20

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 ...


13

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-...


12

Mathematics constests are a kind of game or puzzle, like chess, poker, sudoku, etc. Not all mathematics adapts well to the context of a competition in a limited amount of time. While it's true that research mathematicians can be very competitive (in Yau's recent autobiography, he several times describes mathematics as a competitive activity - note though ...


11

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 ...


10

I don’t think it is remotely reasonable to include the areas you mentioned, since (in the U.S., at least) all those that you mentioned are at least 3rd and 4th year undergraduate subjects. Indeed, functional analysis is usually a 2nd year graduate course (often not even required for a Ph.D. in math), and mathematical physics topics such as quantum mechanics ...


7

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 ...


7

There are a lot of resources for problem solving online. One place to look is MIT's OpenCourseware; see their Problem Solving Seminar: Course Description This course, which is geared toward Freshmen, is an undergraduate seminar on mathematical problem solving. It is intended for students who enjoy solving challenging mathematical problems and who are ...


7

The Art of Problem Solving have designed exactly such a curriculum, which is also meant to prepare students for the AIME and AMC. I also very warmly recommend the Gelfand series of books, starting with Algebra; which was originally designed for his correspondence school for talented math students. Finally, this little book on problem solving by Ravi Vakil ...


7

The book by Masha Gessen on Grigory Perelman has a fascinating description of the Russian math camps, specialized schools (e.g., Specialized Mathematics School Number 239 in Leningrad), and the instructors who excel in training the future IMO candidates. One such instructor was Valery Ryzhik (School 239), but the most influential was Sergei Rukshin: "......


6

I know they contain algebra, geometry, combinatorics, and number theory, but is there any other well-defined subject that is tested in these competitions? Consulting Gelca and Andreescu's (2007) Putnam and Beyond, there are also: methods of proof real analysis trigonometry probability and, within each of these "well-defined subjects," many sub-categories....


6

The (university) course I took on problem solving and mathematics competitions had several problem sets with competition-level problems that were sorted into problem-solving strategies. For example, during week 3, we solved problems using the Extremal Principle and got familiar with it. In another week we solved Pigeonhole Principle problems. You might get ...


6

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, ...


5

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 ...


5

I teach a lot of contest math and I have very mixed feelings. Contests reward repertoire, perspicacity and speed. These are all certainly useful at undergrad, but the emphasis given to them feels wrong. On the other hand, there is a decent overlap between students who can assail these problems and students who will develop the other qualities and attitudes ...


5

I highly recommend Paul Zeitz, The Art and Craft of Problem Solving. The books coauthored by Richard Rusczak on problem solving are probably excellent, though I haven't yet looked at them myself: The Art of Problem Solving, Volumes 1 and 2. And then there are the math circle books, many of them geared to competition preparation... You might also find ...


4

With acknowledgement to @RoryDaulton - there is an archive for the Putnam Math Competition. I believe this is exactly what you requested. Free. Let us know if you have trouble accessing. You can also download the entire book The William Lowell Putnam Math Competition 1985-2000. Note: the copywrite page offers “Reproduction. The work may be reproduced by any ...


4

You could have them work through problems on aops. It contains AMC questions as well as video lectures on the problems. Furthermore it has books and other resources you may be interested in.


4

I'd like to highly recommend George Pólya's "How to Solve It". This 1945 book contains great theory and advice on how to approach hard problems in math.


4

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 ...


4

First of all, it might be worth pointing out that algebra can be split in inequalities, polynomials and functional equations. Not all problems fall in one of these three categories, but I think that approximately four-fifths does. The others are mainly about sequences. Some introduction to graph theory (as a part of combinatorics) would also be nice. There ...


3

I figured that I can expand my comment into an answer, even though it is still more a comment than an answer. You say that math contest questions "involve a lot of small tricks that one hardly ever needs in math research", and because of this you find contest questions useless for future career as a mathematician. But I do not think that math contests are ...


3

"Why it is said that Finland has a particularly good education system, but Finland's performance on international mathematics competitions is quite often at relatively intermediate level?" Because the first part is a comparison of population averages and the latter part is a comparison of population extremes. If the distributions vary in shape, than you ...


3

I don't know if such a list exists, but I can recommend other math competitions - specifically the waterloo contests: http://www.cemc.uwaterloo.ca/contests/past_contests.html#pcf Hope this is helpful in some way.


2

You can organize it as a low-tech version of StackExchange! Collect questions (anonymously) and present them to the class, letting them upvote the questions they feel are relevant. Then collect answers (anonymously) and present them for evaluation. Students should upvote those answers they feel help them understand better. The reason anonymity is important ...


2

In addition to the perfect suggestion for the Art of Problem Solving, I would recommend getting your hands on the mathcounts materials, some other contest problem books, and old AMC exams. You may also be interested in books by the USA Olympiad coach Titu Andrescu or the new Coach Po-Shen Loh, at his website expii.com.


2

Knowing your math subjects is one thing, but experience with creative problem solving itself is key. Polya's 'How to Solve It', while a bit dated in its language, is still a good reference for anyone interested in improving as a problem solver or helping others to improve. This book is light on example problems or mathematical work, but talks generally ...


1

My general idea of math results is that it only shows how good one is at that thing, and it's not very easily comparable. For example, you could be good at calculation and go extremely well in tests (that could be checked with a calculator) but be very weak on concepts. Likewise, you could be very good at concepts but not be good at, say, answering multiple ...


1

My answer here to a different question has a link to some "college-level contest" problems that are arranged by topic as you request. The issue is that these contest problems are not really "Algebra I" exactly. So this may be helpful only to future readers, not to you personally, because they are sorted by topic just well enough that you will find that "...


1

Any math olympiad is about more than its topics. You will definitely want to include books that help students learn to problem solve. One book I highly recommend is The Art and Craft of Problem Solving, by Paul Zeitz. I might have ideas on the particular topics, but I'll edit later for those. I'm hoping others will give you more than I can.


1

"The problems themselves are challenging for myself." That's not a bad situation to be in. Work through the problems and figure out why they are challenging to you, Then teach the students what you have learned. Chinese students are well trained and disciplined by their parents. They will pretty much follow where you, their teacher, leads them. Pretend ...


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