Singapore Mathematical Olympiad (SMO) Syllabus

Structure by Section

The competition has three categories based on school level: Junior (up to Secondary 2), Senior (up to Secondary 4), and Open (up to Junior College 2). The syllabus is cumulative, meaning topics from the Junior section are assumed knowledge for the Senior and Open sections. The main areas tested are Algebra, Geometry, Number Theory, and Combinatorics, with increasing complexity at each level.

Algebra

• Polynomials Includes Vieta's formulas, roots of unity, and polynomial division.
• Inequalities Covers AM-GM, Cauchy-Schwarz, rearrangement, and other classical inequalities.
• Functional Equations Focuses on substitution methods and identifying properties like injectivity and surjectivity.
• Sequences and Series Explores recurrence relations and properties of arithmetic and geometric progressions.

Geometry

• Euclidean Geometry Properties of triangles (incircles, circumcircles), cyclic quadrilaterals, and power of a point.
• Trigonometry Advanced identities, trigonometric substitutions, and their use in geometric problems.
• Coordinate Geometry Using coordinates to solve geometric problems, especially for the Open section.
• Vectors Applying vector methods to prove geometric properties, primarily in the Open section.

Number Theory and Combinatorics

• Divisibility and Primes Fundamental Theorem of Arithmetic, properties of prime numbers, and GCD/LCM.
• Modular Arithmetic Congruences, Fermat's Little Theorem, Euler's Totient Theorem, and Chinese Remainder Theorem.
• Diophantine Equations Finding integer solutions to polynomial equations.
• Combinatorial Counting Pigeonhole Principle, Principle of Inclusion-Exclusion, and binomial coefficients.
• Graph Theory Basic concepts like vertices, edges, and degrees, often appearing in combinatorics problems.