Do techniques in contest math differ from those used in solving advanced (but *not research level*) course problems?

Question

How does contest math differ from challenging course/textbook/test problems (not research math)?

There are several good posts on math.SE describing how contest math differs from research math. Most of these emphasize that contest math problems are short, time limited, well defined, and have known answers, whereas research problems are open, long duration, change as you work on them, and don't have known answers.

The consensus on math.SE seems to be that practicing contest math, therefore, isn't helpful for other "proper" math - that is, math aside from contests.

My question is: Challenging course problems are also short (solved in at most a few hours), well defined, and have known answers. Yet, they seem very different than contest math. I'd like to understand the difference.

One difference is that good textbook problems involve new concepts, which are often absent (and assumed known) from contest math. But this isn't entirely accurate: many good course problems (and even famous research) involve technique. Coming up with a good technique is 100% proper math (not simply a contest trick).

Are the techniques used in contest math different from used in other (non-research level) math? How?

One point: It feels that contest math techniques are very specific to the problem at hand, whereas other math techniques generalize better. But it's hard for me to argue this point: Many contest techniques generalize well, and many "proper" math techniques are of limited use. Besides, shouldn't the skill of devising a technique be independent of how many cases it is used for?

Another point: It feels that contest math involves removing intentional obfuscation, whereas proper math involves easily generalized examples. But here too: Often a tough problem requires seeing that it is really something simpler in disguise. Why is this only valuable in constest math?

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Relevant math.SE posts quotes:

• In Is it worth it to get better at contest math?, the top answer is "No, it is completly useless. [Contest math is] timed, require no advanced mathematics, often solutions are ad-hoc/brute-force-ish.... Its relevance for research is comparable to that of beeing able to recite the digits of Pi."
• In Is it worth it to get better at contest math?, Terrence Tao is quotes as "But mathematical competitions are very different activities from... mathematical research... [which] require[s].... the much more patient and lengthy process of reading the literature, applying known techniques, trying model problems or special cases, looking for counterexamples, and so forth."
• In Tricks in research vs. contest math, Amy Lin writes "Contest math and research math differs in that contest math has a known answer and time limit while research math often does not"

Answer 1

The linked posts talk about the difference between contest math and research, but both have an important similarity that coursework lacks.

Homework problems, exam problems, and textbook problems are typically testing you on some specific knowledge you've got. Sometimes they're completely routine: you just learned integration by parts, and now you have some integrals that you can solve by integration by parts. Even when they're not as straightforward, a homework problem typically tests you on the material you've learned that week, an exam problem tests you on the material you've learned that semester, and a textbook problem tests you on the material you learned in the preceding chapter.

On the other hand, contest math (and research) requires you to bring your all. Even if we're looking at a middle school contest with problems you're expected to solve in less than a minute (and with no advanced mathematical concepts), there are dozens of different ideas from different areas of math that you might encounter. There are no rules about what techniques you must use, and even the formulation of the problem can be deceptive about the subject - you might turn a probability problem into a geometry question, use coordinates to reduce geometry to a system of equations, or find an interpretation that turns an algebra problem into a question about probabilities.

This goes hand in hand with a sharp drop in the amount of... not difficulty, because a difficult textbook might have problems as hard as any contest problem, but maybe hand-holding. (I'm open to suggestions for more precise terminology.) Consider:

• On a homework assignment, or on an exam, the implicit promise is that a student who's paid attention and worked hard will solve all the problems and get an A. Textbooks vary more, but generally it's also true that if you've read the textbook up to that point, you should be able to solve the exercises.
• The implicit promise made by a contest problem, on the other hand, is merely that there is a solution out there. Again, difficulty varies; however, the average contestant is expected to struggle with the majority of problems. (A notable example is the Putnam competition, where the median score is proverbially $0$ or $1$ points almost every year it runs.) With a good contest problem, the first step to solving it is the insight into what you have to do in order to reduce it to a textbook-like problem.
• Of course, research math makes no implicit promises at all. I think the jump from contest math to research math is similar to the jump from coursework to contest math. If I had to compare it in the same language, I would say that the first step is still the insight; however, nine times out of ten, after you have the insight, you will realize that you went in the wrong direction and now you're back at square one.

Contest math brings a lot of attached baggage: time limits, competitiveness, and so forth. In some specific subjects, memorizing tricks is a near-requirement to do well. For these reasons, it is not for everyone.

But when you're not ready to do research math - when you're not in an environment that will support you doing research math - contest math is very very different from the alternative, and it is a difference in the same direction.

Answer 2

Most mathematics textbooks have problems that in addition to the main purpose of teaching concepts, are meant to improve your problem-solving skills. The harder problems in this category are very similar to the problems in math contests but more specific to the particular field.

Some people bring up the time pressure as something that sets contest math apart from textbook material, but I faced the same time pressure when I was taking exams in a lot of my math courses.

I may be in the minority to believe so, but in my experience, everyone I knew who was good at math contests did exceptionally well in their university courses. The converse was not true. Many people did well in university courses and never showed any interest in contests.

It is hard to argue that the skills one learns in becoming good at contests are useless when taking courses. The question may be, is the added value worth the time needed to become good at contest math? That depends on your goal. A lot of people enter math competitions because they enjoy it. If you don't, studying linear algebra is a better way to improve your linear algebra than doing a math contest. But if you do enjoy it, I'm sure you will gain skills that are useful in your future math career along the way.

Finally, let me say a few words about techniques used in contest problems. The techniques used in contests are more elementary and more general. As you take more courses, you will learn about powerful techniques that are applicable to specific subfields, and you often learn how to use them in the context where you know beforehand that such techniques should be used. In contrast, the difficulty of many math contests is in finding which technique is applicable.

Let me give a very clear example. I have seen many contest problems that use induction. Induction is a technique used in many fields of math. It could be helpful with your courses, and you might even use it in research. When you first learn induction in a math course, those problems come with instructions like "use induction to prove [...]." So you know immediately that you need to use induction. In contrast in a contest, you have to come up with the idea of using induction and on what variable. This is also the case if you need to do induction in research, or when it comes in handy in solving a homework problem in an upper-level math course. In that sense, at least in some cases, contests may prepare you better for using more elementary techniques than courses that teach those techniques. But as you do higher-level courses, you will get similar practice with your elementary techniques and will learn techniques that are more specialized.

Answer 3

This is a really stupid discussion. Most IMO gold medallists (and maybe even some silver medallists) are much better at math than most people who are completing their undergrad in math. note that better does not mean that they know more stuff (though a lot of imo gold medallists probably know as much math as the average top math school 4th year undergrad), it literally means they are better at math, and usually they are much more passionate about it also. Obviously not all IMO people go on to become research mathematicians, but any one of them who continued to work hard and be passionate, basically who lived their life like olympiad days would become a good mathematician anyways. It's kind of like not all players who get drafted high in the NBA become NBA legends but there is a reason they were drafted high and if they continued that, they would succeed in NBA also...

On your point about textbooks, I am pretty sure any IMO gold medallist would be able to solve as many problems in a textbook as anyone else, so this whole discussion is moot.

In general, no offence to you, but I find everyone who has this discussion really annoying,(and i think most of these people dont even know what the imo actually is) math contests are a great way for people to learn (real) math, enjoy math and work hard, the people who actually prepare for olympiads, learn cool math and are passionate about it are much better prepared for 'college textbooks' than people who have stupid discussions about how a top 50 high school math kid just remembers a bunch of tricks and is not actually creative.