Olympiad mathematics is a particular discipline. It is not the same as school mathematics, and it is not the same as research mathematics. It sits between them — more creative than the former, more constrained than the latter. The problems are finite and solvable, but they require genuine ingenuity, not computation.

This page is a structured guide to the resources I have found most useful. It covers all four major olympiad areas — algebra, number theory, geometry, and combinatorics — along with general problem-solving, online platforms, structured programmes, and communities. I have tried to be honest about what each resource is good for and what it is not.

One note upfront: the resources below assume you want to actually get good at olympiad mathematics, not merely to have read the books. Reading without solving problems is almost worthless in this discipline. Buy a notebook. Work problems by hand. Come back to the solutions only after a genuine attempt.

Another thing to note is that I am not someone with alot of experiences in this field and neither I am perparing for any formal competitions such as the IMO, INMO etc. I am someone who studied it just for the sake of learning and these are the resources which I found the most useful.

Where to Start: General Problem-Solving

Before any area-specific study, you need a foundation in mathematical problem-solving as a craft. These books build that foundation.

Art of Problem Solving, Volumes 1 & 2 — Richard Rusczyk, Sandor Lehoczky

These are the entry point for almost everyone in the olympiad community. Volume 1 covers the fundamentals — number theory, algebra, geometry, counting — at a level accessible to a strong middle-school or early high-school student. Volume 2 goes further into each area and introduces more sophisticated techniques.

They are not glamorous books. The problems are not IMO-hard. But they build the vocabulary and the habit of proof-based thinking that everything else requires. If you are genuinely starting from scratch, begin here. If you can solve most of Volume 2 problems without much struggle, move on.

A mild criticism: the exposition in Volume 1 can feel slow and procedural. If you find yourself ahead of the material, skip forward. These books are designed to be complete, not lean.

The Art and Craft of Problem Solving — Paul Zeitz

This is a different kind of book. Zeitz is less interested in teaching you specific techniques and more interested in teaching you how to think about problems — when to try an invariant, when to look for a monovariant, when extreme cases are informative, how to generalise from examples.

The philosophy chapters at the beginning — on psychological strategies, on exploration, on the difference between a tactic and a strategy — are worth reading even if you skip everything else. No other olympiad book says these things as clearly.

The problem difficulty varies considerably, and some of the harder problems are genuinely difficult for a high-school student. Do not be discouraged by these. The book is not meant to be completed in one pass.

Problem Solving Strategies — Arthur Engel

Engel’s book is not gentle. It is organised by technique — pigeonhole principle, invariants, colouring arguments and many others — and each chapter contains a large number of problems ranging from accessible to hard, all solved with minimal exposition.

This is a book for someone who already has a foundation and wants to see a large number of ideas in concentrated form. The solved examples are the main attraction. Engel does not hand-hold, but what he shows you is genuinely useful.

If you are preparing for IMO or equivalent competitions, Engel is essential — not for the exposition, but for the sheer density of technique it covers.

Mathematical Circles: Russian Experience — Fomin, Genkin, Itenberg

A product of the Russian mathematical olympiad tradition, this book reads differently from the others on this list. It is structured around mathematical circles — informal gatherings of students working through problems together — and it reflects the spirit of that format: curious, exploratory, not particularly concerned with being systematic.

The content covers elementary topics — divisibility, combinatorics, games, basic geometry — in a way that develops genuine mathematical intuition rather than technique catalogues. The problems are well-chosen and the discussions between them are worth reading. This is an excellent companion to AoPS Vol. 1 for someone early in their olympiad journey.

Excursions in Mathematics — Modak

A classic Indian olympiad resource, particularly well-suited to students preparing for RMO and INMO. The treatment of number theory, geometry, and combinatorics is clear and problem-rich, with Indian competitions well-represented in the problem sets. Less widely known internationally than Engel or Zeitz, but more calibrated to the Indian olympiad environment specifically.

Number Theory

Number theory is the most self-contained of the four olympiad areas. Elementary number theory — divisibility, congruences, prime factorisation — requires less background than any other area, which makes it a natural starting point. Olympiad number theory, however, quickly involves ideas from abstract algebra (p-adic valuations, lifting the exponent) that require more preparation.

Modern Olympiad Number Theory — Aditya Khurmi

The best comprehensive treatment of olympiad number theory currently available. Khurmi covers the full range from elementary divisibility to advanced topics — LTE, order, primitive roots, multiplicative functions — with clear exposition and excellent problem sets. The writing is modern and the problem selection reflects current competition trends.

This is the book I would recommend first for any serious olympiad number theory study. It is freely available online. Read it carefully and do the problems.

