Olympiad problem-solving technique

Symmetry

If the problem doesn't care, neither should your answer.

When a problem is unchanged under swapping $a \leftrightarrow b$ , its extremes tend to live at $a = b$ (or, treacherously, at the boundary, so always check both). Symmetric expressions like $x + y$ and $xy$ deserve their own names.

Strategy stealing and mirroring win games: respond to every move with its mirror image and the invariant symmetry carries you to victory.

Where it appears on Lemma: Level 1 algebra, Level 4 inequalities, Level 6 games.

Train it on Lemma

Symmetry unlocks at Level 2 of Lemma's eight-level ladder, with lessons that teach it and drills that make it stick. 125 lessons, 1206 curated problems and unlimited generated practice at six difficulties. Free to start, no card, and every paid plan opens with 3 free days.

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