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Pigeonhole principle — AMC & AIME Problems

Practice every real AMC 8, AMC 10, AMC 12, and AIME problem that uses Pigeonhole principle — a combinatorics concept — each with a professionally curated solution, many with a video solution too, from LIVE by Po-Shen Loh. Browse all concepts or the full past-contest archive.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

Filter by difficulty

These problems’ difficulty ratings come from real solve attempts on our platform. Tap a band or a contest to filter the list below, and tap a selected band again to include all difficulties. See the full difficulty guide.

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AMC 8

1988 AMC 8 Problem 16 • 1990 AMC 8 Problem 19 • 1994 AMC 8 Problem 21 • 2005 AMC 8 Problem 16 • 2026 AMC 8 Problem 22

AMC 10

2006 AMC 10A Problem 20 • 2010 AMC 10B Problem 3 • 2011 AMC 10B Problem 11 • 2019 AMC 10A Problem 4

AMC 12

1961 AMC 12 Problem 39 • 1976 AMC 12 Problem 12 • 1986 AMC 12 Problem 17 • 1994 AMC 12 Problem 19 • 2018 AMC 12A Problem 12 • 2018 AMC 12B Problem 10 • 2019 AMC 12A Problem 3

More combinatorics concepts

arrangements with restrictionsbasic countingbijectionBurnside’s LemmaCatalan Numbercircular arrangementscombinationscombinatorial gamecomplementary countingcounting integers in a rangecounting intersectionscounting pairscounting regionscounting shapes in figuresderangementsdouble countingfencepost countinggenerating functionsgraph theoryinclusion-exclusionlattice pathsmultiplication principlemultiset permutationspartitions and compositionsPascal’s Trianglepermutationsrecursive countingstars and barssubsetstilingtree diagramVandermonde’s ConvolutionVenn Diagram