or do contests?

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    $\begingroup$ How about this question: Do exercise for personal fitness, or train for the Olympic shotput: which is better? $\endgroup$ – Gerald Edgar Nov 30 '19 at 22:27
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    $\begingroup$ @GeraldEdgar Personal fitness and Olympic shotput require an entirely different amount of efforts; training for a competition and taking an advanced class requires roughly the same amount of effort. $\endgroup$ – Ma Joad Nov 30 '19 at 22:57
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    $\begingroup$ @MaJoad I think you miss Gerald's point: Which option - (1) exercise for personal fitness, vs. (2) train for the Olympic shotput, is going to help you most in the long run? (Hint: (1) is conducive to lifelong fitness, while (2) may allow you to peak for one or two Olympics, but not much beyond that.) $\endgroup$ – amWhy Dec 1 '19 at 0:08
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    $\begingroup$ I find it disheartening that Euclidean geometry is considered obsolete. It's still very useful in many capacities: at work as an engineer, or even just doing basic home maintenance! But the most important lesson is how to think logically. How do you prove something given only a certain set of rules? And that... well that is useful in almost every future career and life circumstance! So while we may have computers to do much of the heavy lifting, geometry in the classroom is anything but obsolete!! $\endgroup$ – corsiKa Dec 2 '19 at 3:53
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    $\begingroup$ I'm wondering, Ma Joad, why think think there is only one correct answer: either math competition, or else, advanced math classes?? Each ambitious student is different. There is no "one size fits all" answer to your question. $\endgroup$ – amWhy Dec 2 '19 at 15:52
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I'm not sure whether your question aims at educators or at the students themselves, but my answer actually would be very similar.

Having as well as being a good, perhaps gifted student is a treasure. It's fun both to teach and to be one. The subject is fun. Conversations are fun. New angles are appearing which were not obvious, perhaps not even to the teacher. If the talent is substantial it is likely that the student will pursue a higher education and later a profession which will be both reasonably fulfilling and reasonably lucrative. The only thing that can go wrong is that the student loses interest, that the subject is no fun any longer.

My advice is to let the student choose. Tease them, make offers, prepare material, let them participate in an olympiad, and also expose them to college-level math through tutoring or simply through text books. And then let the student choose their topics and their way, and support them without overtaxing them. The future will be bright no matter what; perceived usefulness for a career should be secondary, not least because we don't know the future ("leave those large prime numbers alone and study something useful!").

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Short answer: The skewed content is not a good reason for avoiding IMO-style contest training, because if the training is done right then the students will be led to explore mathematics and would never have a mistaken picture that mathematics is mostly about IMO topics. (I of course compare between good IMO-style training and good teaching of university-level mathematics, because there is no point comparing lousy training with anything.)

Long answer: (Note that the question has been edited slightly so my responses were to the original points.)

I believe there are concrete mathematical pedagogical issues relevant to your question. A major point is that IMO-style mathematics is a significant step up from the blindly rote-learnt mechanical 'math' that pervades many high-school education systems (both in the syllabus and in the delivery). This is because there is a proper focus on rigorous logical reasoning in the form of proofs. I think that this is the most important reason to value IMO-style problems, and should be properly emphasized in any competent training on IMO-style problems (whether or not with the goal of participating in contests):

  1. Students should be taught logical reasoning in conjunction with how it is used to solve IMO-style problems (starting with easy problems but with fully rigorous solutions). The objective is that they should be able to carefully, accurately and confidently identify what exactly is proven, what they wish to prove, and what assumptions they used, throughout the solving process, and similarly perform the same analysis at any point in someone else's proof.

  2. Students should be trained to write logically valid proofs, not just to write handwaving arguments or a messy bunch of statements or equations with imprecise logical structure. If any deductive steps are skipped in a proof, the student must be able to produce upon being asked a correct subproof to justify the jump.

  3. Students should also be trained to analyze others' proof attempts and correctly verify them or identify logical errors or gaps in the argument. Even if the teacher claims that a proof is correct, but actually is wrong, the students should be able to confidently reject the teacher's claim. In other words, an incorrect argument from authority must lose to the students. This is a litmus test that the students actually know what they are doing.

I also want to address several misconceptions revealed by some points in your question about competitive mathematics. You are definitely right on certain points, such as the fact that contest mathematics tend to be restricted to a small 'syllabus'. This is both a disadvantage (as you noted) and an advantage, because contest problems are usually much more interesting than 'standard' mathematics. But in my experience not all your points hold.

Nobody disagrees that 16th-century tricks of Euclidean geometry are not of much use to any professional mathematicians today. Some even doubt if it should be part of the high-school curriculum at all.

