Recommendations (Feb '22 version)

(Draft, I'll update it soon and when final, I'll update this to a page instead of post.)

Here is a relatively comprehensive list of resources, books etc. that I believe are the most helpful to students. I have tried to divide it based on difficulty level and subject as much as I can. Hopefully, this list is helpful and I've added a few comments on why I like a particular resource. Okay, let's get started!

One thing to note is as important as improvement is, I believe the first goal with Mathematical Olympiads has to be that you thoroughly enjoy it and improvement will come heavily as a consequence. Listen to Baba Ranchoddas people.

With this in mind, these are not only the books that I learnt the most from but also the ones I enjoyed the most. (Very few books that I have added are ones that I have not read myself.)

The definitions of grade levels and difficulty is a bit arbitrary since I myself had done none of this before 10th grade.

I am well aware of most books that I have omitted from my recommendations and the decision is conscious.

Just for fun (Pre 7th grade or <RMO)

• Moscow Math Circles
• Ian Stewart's books
  
  I have myself only read one of Ian Stewart's books but it was amazing and I have heard from many people that the others are amazing too. I'll soon read more :)
  
  I've only gone through some sets of Moscow Math Circles and not its entirety but can say with confidence that it is a wonderful book for those just seeing non school mathematics.

Beginners

• Mathematical Circles by Fomin et. Al.
  
• Past RMO papers
• Geometry Puzzle Pages
• BMO(British Math Olympiad) Round 1
• Problem Primer for the Olympiad by CRP
  
  These books are wonderful for starting out and these were the first books I did in my 10th grade before RMO and qualified it comfortably only relying on them. I was also able to enjoy a lot due to these books and I was quite excited to learn further. 
  
  Mathematical Circles has to be the one of the first books any student getting into Olympiad mathematics. Even if you don't end up doing Olympiads but you only do this book for fun, you would've learnt a lot and be much happier :)
  
  The Geometry Puzzle pages are so pretty and so UwU! These problems are just beautiful! Everyone should try them and realize the joy of pretty geometry :)

Intermediate (INMO ish)

• Past INMO papers (>=2018)
• Initial IMOSL problems (X1-X3)
• BMO Round 2
• St. Petersburg Math Olympiad, Russian Math Olympiad (Grade 8,9)
• Art and Craft of Problem Solving by Paul Zeitz
  
  I have mostly only added problem sources here as there is really no need for new theory at this stage and one can do well on INMO by just practicing good problems.
  
  I also included Art and Craft of Problem Solving as it's a pretty good book and the only one that seems to fit here. The process of solving a problem is explained very well.

Advanced I

This is the first time that you might want to study subjects from different books to improve further and see more interesting theory!

Number Theory:

• Olympiad Number Theory Through Challenging Problems by Justin Stevens
  
  This is an amazing short book introducing many key ideas in beautiful ways. I highly recommend it and a PDF seems to be freely available on google search.
  PDF link
• 104 Number Theory Problems by Titu Andreescu
  
  This is one of the first number theory books that I had done and it is a brilliant collection of pretty problems. I highly recommend trying these problems.
  
  

Combinatorics :

• Problem-Solving Methods in Combinatorics by Pablo Soberón Bravo
  
  This book is so well written and an extremely fun read. There's no other book really needed and this is sufficient. You can mostly rely on problems from Russian contests and ISL outside this.

Geometry :

• 106, 107 Geometry Problems by Titu Andreescu
  
  These two books are quite well written and the problem sets are well amazing. Sadly, I had not done them in my Olympiad journey but did look at the books later and they are quite fun.
  
  
• Geometry at its best by Eric Shen 
  
  These are 30 very pretty problems from various different contests compiled by Eric Shen and are quite fun to solve and go through :)
  
  
• For bash : Euclidean Geometry in Mathematical Olympiads by Evan Chen
  
  Evan is one of the best at bashing and if it is something that you really want to learn (I suggest that you don't) then Evan has a very good exposition. The chapter on Barycentric Coordinates is very well done.
  
  
• Common configurations: Try not to worry about them too much, if they are common enough, they will appear frequently in the problems you practice. If you like any configuration coming up particularly, you can look it up and explore.

