Curriculum · Olympiad Topics
Olympiad Topics
Invariants, extremal arguments, functional equations, graph theory — the proof-based ideas that separate qualifiers from medalists.
Combinatorial Essence
Invariants & Monovariants
The olympiad opener: find what a process preserves, and impossibility proofs write themselves.
- Identify invariants of a process
- Use parity and coloring arguments
- Deploy monovariants for termination
prerequisite: pigeonhole-principle, modular-arithmetic
The Extremal Principle
Consider the largest, the smallest, the closest pair — and let extremity do the work.
- Choose the right extremal object
- Derive contradictions from extremality
- Combine with induction
prerequisite: invariants-monovariants
Graph Theory, First Steps
Vertices, edges, degrees, connectivity — and the handshake lemma's surprising reach.
- Model problems as graphs
- Apply the handshake lemma
- Argue about connectivity and cycles
prerequisite: counting-fundamentals
Generating Functions
Counting with power series: convolution, partitions, and closed forms.
- Encode sequences as series
- Multiply generating functions to convolve
- Extract coefficients
prerequisite: binomial-theorem, sequences-telescoping
Algebraic Arts
Functional Equations
Substitution discipline, injectivity and surjectivity, and the Cauchy equation.
- Make systematic substitutions
- Prove injectivity/surjectivity
- Recognize Cauchy-type equations
prerequisite: functions-intro
Recursive Sequences
Linear recurrences, characteristic roots, and periodicity hunting.
- Solve linear recurrences
- Find and prove periodicity
- Translate recursions into closed forms
prerequisite: sequences-telescoping
Olympiad Inequalities
AM-GM, Cauchy-Schwarz, and the strategy of smoothing and normalization.
- Apply AM-GM and Cauchy-Schwarz
- Normalize cleverly
- Find equality cases first
prerequisite: quadratics-vieta
Geometry at Depth
Cyclic Quadrilaterals & Angle Hunting
Ptolemy, cyclic characterizations, and the concyclic points nobody mentioned.
- Prove points concyclic
- Apply Ptolemy's theorem
- Chase directed angles
prerequisite: circle-theorems, power-of-a-point
Advanced Configurations
Radical axes, homothety, and the classical configurations of olympiad geometry.
- Use radical axes
- Spot homothety centers
- Recognize classical configurations
prerequisite: cyclic-quadrilaterals
Proof Writing
The craft of the written proof: structure, rigor, and the reader's experience.
- Structure proofs clearly
- Close logical gaps
- Write for a grader
prerequisite: invariants-monovariants