Olympiad Number Theory Through Challenging Problems — Justin Stevens

A shorter, more focused treatment. Stevens moves quickly and the problem difficulty is high throughout. This is not a first resource — come to it after Khurmi. As a problem set for someone who already knows the theory but wants harder practice, it is excellent.

Number Theory — Naoki Sato

Sato’s notes are free, concise, and precise. They cover the standard olympiad number theory content without any fat. The exposition is clean. If you want a compact reference for standard results — Euler’s theorem, CRT, quadratic residues — Sato’s notes are faster to navigate than a full book. Particularly useful for a quick review before a competition.

A word of caution: Sato’s notes are not a substitute for a full treatment like Khurmi. They assume you have seen the ideas before. Use them as a reference, not a primary source.

Algebra

Unlike areas of mathematics that are built on a fixed framework, olympiad algebra is better understood as a collection of ideas, techniques, and results drawn from multiple fields. It is the area where technique catalogues matter least and structural intuition matters most. The problems — inequalities, functional equations, sequences — require you to recognise what is happening algebraically before you can choose an approach. Mechanical technique application fails more often here than in any other area.

Pang-Chung Wu’s Functional Equations Book

Functional equations are one of the trickiest olympiad topics precisely because there is no single method — the approach depends almost entirely on the structure of the equation. Wu’s treatment is the most systematic available at the olympiad level. It categorises functional equations by type, provides worked examples, and develops the habit of checking injectivity, surjectivity, and substitution strategies methodically.

If functional equations are a weak point — and for most students they are — this is the book. It is free online.

Secrets in Inequalities — Pham Kim Hung (Volumes 1 & 2)

Inequalities are the area where olympiad algebra most often overlaps with analysis, and this book reflects that. Volume 1 covers classical techniques — AM-GM, Cauchy-Schwarz, Schur, SOS, mixing variables — with extensive worked examples. Volume 2 is harder and covers more advanced methods.

The coverage is comprehensive to a fault — Hung includes so many techniques that it can be difficult to develop instinct for which one to apply. The best way to use this book is to work through problems first and come to the exposition only to understand why your approach succeeded or failed, not to build a catalogue to memorise.

For an alternative perspective on inequalities, Michael Rozenberg’s work on AoPS forums is excellent and freely available.

Geometry

Geometry is the area where the gap between knowing theorems and being able to use them is widest. You can memorise every theorem in EGMO and still struggle to solve problems, because geometry requires you to see configurations — to notice that a particular point is the circumcentre, that an angle is the angle in the alternate segment, before you can proceed. This seeing is developed through practice, not through reading.

The books below are among the best at teaching both the theory and the seeing.

Euclidean Geometry in Mathematical Olympiads (EGMO) — Evan Chen

The standard reference. Chen’s book covers the full range of olympiad geometry — from basic angle chasing through projective geometry, inversion, and complex numbers — with exceptional clarity. The problem sets at the end of each chapter are well-chosen, progressing from accessible to genuinely hard.

What distinguishes EGMO from other geometry books is the emphasis on directed angles — a technical choice that eliminates a large number of configuration errors that plague geometric proofs. If you have been doing geometry without directed angles, reading the relevant sections of EGMO is worthwhile even if you already know the underlying theory.

The book is available from MAA. It is worth the price. Chen also maintains a free excerpt on his website at web.evanchen.cc.

A Beautiful Journey Through Olympiad Geometry — Stefan Lozanovski

A freely available PDF that covers olympiad geometry from the basics to advanced topics, with particular attention to the intuition behind each technique. Lozanovski’s writing is warmer than Chen’s — he explains not just what a technique does but when you would think to reach for it.

For a student who finds EGMO moving too fast, Lozanovski is an excellent companion. For a student who has worked through EGMO, the problem sets in Lozanovski offer additional practice at the same level. Available free at olympiadgeometry.com.

Lemmas in Olympiad Geometry — Titu Andreescu, Sam Korsky, Cosmin Pohoata

A more advanced book, focused specifically on the configurations and lemmas that appear repeatedly in hard geometry problems — the radical axis, spiral similarities, the angle bisector and its relatives, the nine-point circle, pole-polar relationships. Each chapter takes a lemma, proves it carefully, and then shows how it appears in competition problems.

This is not a first book. Come to it after EGMO. The value is in seeing how professional olympiad problem-solvers decompose hard configurations into known lemmas — a skill that is not taught explicitly elsewhere.

