Olympiad problem-solving techniques

The 29 techniques that solve competition mathematics, from the pigeonhole principle, invariants and induction to AM–GM, Cauchy–Schwarz, Vieta jumping and inversion. Each one is taught inside Lemma's 125-lesson ladder and drilled until it is a reflex; each page below explains the idea, when to reach for it, and where it appears in real olympiad problems.

• SymmetryIf the problem doesn't care, neither should your answer. · unlocks at Level 2
• TelescopingWrite each term as a difference and watch it collapse. · unlocks at Level 2
• Working BackwardsStart at the end. The end knows more. · unlocks at Level 2
• Angle ChasingLabel everything, then keep going. · unlocks at Level 3
• Pigeonhole PrincipleMore pigeons than holes means a shared hole. · unlocks at Level 3
• Similarity & RatiosAngles give you lengths. · unlocks at Level 3
• Bounding & SqueezingTrap the answer between two walls. · unlocks at Level 4
• Classical InequalitiesEquality tells you where to aim. · unlocks at Level 4
• Coloring ArgumentsPaint the board so the impossible shows its color. · unlocks at Level 4
• Constructive Counting & OvercountingBuild the object step by step, then divide by what you repeated. · unlocks at Level 4
• Double CountingCount it twice, set the answers equal. · unlocks at Level 4
• Modular ThinkingWhen the numbers are too big, shrink the universe. · unlocks at Level 4
• Ceva & MenelausConcurrency and collinearity are products of ratios. · unlocks at Level 5
• InductionTwo finite steps climb an infinite ladder. · unlocks at Level 5
• InvariantsFind what the chaos cannot touch. · unlocks at Level 5
• Power of a PointOne number controls every line through P. · unlocks at Level 5
• Symmetry Breaking & WLOGOrder the variables before you start. · unlocks at Level 5
• The Extremal PrincipleLook at the largest, the smallest, the closest, the first. · unlocks at Level 5
• Algorithmic & Greedy ConstructionsDescribe a procedure, then prove it terminates and works. · unlocks at Level 6
• Infinite Descent & Vieta JumpingNo smallest counterexample can survive. · unlocks at Level 6
• The Extremal Principle in GraphsTake the longest path, the vertex of largest degree. · unlocks at Level 6
• The Incircle ConfigurationTangent lengths turn sides into s − a. · unlocks at Level 6
• Convexity & JensenBalance minimises what curves upward. · unlocks at Level 7
• Generating FunctionsTurn a sequence into a single function. · unlocks at Level 7
• Smoothing & NormalisationNudge the variables toward equality and watch the bound improve. · unlocks at Level 7
• The Ravi SubstitutionKill the triangle inequality by construction. · unlocks at Level 7
• Complex Numbers & Coordinates in GeometryWhen the synthetic route stalls, compute. · unlocks at Level 8
• Phantom PointsDefine the point you want, then prove it is the one you have. · unlocks at Level 8
• TransformationsMove the picture until the problem is easy. · unlocks at Level 8