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Three Problems on Inequalities in the International Mathematical Olympiad

EasyChair Preprint no. 2981

11 pagesDate: March 17, 2020

Abstract

Inequalities receive less attention compared to equalities because of a greater difficulty in their study, however they are no less important. Three problems that were proposed for IMO (International Mathematical Olympiad) are discussed in detail. They can be incorporated into the training of high school students for this type of competition or in classes for college students. In the first one, an infinite and strictly increasing sequence of positive integers is used. The inequality involved is always satisfied by a single term in the sequence. The second one explores a sequence of positive real numbers. The solution of the challenge is obtained by the repeated use of the Arithmetic and Geometric Means inequality. The third problem deals with an infinite and non-increasing sequence of positive real numbers. To solve it, we use the CauchySchwarz inequality and the sum of a geometric series.

Keyphrases: Arithmetic and Geometric Inequality, Cauchy-Schwarz inequality, International Mathematical Olympiad, sequences, teaching