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 — 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. 78 lessons, 624 curated problems and unlimited generated practice at six difficulties. Free to start — no card, no trial clock.

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TelescopingWrite each term as a difference and watch it collapse.
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