Combinatorics

Combinatorics is the hardest area to systematically prepare for, because it is the least dependent on technique and the most dependent on genuine creative insight. Two combinatorics problems at the same difficulty level may require completely different approaches, and the catalogue of applicable techniques is much smaller than in geometry or number theory. This means you can prepare less by reading and more by solving.

That said, there is structure here. Counting arguments, pigeonhole, invariants, extremal principles, graph theory, probabilistic arguments — these are the tools. The books below cover them.

Olympiad Combinatorics — Pranav A. Sriram

Freely available online and arguably the best dedicated olympiad combinatorics resource at the moment. Sriram covers graph theory, extremal combinatorics, probabilistic methods, algebraic combinatorics, and more, with problems from recent olympiads throughout. The treatment is serious and does not condescend.

The probabilistic method chapter is particularly good — this technique appears in hard problems and is rarely covered at the olympiad level elsewhere. Essential reading for anyone targeting IMO or equivalent.

102 Combinatorial Problems and related books — Titu Andreescu

Andreescu’s combinatorics books — including A Path to Combinatorics for Undergraduates (co-authored with Zuming Feng) — are more accessible than Sriram and serve as better entry points. The problems are well-chosen and the solutions are readable. If Sriram feels too advanced, start here and come back.

Andreescu’s books are consistently good across all four olympiad areas. The combination of clear exposition and strong problem sets makes them reliable starting points.

Online Platforms

Art of Problem Solving — artofproblemsolving.com

The central hub of the anglophone olympiad community. The forums contain solutions and discussions for essentially every problem from every major competition. The community is large and generally serious. If you have a question about a problem, search AoPS before asking anywhere else — the answer is usually there.

For contest archives, the AoPS wiki is the most comprehensive available.

Evan Chen’s Website — evanchen.cc

Chen posts an enormous quantity of useful material here: handouts on essentially every olympiad topic, his past olympiad papers, the OTIS Excerpts (a freely available problem set). His beginner’s page is the best starting guide I have seen for someone new to olympiads. His olympiad handouts page is worth bookmarking and returning to regularly.

Programmes

OTIS (Olympiad Training for Individual Study) — Evan Chen

web.evanchen.cc/otis.html

A proof-based olympiad training programme structured as handouts and problem sets across three difficulty tracks. Chen is an IMO gold medalist and a PhD mathematician — the programme reflects that background. The material is serious and the feedback is genuine.

I am currently in OTIS. The quality of the problems is high, and the structure forces consistency in a way that self-study alone often does not. Applications are open each year — the programme requires an application demonstrating proof-writing ability. If you are targeting USAMO or equivalent and you can write proofs, apply.

The SOPHIE Fellowship

A programme for mathematically talented students, focused on development through problem-solving and mentorship. Worth considering if you are at the stage where you are looking for structured guidance beyond self-study. Details and applications available through their official channels. Here is the link to apply https://sophiefellowship.in/

Communities

The quality of the communities you learn in matters. The following are worth finding:

Discord servers. Several large olympiad Discord communities exist and are active. The MODS is the largest. There are also more specialised servers for specific competition systems — Indian olympiad, USAMO, European competitions — which can be found easily. The quality of discussion varies; the best servers have active members who are willing to engage seriously with problems rather than just post solutions. Here is a link to one such group https://discord.gg/mods

Telegram groups. Particularly active in the Indian olympiad ecosystem. Groups focused on RMO and INMO preparation are easy to find and tend to have daily problem discussions. Search for groups by competition name. The culture is competitive and the problem exposure is high. Here is a link to one such group https://t.me/dailymathsproblems

Local mathematical circles. If you have access to one — through a university, through a school or a programme similar to what I used to operate — use it. The feedback loop from working with more experienced people in person is faster than anything a book or online community can provide. Russia built its olympiad tradition through mathematical circles. There is a reason for that.

An Honest Note on the Process

The books above, taken together, are more than enough material for years of serious study. The constraint is never the availability of resources. It is consistency — the willingness to sit with problems that do not yield quickly, to return to the same problem after failing it, to keep a notebook of mistakes and patterns.

Geometry in particular rewards accumulated pattern recognition more than any other area. You will not get good at it without doing hundreds of problems and actively cataloguing what you notice. This cannot be shortcut.

Number theory is the most learnable in the sense that the theory is compact and the techniques are finite. You can reach a high level in number theory faster than in any other area if you study the right things in the right order.

Algebra and combinatorics — especially combinatorics — are the most resistant to systematic preparation. The only real approach is to do more problems and to think carefully about why each solution works at a structural level, not just what the clever step was.

Whatever area you are weakest in: that is where to start.

Last updated March 2026.