奥数
热度 1 已有 168 次阅读 2022-5-10 18:59 系统分类:成长记录
ALGEBRA
Main topics: Polynomials, equation-solving, inequalities, functional equations
Parts I found helpful: Basic theory of polynomials, basic algebraic manipulation techniques, rough idea (not proficiency) of how to approach inequalities and functional equations.
Parts I did not find helpful: Basically, the huge amounts of practice that went into acquiring proficiency in functional equations and inequalities.
Let me explain this a bit more. Functional equations is an Olympiad math topic that isn't explicitly covered in either school math or college math. But the main ideas behind functional equations: plug in specific values in an expression that is identically true to deduce information about the objects involved -- is extremely useful through mathematics, and it becomes more and more useful in higher mathematics. The topic most directly connected to functional equations is differential equations (in fact, differential equations are a type of functional equation that relate local behavior). But the idea of plugging in specific values into general conditions is a staple throughout much of mathematics, and functional equations is probably the part of Olympiad math that best drills this in.
On the other hand, getting really good at functional equations entails learning lots of tricks, and these tricks are of little use to math study and research.
The basic idea of inequalities is very helpful to understanding the stuff about L^p norms one sees in analysis. But again, most of the tricks used for handling inequalities aren't of direct relevance to higher mathematics (though perhaps some branches of higher mathematics do rely on inequalities a lot -- I avoided those).
GEOMETRY
Main topics: Planar Euclidean geometry, including many facts about triangles and circles. We could also optionally use coordinate geometry, complex numbers, and trigonometry, so I learned all these techniques.
Parts I found helpful: 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.I experienced this latter effect significantly. Triangle geometry really drove in the idea of mathematics as a collection of facts that beautifully come together rather than a collection of techniques.
Parts that I did not find helpful: Most facts about triangle geometry and circles -- here I mean the facts themselves rather than what they taught me about the nature of mathematics or what they inspired me to. I haven't used any facts about triangle geometry in my undergraduate or graduate studies, even though I would have loved to (given I'd spent so much time learning the stuff).
NUMBER THEORY
Main topics: Elementary number theory (not using any abstract algebra ideas explicitly), including congruences and Diophantine equations
Parts I found helpful: 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.
Parts I did not find helpful: Some of the techniques to solve Diophantine equations are artificial. These would still have been useful to me if I'd chosen a particular specialty that involved tackling such equations, but they don't show up in most research (including most of algebraic number theory and analytic number theory).
COMBINATORICS
Main topics: Counting rules (enumerative combinatorics) and existential combinatorics (e.g., Ramsey theory)
Parts I found helpful: Most of it, because dealing with abstract mathematical structures often requires using abstract counting procedures, and existence arguments, even to get a sense of what's going on. It also helped me with understanding algorithms and prepared me for computer science somewhat, although a discussion of what would be beyond the scope of the question.
Parts I did not find helpful: Nothing, really. But just as with number theory, some of the combinatorial constructions require too much cleverness and acquiring proficiency with that doesn't transfer much to the rest of math study and research.
