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Invariant — AMC & AIME Problems

Practice every real AMC 8, AMC 10, AMC 12, and AIME problem that uses invariant — a problem-solving techniques 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 of these problems are used with official legal permission of the Mathematical Association of America (MAA).

AMC 8

1997 AMC 8 Problem 21 · 1998 AMC 8 Problem 25 · 2004 AMC 8 Problem 5 · 2007 AMC 8 Problem 22 · 2022 AMC 8 Problem 11

AMC 10

2000 AMC 10 Problem 17 · 2003 AMC 10B Problem 15 · 2013 AMC 10B Problem 17 · 2016 AMC 10A Problem 13 · 2024 AMC 10A Problem 6 · 2024 AMC 10B Problem 16

AMC 12

1999 AMC 12 Problem 20 · 2004 AMC 12A Problem 7 · 2007 AMC 12B Problem 10 · 2013 AMC 12A Problem 6 · 2013 AMC 12B Problem 10 · 2013 AMC 12B Problem 16 · 2013 AMC 12B Problem 18 · 2015 AMC 12A Problem 13 · 2016 AMC 12A Problem 10 · 2019 AMC 12B Problem 19 · 2021 AMC 12B Spring Problem 22

AIME

2004 AIME II Problem 15 · 2007 AIME I Problem 14 · 2012 AIME I Problem 11 · 2018 AIME I Problem 10 · 2021 AIME I Problem 10

More problem-solving techniques concepts

bounding to limit casescaseworkextremal argumentinductionoptimizationpairing and groupingpattern recognitionprocess simulationsmall casessymmetrysystematic listingwork backwards