Algebra :

• Functional Equation by Pang-Cheng, Wu
  
  This is another gem that a student has written and is one of the best books out there for exploring FEs and getting good intuition on them. The PDF is also publicly available! 
  AoPS Thread for PDF
  
  
• Secrets in Inequalities by Pham Kim Hung
  
  This is a beautiful book but might be a little too hard so I recommend trying to get comfortable using AM-GM and Cauchy Schwarz type inequalities.
  If you are simply looking for a problem collection to practice these, I have a huge one here. Doing part A of that should be sufficient to start.
  
  
• Polynomials : 
  First begin by getting comfortable using ideas like Vieta's formulas and others on problems. If you are comfortable with these and can consistently solve A1-A2 level problems on ISL. I recommend reading my handout.
  
  It's not written in the best way and is hard to read at times and is quite dense as it covers a lot of ground over just 50 pages so please be comfortable before getting into it. PDF
  
  

Advanced II

• Number Theory by Naoki Sato PDF
  
• Late ISLs i.e. >=X5, TSTs of different countries, Russia Grade 11
• Problems from the Book
• Straight from the Book
  
  

General Advice:

• Try to keep the focus primarily on enjoying this journey of doing math and exploring it rather than test scores.
• The previous point takes precedence over everything so don't worry about finding the "best" resources every time. If you are enjoying some particular book or some particular notes, that's usually more than enough. The focus should be on the math and not looking around for resources :)
• Evan's advice on reading solutions, please keep point 5 in mind particularly: Evan's Blog
• Evan's notes on writing proofs : English.pdf
• Focus more on doing problems than learning theory as there's not really that much theory needed.
• If you ever want to learn a topic and don't know the right book, just google and you should be able to find a decent enough exposition on it in some handout.
• Don't use theorems and results that you do not understand or can prove by hand on your own. This seems to be a vicious cycle as you start losing intuition on various things are true even if they work for much more elementary reasons.
  
  For example, it's rarely justified to use something like Kobayashi's theorem or Catalan conjecture or even Dirichlet's theorem as these are quite advanced and require a lot of theory to be built before their use. If you don't think you understand that theory, try to avoid using these nukes as well. 
  
  We should not bring conclusions on problems that can simply be tackled with order based arguments through something like Kobayashi's theorem, it advertises the wrong kind of intuition.

Other Useful Links:

• Canada IMO Training Homepage
• Online Math Club
• Contest Collections
  
• Yufei Zhao's Home Page
  
• Evan's Olympiad Home Page
  
• Alex Remorov's Home Page
  
• Po Shen Loh's Handouts Page
• Catriona Agg's Twitter (For pretty geometric puzzles)
• Evan's blog
• IMO Math

Useful Training Programs/Communities

• OTIS : OTIS has an amazing collection of handouts compiled by Evan and the problem collections in all handouts are brilliant. I especially loved the Combo ones. I highly recommend joining just for these handouts. There are various other things like the community but the combo handouts take center stage for me! Website
• OMC : At OMC, we have regular sessions every week on various topics from many great speakers and the sessions are usually quite fun and relaxed. The difficulty level also varies from 0 pre-requisite topics to introductory college topics. Website
• Sophie Fellowship : This is quite a restrictive program that we have only started recently so things are still being figured out but due to the team of people working, this program is quite unique. We would be excited to take more students. Website
• CAMP : This has been only run twice till now but is an opportunity for students to take part in an online camp with many amazing people taking sessions on Olympiad topics and many other activities happening. Website
  
• Maths Beyond Limits : This is one of the most fun camps that I have taken part in and you get to interact with various people from all across the world. These students are not just quite talented but as it is on site, you get to take part in various activities and form friendships. Website
• Camps in the USA: There are various camps that are conducted in the US like the Ross Program, PROMYS, USA-Canada MathCamp, MathLy, AwesomeMath etc. You might also want to check these out and see if they seem attractive to you. 
  P.S. Promys India is also on the charts and should be started in India from 2023.

Suggested Contests/Problem Sources:

(Not ordered)

• Anything Russian: Tournament of Towns, St. Petersburg, All Russia, Sharygin
• Canada National Olympiad
• Japan MO Finals
• IMO Shortlists
• Taiwan Competitions
• KoMaL
• Kurschak
• Romanian Masters of Mathematics (RMM)

Long post over: If you would like to discuss something or any queries etc, please feel free to comment or shoot a mail to rohang@cmi.ac.in

p.s. Check out The Championship of Mathematical and Logical Games! IFOMG