For context, I'm a sophomore high school student. I'll be done with undergrad Abstract Algebra, Topology, and Analysis by the end of the school year, and I'll be burning through the necessary graduate Analysis, (Rudin) and Algebra (Lang) over the summer for Hartshorne, and doing some Differential Geometry from Tu along with Hartshorne.

I'm hoping that by the time I make it to Diophantine Geometry, I'll be able to use it to solve Olympiad number theory problems (Diophantine Equations in particular). However, that's something I can't really find out for myself until I actually learn basic Diophantine Geometry, which will take at least two years.

So, in order to get around that, I thought "Why not ask people who know Diophantine Geometry?"

Although this question is mainly about Arithmetic Geometry, answers about other high-level math subjects applicable to Olympiad problems are welcome.

And here we are. By the way, I also enjoy the Algebra by itself, it's not like I want to slog through more for the express purpose of solving competition math problems. I just wanted to know if it was an alternative or supplement to studying classical methods.

  • $\begingroup$ I'm not sure whether or not this should be marked 'soft question'. Whether or not previous contest problems have arithmetic geometric solutions seems rather objective, but the entire question is rather personal, so it has that 'soft question' feel. $\endgroup$ – John Clever Nov 7 '20 at 14:01
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    $\begingroup$ If your goal is to increase your olympiad problem-solving ability, I would focus on prior olympiad contests and related books. The applicability of graduate-level math to olympiad problem-solving is near zero. In the other direction, your olympiad problem-solving experience and skill will improve your ability to tackle math problems at the graduate level. $\endgroup$ – quasi Nov 7 '20 at 14:06
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    $\begingroup$ While high level knowledge could help, contest setters usually ensure that 1) It doesn't provide that significant of a boost, and 2) There are elementary/rudimentary methods that will work. I do encourage you to learn high level topics, but don't do so with the main goal of improving olympiad problem solving. Having said that, If you really want a high level topic that's more applicable to olympiads, give algebraic number theory a look (assuming you know the prerequisites). $\endgroup$ – Calvin Lin Nov 7 '20 at 14:13
  • $\begingroup$ @quasi That's what I suspected, although I was greatly hoping I was wrong. Wouldn't it be great if number-theoretic problems in olympiad geometry became trivial with high-enough level number theory, just as previously difficult area-under-the-curve problems became trivial with integral calculus? Alas, it was not to be. To answer your previous assumption, no, the goal of my learning it was not to increase my olympiad problem-solving ability. The original idea for this question came from me wondering "hmm, why aren't you allowed to participate in the IMO if you've gone to college?" $\endgroup$ – John Clever Nov 7 '20 at 14:14
  • $\begingroup$ @quasi I suppose my motivations are in the reverse: "Is the math that I plan to learn applicable also to olympiad problems?" $\endgroup$ – John Clever Nov 7 '20 at 14:38

Algebra and analysis: you should definitely know about the derivative and limits of sequences, though I doubt that you will need integration; continuity can help with functional equations. It doesn't hurt to have an understanding of fields and rings if you want to really understand what is going on with polynomials, like Gauss's lemma and Bezout's lemma for polynomials and irreducibility criteria.

Geometry: projective and inversive geometry; Barycrentric coordinates can be made formal using higher mathematics. Euclidean geometry is not a part of higher math. Be familiar with determinants.

Combinatorics: generating functions, including the derivative. There is occasionally a bit of graph theory though nothing very advanced. On that note, I guess knowing the general version of Ramsey's theorem isn't a bad idea. The probabilistic method is good to know. Solving homogeneous linear recurrences in general requires some linear algebra. Combinatorial Nullstellensatz and Polya enumeration theorem are probably as sophisticated as olympiad combinatorics gets.

Number theory: There are some non-trivial number theory theorems that pop up in olympiads, such as Thue's lemma, Lifting the Exponent lemma, Hensel's lemma, Zsigmondy's theorem, the solution to Pell's equation. Gaussian integers is also a possibility, as are aspects of algebraic number theory. You might consider checking out some of Titu Andreescu's books on number theory as he likes to occasionally bring in higher level stuff into his curriculum.

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    $\begingroup$ +1 A) Other areas with several significant applications include linear algebra, group theory (not just applied to NT), and fixed-point theorems. B) I consider several items listed here as "standard IMO olympiad approaches" rather than "high level math" (but those classifications can be debated) $\endgroup$ – Calvin Lin Nov 7 '20 at 17:30

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