Well, here are some that I think professional mathematicians ought to know.

The compass-and-straightedge construction of the square-root of a given ratio of lengths is directly relevant to the proof that the constructible reals are precisely those that can be obtained by a sequence of quadratic extensions. This result is an important one in the history of Galois theory, and I think it should be taught in any introductory course to field theory.

Another one is the proof of the Pythagoras theorem, which is needed to give a real motivation for the modern notion of Euclidean spaces. Related are all the applications of circle and conic geometry in classical mechanics and optics, which is very much a part of applied mathematics.

Yet another often overlooked tool in geometry is inversion, which is closely related to the Mobius transformations in complex analysis, as well as the more general conformal mappings.

In general, competition stuff are irrelevant to the research of most of the professional mathematicians (and of course, more irrelevant for other people).

Let's look at the IMO syllabus specifically, which roughly covers elementary number theory, combinatorics, graph theory, inequalities, Euclidean geometry and functional equations. In my own experience, all the concepts in number theory, combinatorics and graph theory in the IMO syllabus also show up in any university-level courses in those topics, and these form foundations for higher-level courses. It is true that many things in the other topics do not show up in university courses, but still many do.

For example, the AM-GM inequality is a basic one that every mathematician must know, and higher mathematics also relies on some IMO-level inequalities such as Jensen's, Bernoulli's, Cauchy-Schwarz, Chebychev, Holder's, and the power mean inequality. The concept of smoothing is also a very important and productive one, as are the various other generic techniques commonly used in IMO including homogenization, re-parametrization, splitting of the domain, even if we scrupulously avoid whacking techniques that many IMO students learn such as Lagrange multipliers.

I have given some examples of Euclidean geometry applications above, but one major advantage of learning Euclidean geometry is that it is a beautiful and vast playground suitable for learning rigorous logical reasoning with focus on propositional logic, as almost all Euclidean geometry problems can be solved in pure propositional logic.

As for functional equations, I would say that some of it (such as learning the solution of well-known functional equation $f(x+y) = f(x)+f(y)$ for continuous $f : \mathbb{R} → \mathbb{R}$) provides the IMO student a glimpse of real analysis. The concept of transforming the function itself is also a very important one (such as in reducing the functional equation $f(x·y) = f(x)·f(y)$ for continuous $f : \mathbb{R}^+ → \mathbb{R}$ to the previous one), and such techniques show up in all branches of mathematics.

Although students are likely to learn some knowledge beyond high-school if they manage to get to IMO level, in general, for most students who are training for competitions, the contests tend to limit them within the scope of high-school conte[s]ts. This, again, creates a biased picture of the subject in their mind.

You may be right that training for only high-school contests, if they do not involve proof questions, would create an inaccurate picture of mathematics. However, in many countries there are high-school contests with final stages involving proof questions. It would be sad if students merely trained to get some results on only multiple-choice or short-answer questions. As for IMO level, most IMO participants I ever got to know did not ever stay within the 'confines' of IMO-style mathematics, but on their own delved deep into other topics in mathematics as well.

Some people do not like the fact that maths is made into a "sport" - the long hours of training are laborious, but does not necessarily lead to deep thoughts.

I do not agree with excessive training in only one thing, whether it is mathematics or not, if other important subjects are neglected. On the other hand, long hours of training is typically necessary to become an expert at anything. Some people say that an average person needs roughly 1000 hours to become reasonably good at something, and 10000 hours to become an expert, and I think it is roughly right.

students who are good at maths would better not pursue their interests through competitions; instead, they can learn higher level university-level maths in advance.

I have some gripes with contest mathematics (such as the fact that many of them including the IMO use problems that have appeared before), but I would say that there is much benefit in learning and even training for IMO-style mathematics. As I mentioned earlier, contest mathematics is usually far more interesting than modern mathematics to a high-school student, so there is ample motivation to explore.

Also, in my opinion, it is a rather 'clean' playground. Earlier I mentioned Euclidean geometry as being suitable for learning propositional logic. Even the other IMO topics are more suitable than modern mathematics for learning first-order reasoning.

For example, it is easy to learn induction, strong induction, structural induction, well-ordering, and the extremal principle in the context of combinatorics and graph theory, and there are numerous cute problems involving these tools in their solutions. I personally consider full first-order induction as a key part of logical reasoning, but sadly numerous undergraduate math majors cannot use it properly.

