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I may one day be interested in teaching students who are preparing for, say, the IMO or other math competitions. However, I haven't found a particularly clear statement about exactly what material is on it. I know they contain algebra, geometry, combinatorics, and number theory, but is there any other well-defined subject that is tested in these competitions?

I know there are a number of resources for preparing students and teachers for these competitions, but reading a document that is exclusively about the competition feels too narrow--I'd rather learn the subjects that are tested instead, and then apply that knowledge to teaching the competition material.

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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.

Although non-exhaustive - and intended for a particular mathematics competition, i.e., the Putnam - I believe this provides a reasonable skeleton. Moreover, drawing again from this same reference, here are the chapter-by-chapter breakdowns of content.

(Click the first image for a higher-resolution version.)

enter image description here

enter image description here

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    $\begingroup$ [One nice feature of this book, by the way, is that it contains solutions for all of its problems!] $\endgroup$ – Benjamin Dickman Feb 13 '16 at 2:46
  • $\begingroup$ The OP has the tag "secondary education," so information about a college-level competition such as the Putnam is not on-point. $\endgroup$ – Rory Daulton Feb 13 '16 at 11:58
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    $\begingroup$ @RoryDaulton I strongly disagree with your assessment of this as "not on-point." The OP asks about topics to read into in preparation for "the IMO or other math competitions." I might also note that some high school students do take the Putnam; not many, but nor are there many preparing for the IMO (and IMOers are very likely to take the Putnam, if not in high school then by their first year in college; to score well necessitates tremendous preparation beforehand). Still, a single post/book necessarily excludes topics; if you are aware of other lists, then I hope you will post them, too! $\endgroup$ – Benjamin Dickman Feb 13 '16 at 17:03
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    $\begingroup$ Solutions are for faint-hearted problem solvers. I want my math books to torture me :) $\endgroup$ – NiloCK Feb 14 '16 at 20:36
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    $\begingroup$ @NiloCK Tear 'em out! Easier to destroy than to create (plus, I happen to like the added challenge of finding alternative solutions: especially ones that [I think] are more "elegant" than the book's!). $\endgroup$ – Benjamin Dickman Feb 14 '16 at 20:40
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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 aren't many problems about it, but you don't want to get an graph question on the IMO and barely knowing what a graph is. Those graph questions are usually not too hard if you have some experience with them.

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    $\begingroup$ If you check my earlier answer, then you will see (by viewing the first image in its high-res version) that algebra is broken down into several parts (although functional equations are placed later on, in real analysis). Similarly, combinatorics contains sub-categories (e.g., Euler's Formula for planar graphs... though there could certainly be more about graph theory!). $\endgroup$ – Benjamin Dickman Feb 14 '16 at 20:11
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    $\begingroup$ @BenjaminDickman I know. However, your answer (or at least the book) is more focused on contests like Putnam. For example, I think it is a bad idea to cover functional equations with differential equations. They require really different techniques. $\endgroup$ – wythagoras Feb 14 '16 at 20:14
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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 about the processes which serve us when confronted with difficult problems - reduction to smaller case, finding analogy with simpler problems which we're already able to solve, etc. I'd say that it's required reading in this area.

It's been a while since I read it, but I really enjoyed and benefitted from Paul Zeitz's 'Art and Craft of Problem Solving'. This book contains a fair number of problems which exemplify different specific strategies for solving problems - finding and exploiting a symmetry, induction, etc. This one would be useful for preparing lectures / lessons on specific mathematical topics and techniques.

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    $\begingroup$ (An irresistible side-note: One of my favorite problems is #3.4.31 in "A&CoPS" (2e, p. 107) by Paul Zeitz. Essentially, it says: There are 23 people with integral weight, such that whomever you remove, the remaining 22 can be partitioned into two groups of 11 - each with the same total weight. Prove that all 23 people must weigh the same amount. A few years ago I began to track down this problem's history, from the original formulation with integral weights through the abstraction to weights in $\mathbb{C}$. I recorded my findings in MO105400...) $\endgroup$ – Benjamin Dickman Feb 14 '16 at 20:47

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