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# Contest mathematics learning benefits

This page lists benefits of learning the subject contest mathematics. In other words, it tries to answer the questionWhy should I learn contest mathematics?|See all pages on the benefits of learning specific subjects

This page discusses the benefits of learning contest mathematics. It is intended as a resource to help people

## Contents

## 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

See also the Quora question How relevant have you found your contest math experience to math study and research?. The information below is quite similar to Vipul Naik's answer.

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.

See also the blog post Great Mathematicians on Math Competitions and "Genius" that includes quotations of statements by many mathematicians regarding the value of contest math.

### Preparation for the SAT

Some people have claimed that the lower levels of contest math preparation help increase one's *potential* to score highly on the SAT (quantitative section), even though realizing the potential requires preparation specific to the SAT. However, the statement has not been independently verified. Please also see the problem-solving skills section below.

## Developing problem-solving skills

Contest math requires people to develop the skill of approaching a problem using a general collection of heuristics and a base of factual knowledge rather than by applying a pre-specified procedure mechanically. This enables the development of the general skills as well as the intellectual temperament to approach problems using a trial-and-error, heuristic-based approach.

The extent to which there exist *general* problem-solving skills (that transfer across domains) as well as the extent to which these skills can be *trained* through practice in one area is much debated. It is likely, however, that because school education focuses *so little* on the development of these skills, a decent effort at contest math preparation can significantly increase one's exposure to problem-solving skills. That being said, people who have already been exposed to situations that have called upon extensive use of their problem-solving skills may have less to gain from contest math preparation.

## Signaling quality to colleges

- Getting to a high level in contest mathematics (such as clearing a regional Olympiad, or clearing the first stage of a national Olympiad) can be a powerful asset in college admissions. The objectivity of the indicator makes it better than most other high school extracurricular activities.
- Some colleges, such as MIT, accept American Mathematics Competitions (AMC) scores as part of the application as information about the mathematical skill level of contestants.