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    $\begingroup$ Just some points to clarify: 1) Of course everyone ought to know Pythagoras. Things which are unnecessary are those "tricks" (e.g. constructing 6 extra lines before you can provide proof). Same thing for AM-GM. Of course, they are useful, but not in the same way as they are in competitions. 2) The rule and compass construction is an application of Galois theory, not the theory itself, so it is not essential. Again, everyone ought to know rule-compass construction, but I don't believe that the level of complication in IMO is necessary for research. $\endgroup$ – Ma Joad Nov 30 '19 at 23:09
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    $\begingroup$ @Namaste Yes. My focus is on whether we should take advanced class instead of competitions. $\endgroup$ – Ma Joad Nov 30 '19 at 23:15
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    $\begingroup$ Note the reason why contests are a skewed representation of the subject: one cannot deny that none of analysis, algebra and geometry is covered. However, they are the biggest areas of research. This has nothing to do with logical reasoning training. I just mean the content is skewed. $\endgroup$ – Ma Joad Nov 30 '19 at 23:19
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    $\begingroup$ Are you sure about that $f(x+y)=f(x)+f(y)$ for continuous $f:R\to R$? $f(x):=x^2$ is continuous but $f(2+2)=f(4)=16 \neq 4+4=f(2)+f(2)$ $\endgroup$ – JJJ Dec 1 '19 at 2:43
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    $\begingroup$ @JJJ: Look up the wikipedia article on "functional equation"... The example I gave is even a well-known one that appears in that article. $\endgroup$ – user21820 Dec 1 '19 at 6:54
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One rather major argument in favour of contest-style mathematics is its ability to cultivate problem-solving abilities in students while not requiring much difficult machinery. It is of course undeniable that the ways of thinking in contest math are much different from the ways of thinking in university mathematics, but what both have in common is a requirement to be able to observe patterns, make deductions based on known facts, deduce results systematically and formally, et cetera. The problem solving abilities which contest math can cultivate can carry over naturally to university mathematics, and I think that is one of the main values in contest math for those seeking higher education in mathematics.

So regarding the point that "tricks" used in contest math (e.g. most of Olympiad Euclidean geometry) are irrelevant in university math---that's true, but I think it rather misses the point. If one wants to investigate mathematics as a subject, obviously one should not restrict themselves to Olympiad topics or Olympiad methods. It's a bit like trying to solve a Number Theory question of research interest using only elementary contest-level mathematics; it's completely hopeless. The point of contest math is, instead, as a greenhouse or playground of sorts, to perfect one's mental acumen before being introduced to more advanced tools and machinery to tackle mathematics in general.

Back to the question of which is better: I think I concur with Peter that a student in this position should follow wherever their passion and interest takes them. And they should do so with the recognition that whichever they choose to go for (or even better, a healthy mix of both) will benefit their own mathematical ability. Choosing to do contest math is not as useless as OP seems to imply.

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    $\begingroup$ Strong agree. Friendships aside, by far the most valuable thing I got out of the IMO program was experience in creative problem-solving. $\endgroup$ – Geoffrey Brent Dec 1 '19 at 21:32
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I personally think taking advanced math would be way more fun than preparing for some contest largely based on tricks.

I'm not convinced these contests are very revealing of much for most students. I've had students who went on to be successful in graduate school in nontrivial pure math and yet scored barely above the noise in one of these contests. On the other hand, his peer who was of similar intelligence and mathematical creativity took the contest and smoked it. Why did the second student kill it ? Simple, he was schooled in high school for the math Olympiad. Of course contests do attract elite students and one can always give evidence to correlate contests with excellence later in mathematical life. I doubt that excellence has much to do with the contests. I think it has everything to do with the contestants.

My advice, focus your attention on what is most interesting. If you don't like contest math, find something else interesting in which to invest your time.

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One thing that hasn't been mentioned yet is the social aspect. There are always some people who like to stay alone, but for the large majority, meeting like-minded people and engaging in some sort of activity with them is much more fun than sitting at home and staring at textbooks. Especially at the higher level, where some traveling is involved, competitions are much more than just exams, they are a chance to meet like-minded people for a weekend and spend some time with them.

Contrast that with doing university maths early. Even if there is a nearby university where you can attend some classes, you'll be the odd one out. You might make some friends as well, but fundamentally you are in a different phase of your life. They might go out for a party after class, you'll go home to your parents.

I would not advise against doing university maths early, if the student is really interested in doing so. But even in the best case scenario, where the work can directly be turned into credits and the student does not have to bore himself through repetition of known topics at university, there is not that much gain in progressing through university one or more years earlier. Being ahead of people of the same age is a big thing at high school level, with huge implications and big discussions when students want to skip a grade. But being ahead at the end of university is almost negligible. A 27 year old who just finished a maths PhD is not fundamentally different from a 30 year old who just finished a maths PhD.

