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I am a second year studying math at university, and have taken theoretical linear algebra, group and ring theory, real and complex analysis, and general topology, as well as some miscellaneous technical courses like probability theory, statistical inference, and algorithms. That is to say, I have some basic experience with abstract, proof-based mathematics. I am currently working on two research projects (in applied fields) that require some creativity in problem solving (for example, my computational neuroscience project involves proving certain inequalities to do with statistical inference in the olfactory system), and am finding I am limited in making progress by my problem solving ability. For example, when I take a problem I am stuck on to some friends who are IMO medallists, they can often help me make progress by coming up with creative strategies. I would like to be able to do this myself, and I realize this takes years of work to hone. So I would like to begin now.

I am planning on improving by spending a good amount of time getting familiar with olympiad-style problem solving; I was planning on going through general olympiad books recommended on AoPS, and a lot of past USAMO papers, etc. I am not sure if this is the correct strategy for my goals, so I wanted to hear what others would do in my position. My understanding is that eventually, math olympiads diverge from how helpful they are to research (and can often reduce to knowing the right artificial trick), but the fundamentals of problem solving are useful regardless so it's healthy to be able to do at least basic olympiad problems.

This question is distinct from others like this and this because I am specifically asking for advice taking into account the fact that I already have some mathematical maturity, which may or may not affect how people recommend studying. I have very little experience with contest math. Any advice is appreciated.

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    $\begingroup$ Can you briefly describe a specific problem you got stuck on (as in, write out the inequality or whatever you weren't able to prove), just to give context? $\endgroup$ Feb 28, 2022 at 17:29
  • $\begingroup$ Not without giving a lot of context on the project, since the variables in the inequality are all specific to neuroscience. But broadly when I was stuck my friend rewrote a complicated term in a clever way, and applied Hoeffding and Jensen's inequalities very cleverly to get a bound. Another research project I'm working on is in developmental economics, and I was stuck on an algorithmic problem of telling two road paths apart from camera data, and I describe it here (stackoverflow.com/questions/70696999/…). $\endgroup$ Feb 28, 2022 at 17:34
  • $\begingroup$ If we are talking about diophantine equations, either factorize both sides, or work modulo some number. $\endgroup$ Feb 28, 2022 at 18:42
  • $\begingroup$ who are IMO medallists, they can often help me make progress by coming up with creative strategies --- Isn't this like saying you have trouble with your university gym climbing wall, but when your friends who have scored well in the IFSC Climbing World Championships climb the wall, they often come up with creative strategies? See also the 2nd to last paragraph in this answer. $\endgroup$ Feb 28, 2022 at 20:42

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If you want the fundamentals of problem-solving, one of the best books I can recommend is "The Art and Craft of Problem Solving" by Zeitz.

http://www.gang.umass.edu/~franz/Paul_Zeitz_The_Art_and_Craft_of_Problem_SolvingBookosorg.pdf

Most of the examples the author uses don't go far beyond the Olympiad syllabus, but it's genuine food for thought and material of some relevance, not just cramming for a contest. Also, the second chapter tells you what goes through the mind of someone trying to solve mathematical problems, which you might well like.

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There's something called inquiry-based learning. The point is to find ideas on your own instead of having them delivered to you in the form of lecture notes. You can try learning undergraduate subjects in this revolutionary way.

One of the prime examples of this is the "Linear Algebra Problem Book" by Halmos. I understand that you've already learnt some linear algebra, but do have a look at the book just to see what it's like.

http://dl.hamyarprojeh.ir/Halmos%20P.R.%20Linear%20Algebra%20Problem%20Book%20(MAA_%201996).pdf

It's designed to be a self-contained textbook, where the reader is prompted to confront examples, come up with ideas, and ultimately prove lemmas and theorems without excessive assistance.

Here's a textbook on analytic number theory by Murty, consisting almost entirely of problems:

https://link.springer.com/book/10.1007/978-0-387-72350-1

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Since you seem to be focusing on applied maths, you probably won't want to read these books too deeply. But they may well be of some use.

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