APMO · 1. Inequalities
How to find and write olympiad proofs of inequalities: absolute values and sharp constants, squares, AM–GM and Cauchy–Schwarz, tangent-line bounds, and induction with telescoping, through five fully proved APMO problems.
Use the triangle inequality and build sharp examples for 'largest constant' problems.
Apply AM–GM and Cauchy–Schwarz with their conditions and equality cases.
Design a tangent-line bound that is tight at the equality point.
Prove an n-variable inequality with a two-variable lemma, induction and telescoping.
Write complete proofs that state every sign condition and the equality case.