Contest mathematics
The term contest math will be used loosely to describe the type of mathematics encountered in mathematics contests, including those used for mathematics talent searches on a large scale, and those leading up to the International Mathematical Olympiad.
Contest math differs from high school math both in terms of the nature of topics covered and in terms of the methods of thinking used. It also differs from higher math in these respects.
Specific contests discussed on this page are specific to the United States, but the considerations outlined are fairly general.
Learning mathematics better
Learning math for pleasure
Contest math can be a fun way of learning mathematical topics and techniques not covered in the school syllabus. However, it is not the only possible or even the optimal way of getting pleasure from learning mathematics. See our math reading recommendations for other ways people can expose themselves to interesting and challenging mathematics for fun and learning.
Preparing for a mathematical or mathematics-based career
Some parts of contest mathematics, despite their lack of direct overlap with school math and higher math, are very helpful in preparing for mathematics-based careers. A review of the relevance of different parts of contest math is below:
| Area | Main subtopics | Parts that are helpful in mathematics-based careers | Parts that are not helpful |
|---|---|---|---|
| Algebra | Polynomials, equation-solving, inequalities, functional equations | Basic theory of polynomials, basic algebraic manipulation techniques, rough idea (not proficiency) of how to approach inequalities and functional equations | Proficiency in equation-solving tricks Proficiency in inequalities and functional equations |
| Geometry | Planar Euclidean geometry, including many facts about triangles and circles For some contests (not IMO) basics of coordinate geometry |
Understanding geometric transformations and geometric invariants can be helpful for geometry as seen in higher math. This is not a central topic of contest math, but is a supplementary topic that many students learn because it often provides shorter, more elegant solutions to contest math problems. The intricacies of triangle geometry can build appreciation of how a large factual base can be structured in the mind and combined with techniques to solve problems, even though the actual facts of triangle geometry are not useful. |
Most facts about triangle geometry and circles. |
| Number theory | Elementary number theory (not using any abstract algebra ideas explicitly), including congruences and Diophantine equations | The basic theorems of number theory, as well as facility with manipulating congruences, are very important in abstract algebra and thereby in much of mathematics that grows out of abstract algebra. The basic idea of how to approaching Diophantine problems is worthwhile. | Some of the techniques to solve Diophantine equations are artificial. |
| Combinatorics | Counting rules (enumerative combinatorics) and existential combinatorics (e.g., Ramsey theory) | Important because dealing with abstract mathematical structures often requires using abstract counting procedures even to get a sense of what's going on. | Attaining proficiency in some of the clever combinatorial constructions may not be worthwhile. |
We would generally recommend the following for people interested in potentially pursuing a mathematics-based career:
- Study thoroughly the "parts that are helpful" from the above table. In particular, study most of the underlying theory for number theory and combinatorics, and study relevant parts of algebra. Geometry is somewhat optional, and you should study it in depth if it fascinates you. Consider studying more about geometric transformations and their invariants instead of diving deep into geometric facts about triangles and circles, unless the latter interests you.
- To hone your problem-solving skills, consider taking an Olympiad-focused mathematics class with the Art of Problem Solving. Based on how well you do and how interested you continue to be, you can decide whether to try for something bigger.