Resources
Books and links.
A short list of books that have proved themselves at every level of competition mathematics, alongside the free online resources worth knowing about. Selections favour clarity, depth, and books that reward sustained reading.
Books
Junior
- Introduction to Number Theory — Mathew Crawford, Art of Problem Solving
- Divisibility, primes, modular arithmetic, Diophantine equations, accessible to a motivated 11- or 12-year-old. The standard starting point for competition number theory.
- Introduction to Algebra — Richard Rusczyk, Art of Problem Solving
- Pre-algebra through quadratics with a problem-solving lens. Not a textbook, a thinking manual. The algebra backbone for early competition mathematics.
- Introduction to Counting and Probability — David Patrick, Art of Problem Solving
- Counting principles, permutations, combinations, basic probability. The clearest treatment at this level; combinatorics tends to be underweighted.
- Mathematical Circles: Russian Experience — Fomin, Genkin & Itenberg, AMS
- Problems from the Soviet mathematical-circles tradition. Playful, deep, unlike anything in standard school curricula.
Intermediate
- Introduction to Geometry — Richard Rusczyk, Art of Problem Solving
- Angles, triangles, circles, similarity, trigonometry, coordinate geometry, all through competition problems. The most complete treatment at this level.
- The Art of Problem Solving, Volume 1 — Rusczyk & Lehoczky, Art of Problem Solving
- The classical introduction. Covers the full range of topics at AMC 8 and AMC 10 level with hundreds of problems and complete solutions.
- The Art of Problem Solving, Volume 2 — Rusczyk & Lehoczky, Art of Problem Solving
- Extends Volume 1 into more advanced algebra, number theory, and combinatorics at AMC 12 level. A bridge to the senior pathway.
- Mathematical Olympiad Treasures — Titu Andreescu & Bogdan Enescu, Birkhäuser
- Problems organised by topic with full solutions. Bridges intermediate and senior olympiad mathematics, drawing problems from olympiads worldwide.
- Intermediate Algebra — Richard Rusczyk & Mathew Crawford, Art of Problem Solving
- Polynomials, functions, complex numbers, advanced algebraic manipulation. Bridges introductory algebra and senior-level competitions.
Senior
- The Art and Craft of Problem Solving — Paul Zeitz, Wiley
- A philosophy of problem-solving before a collection of techniques. Zeitz teaches how to think, how to get unstuck, where the key idea lives. The book to read before any olympiad round.
- Precalculus — Richard Rusczyk, Art of Problem Solving
- Sequences, series, complex numbers, vectors, trigonometry at competition depth. Covers much of the advanced algebra appearing in senior-level competitions.
- Geometry Revisited — Coxeter & Greitzer, MAA
- A compact tour of advanced Euclidean geometry: cyclic quadrilaterals, radical axes, inversions, projective geometry. A classic that has launched hundreds of olympiad careers.
- Number Theory: Structures, Examples and Problems — Titu Andreescu & Dorin Andrica, Birkhäuser
- Advanced number theory for olympiad preparation: congruences, quadratic residues, multiplicative functions, Diophantine equations. Thorough, problem-rich.
Olympiad
- Problem Solving Strategies — Arthur Engel, Springer
- Over 1,300 problems organised by technique: pigeonhole, extremal principle, invariants, generating functions. The canonical olympiad problem book.
- Combinatorics: Topics, Techniques, Algorithms — Peter Cameron, Cambridge University Press
- Rigorous but accessible. Graph theory, designs, Ramsey theory, the probabilistic method.
- 102 Combinatorial Problems — Titu Andreescu & Zuming Feng, Birkhäuser
- Problems from team selection tests and international olympiads, with full solutions. Teaches through example rather than exposition.
- Problems from the Book — Titu Andreescu & Gabriel Dospinescu, XYZ Press
- Beautiful, difficult algebra and number-theory problems with elegant solutions. For IMO and team-selection-test preparation.
Free online
- Art of Problem Solving — artofproblemsolving.com
- The global hub for competition mathematics. Active community, books, and a free wiki covering every major competition with thousands of past problems and full solutions.
- International Mathematical Olympiad — imo-official.org
- Every IMO problem since 1959, with shortlists, longlists, and country-by-country results. The authoritative archive at the top of the pyramid.
- AMC and AIME past papers — maa.org/math-competitions
- All AMC 8, AMC 10, AMC 12, AIME, and USAMO papers with full solutions, freely available. The clearest pathway from school mathematics to olympiad-level reasoning.
- Math Kangaroo — mathkangaroo.org
- International junior competition reaching over six million students across more than eighty countries. Past papers free; accessible from age six upwards.
- NRICH Mathematics — nrich.maths.org
- Enrichment problems for ages 5–18 from the University of Cambridge. Strong at the junior level, with deep problems all the way through.
- Underground Mathematics — undergroundmathematics.org
- Beautiful, deep problems at advanced school level. Strong on the analytical thinking that underpins olympiad algebra.
- Evan Chen — handouts and books — web.evanchen.cc
- Olympiad handouts, the OTIS curriculum, and a free draft of "Euclidean Geometry in Mathematical Olympiads." One of the most respected resources in the global olympiad community.