British Mathematical Olympiad Syllabus
The BMO does not have a formal syllabus like a school exam. Instead, it tests deep understanding of core mathematical concepts, applied in creative and unfamiliar ways.
An Unwritten Syllabus
The BMO tests problem-solving aptitude using pre-university mathematics. The focus is on ingenuity and rigorous proof, not on knowledge of advanced topics. The material is broadly covered by GCSE and A-Level Further Mathematics, but the questions require much deeper thinking and creativity. Success depends on how you use the tools you have, not how many tools you know.
Number Theory
- Divisibility and Primes The fundamental theorem of arithmetic, prime factorization, and greatest common divisors.
- Modular Arithmetic Congruences, Fermat's Little Theorem, Euler's totient theorem, and the Chinese Remainder Theorem.
- Diophantine Equations Finding integer solutions to polynomial equations, often using modular arithmetic or factorization.
Geometry
- Euclidean Geometry Circle theorems (power of a point), similar and congruent triangles, cyclic quadrilaterals, and concurrency.
- Trigonometry Using the sine rule, cosine rule, and various identities as tools in geometric proofs.
- Coordinate Geometry Applying coordinates and algebraic methods to solve problems about points, lines, and circles.
Algebra and Combinatorics
- Polynomials and Inequalities Vieta's formulas, roots of polynomials, AM-GM, and Cauchy-Schwarz inequalities.
- Functional Equations Determining functions that satisfy specific equations through substitution and analysis.
- Counting Principles The pigeonhole principle, inclusion-exclusion, double counting, and recurrence relations.
- Graph Theory Basic concepts like vertices, edges, degrees, paths, and colouring problems.