Classical Inequalities

Equality tells you where to aim.

AM–GM and Cauchy–Schwarz between them settle a large share of olympiad inequalities. AM–GM: the arithmetic mean is at least the geometric mean, with equality exactly when all terms are equal. Cauchy–Schwarz handles sums of products, and in Engel form ( $\sum a_i^2/b_i \ge (\sum a_i)^2/\sum b_i$ ) it is devastating on fractions.

Always locate the equality case first. It tells you which grouping to use, and if your chain of inequalities cannot reach equality where the problem does, you have bounded too crudely.

Smoothing and normalisation are the usual companions: fix a symmetric constraint, then argue the extremum is symmetric.

Where it appears on Lemma: Level 4 (AM–GM) and Level 7 (Cauchy–Schwarz).

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