Smoothing & Normalisation

Nudge the variables toward equality and watch the bound improve.

For a symmetric inequality under a constraint, take any two unequal variables and move them toward each other while preserving the constraint. If the expression improves at every such step, the extremum sits where all variables are equal — which is usually the equality case you guessed.

Normalisation is the companion: a homogeneous inequality is unchanged by scaling, so you may simply declare $a+b+c = 3$ or $abc = 1$ . This removes a degree of freedom for free and is almost always the right first line.

Together they turn many intimidating symmetric problems into a one-variable calculation. Used constantly by Engel and in the Iranian/Russian training traditions.

Where it appears on Lemma: Level 7 (inequality toolkit) and Level 8 olympiad sets.

Train it on Lemma

Smoothing & Normalisation unlocks at Level 7 of Lemma's eight-level ladder, with lessons that teach it and drills that make it stick. 78 lessons, 624 curated problems and unlimited generated practice at six difficulties. Free to start — no card, no trial clock.

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