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A Gentle Introduction to Inequalities: A Casebook from the Fullerton Mathematical Circle
Adam Glesser, Matt Rathbun, and Bogdan Suceavă
Run for nearly a decade, the Fullerton Mathematical Circle at California State University, Fullerton prepares middle and high school students for mathematical research by exposing them to difficult problems whose solutions require only age-appropriate techniques and background. This work highlights one of the avenues of study, namely inequalities. We cover Engel's lemma, the Cauchy--Schwartz inequality, and the AM-GM inequality, as well as providing a wealth of problems where these results can be applied. Full solutions or hints, several written by Math Circle students, are given for all of the problems, as well as some commentary on how or when to assist students, and details about the pedagogical value of certain problems.