Olympiad 2 (O2) Course

Lessons: 40
Fee: $40
Duration: 2
Current Time(s): Sun 07:30–09:30
Workload: 3–4 hours

There will be 2 hours of teaching per week over 40 weeks, and I expect students to spend about 3–4 hours per week reviewing the material and doing homework. The total time spent in this course should be around 220 hours over the year. As the follow-up to the Olympiad 1 (O1) course, this course will continue to develop creative problem-solving skills and proof-writing skills.

Classes

Click on the class to see the week-by-week schedule.

Name	Time	Students	Duration (h)	Status
2026-08-O2	Sun 07:30–09:30	4	2	Starting Soon

Syllabus

Problem-Solving Strategies (5 weeks): data extraction when small cases; more induction, extremal principal, and visualisations; counterexamples and meta-proofs.
Combinatorics (8 weeks): more double counting, bijections, invariants, monovariants, and graph theory; combinatorial geometry; colouring problems and game theory; consideration of combinatorial processes.
Algebra (8 weeks): real and integer polynomials; Cauchy-Schwarz and Jensen’s inequalities; smoothing; consideration of sequence problems; functional substitutions; injectivity and surjectivity; graphing functional equations; periodicity; fixed points; multiplicative functional equations.
Number Theory (8 weeks): more divisibility and modular arithmetic problems, including modular inverses; using quadratic residues and Vieta jumping to solve Diophantine Equations; p-adic valuations; constructions in number theory, including via prime numbers, highly composite numbers, and Chinese remainder theorem; dealing with digits, and dealing with floor function.
Geometry (8 weeks): dealing with equal angle conditions; homothethy (dilations) and spiral symmetries; tips on constructing points and lines; Ceva and Menalaus theorems; moving points around in Geometry problems; trig bashing.
Proofs (1 weeks): getting partial marks

Prerequisites

Before starting this course, students should be familiar with most of the concepts taught in the Olympiad 1 (O1) course. In particular, students should:

Understand what a proof is and be able to write one with an appropriate level of detail.
Use contradiction and induction effectively in proofs.
Be able to use a variety of soft and hard techniques in problem-solving.
Apply a range of theorems and techniques across algebra, combinatorics, geometry, and number theory to prove results, especially in the areas covered in the O1 course.
Understand what graph theory and functional equations are, along with some techniques used to analyse them.
Understand how to use basic techniques to solve inequalities.