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I am looking for word problems that can be tackled by subjects like category theory, commutative algebra, nonlinear algebra, algebraic geometry etc.

Celestine Lawrence
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  • No, math olympiads only assume somewhat advanced high school methods (plus standard combinatorial trickery). No advanced topics (on purpose ! it should be accessible to gifted high school students). – Henno Brandsma Oct 27 '21 at 10:00
  • Yes, and I was hoping there are word problems that are challenging for gifted high school students but quite trivial for practitioners of modern math. Any examples? – Celestine Lawrence Oct 27 '21 at 10:04
  • No, none that I've ever seen.. – Henno Brandsma Oct 27 '21 at 10:09
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    Actually, there are some examples about integer solutions of $y^2=x^3+m$, where completely elementary methods suffice, for some $m$, but which are obvious if you know about Mordell curves. But they are usually not olympiad questions. – Dietrich Burde Oct 27 '21 at 10:18
  • I am looking at word problems like insolvability of quintic for example (https://www.youtube.com/watch?v=BSHv9Elk1MU) – Celestine Lawrence Oct 27 '21 at 10:20
  • If you look at the preparation for IMO in countries like China or USA, they teach the contestants a lot of undergrad or grad mathematics. One example that comes to my mind is the Combinatorial Nullstelensatz (you can google it so see what I am talking about). You can also use some complex analysis (Rouche theorem) to prove that some polynomial is irreducible (look for Yufei Zhao's handbooks) and I am pretty sure that you can use number fields of elliptic curves to solve some low degree diophantine equations appearing in math olympiads. – Aitor Iribar Lopez Oct 27 '21 at 10:21
  • However, let me add also that it is very unlikely that a IMO-level problem can be solved very easily using modern math. Note that each year, hundreds of proffesional mathematicians around the world look at possible solutions for these problems, and discard the ones that could be seen as consequences of higher level theorems. However, if you look for national or regional olympiads, I am pretty sure that a perceptive grad student should be able to simplify most of the problems – Aitor Iribar Lopez Oct 27 '21 at 10:25
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    I heard that the idea for IMO 2017 P6 came from a book of algebraic geometry, and that the IMO 2008 P5 was originally a question about a hypercube. And I know at least a few examples of olympiad problems where I was quite stumped as a high-schooled but I solved it effortlessly with two more years of math general education and overall practise. See also https://math.stackexchange.com/questions/4264555/if-mathrmord-pb-mid-mathrmord-pa-for-all-sufficiently-large-prime-p for a nonelementary approach to a question “harder” than a N7 of an imo shortlist. – Aphelli Oct 27 '21 at 10:31
  • What are "word problems"? There are indeed many competition problems which can be solved with higher mathematics. See e.g. https://mathoverflow.net/questions/277253/imo-2017-6-via-arithmetic-geometry. – Qi Zhu Oct 27 '21 at 11:29
  • I've heard and haven't bothered checking: IMO 2007 P6 was meant to be some combinatorics problem, but it's a lot easier using the Nullstellensatz, which was I guess an oversight since IMO participants are not supposed to need the Nullstellenstaz – Moisés Oct 27 '21 at 16:09

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