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    $\begingroup$ +1 for "Even if there is a nearby university where you can attend some classes, you'll be the odd one out." As someone who has been in such a position, I know from experience it can be lonely to be the only one (or one of a few) doing university mathematics. From a purely social aspect, contest math does seem more appealing. $\endgroup$ – YiFan Dec 2 '19 at 0:16
  • $\begingroup$ @YiFan Taking credit-bearing college classes while in high school can shave one or even two years of college, thus saving both time and money. $\endgroup$ – Rusty Core Dec 10 '19 at 22:13
  • $\begingroup$ @RustyCore I never said that wasn't the case. Indeed, I chose to focus on college-level classes instead of contest math during my high school years, so I'm well aware of the benefits of doing so. But the points mlk raised regarding the social aspect are very important to consider as well. $\endgroup$ – YiFan Dec 10 '19 at 22:26
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Talented students with interest are much better served by learning more math than by being trained at contest math. Where contest math may play a positive role is in getting talented students interested in other kinds of math, something that might otherwise be difficult to do.

In practice students are often guided/directed to contest math rather than learning more advanced math for a number of essentially practical reasons, among which are: the teachers the student knows do not feel comfortable guiding/directing/teaching more advanced math with which they themselves do not feel sufficiently comfortable to teach; the books and other resources to which the student has access are not adequate (no decent library) and the student lacks the autonomy to filter and search herself; teachers have a professional interest in developing teams that do well in math contests; the social aspect of the math team. There are of course other reasons.

Some of these reasons are more compelling than others. The social, team aspect can be a solid reason for some students. The inadequacy of library resources was in the past a serious conditioning factor that now is far less relevant because good resources are available via the web. However, the searching/filtering problem is a serious problem, perhaps more serious now than in the past (if there was a library, it was likely to have some decent books among the few it had). The most serious practical problem is that there is not available a teacher/guide who feels confident acting as such for such a student. It is difficult to learn more advanced math without guidance.

Said all that, I think it is socially more valuable to train students to do more advanced math than it is to train them to do contest math, and students who manifest interest in learning more math are not always well served by being steered to contest math. Here I admit that my personal experience in this regard was very negative. In high school I was interested in learning more math, got a hold of some books on groups and rings from the library (the local library had a lot of the Carus mathematical monographs published by the AMS), and tried to learn from them. I struggled because of lack of guidance. The teacher who could have acted as a guide tried to steer me into contest math, I think both because he felt uncomfortable with groups and rings and the like and because he was interested in having a strong math team. I disliked contest math and didn't particularly like the accompanying social atmosphere, and the whole set of circumstances wound up driving me away from studying math for a 3 or 4 year period that ended only in the middle of my university studies. I wound up a mathematician anyway, and I still like algebra, so I probably would have been better served by learning more math and following my interests. However, the practical problems are serious - looking back I don't see who else around me then could have provided the necessary guidance, and I lacked the autonomy to do it all myself (some students have this autonomy and become really good mathematicians if they somehow get access to the resources necessary to develop their inclinations).

Perhaps the summary is - guide the student where the student wants to go - if the student enjoys contest math, then it can be an excellent vehicle for learning about other kinds of math - but if the student, although showing talent and interest, does not like contest math, better to try to provide access to other stimulating resources. Extreme mathematical talents often manifest themselves early, sometimes in the early teens, and it's a shame if they are not developed.

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I don't know the answer, but clearly understand the question...which means it is a great question.

I think it will depend both on the student (interests, abilities) and the situation. For the situation, it probably includes quality, but also pedagogy (efficient approach) as well as fun factor. Consider the difference between just having a library card and hitting the stacks for some math books or having this generation's Jaime Escalante to entertain you in AP Calculus class. (I have described the advancement option, but same thing might apply for the situation variable of IMO.)

The IMO is better for motivation/fun. The advanced classes are better for usefulness. I haven't a clue how to decide which is more critical, but would maybe try to do a bit of both, allowing student interest to dictate how much time goes where. Since either direction may not succeed, given distractors, perhaps following the one of interest is more likely to end up not being an abandoned experiment.

The other variable that the question seems to interestingly ignore is NON-MATH uses of the time. Advancement in other studies (I realize this forum is full of ex math majors, but they shouldn't assume all "likes math" kids will go into math, versus physics or engineering, etc.). Or time spent on non-academic pursuits.

My gut feel is the IMO thing is probably better. Ideally, it will teach participant some humility, as they will come up on people far better very quickly. And they will still pick up advanced classes fine.

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    $\begingroup$ "The IMO is better for motivation/fun." This will depend greatly on individual factors, it's definitely not more fun for everyone. Representing it as such is misleading. $\endgroup$ – YiFan Dec 1 '19 at 13:17

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