Three Problems on Harmonic Series in the International Mathematical Olympiad

EasyChair Preprint 1519

Abstract

The International Mathematical Olympiad (IMO) is the most prestigious pre-university math competition. In this article we
study in detail three problems proposed in different years and that somehow use the harmonic series. They can be incorporated into the training of high school students for this type of contention or in classes at college level. The first one was proposed for the 2001 IMO and deals with an inequality, that must be shown to be valid for an infinite number of positive integers. The solution considers the fact that the harmonic series grow unlimited. The second one was proposed for the IMO of 1979 and chosen as the first tournament question. In this case the focus is on a partial sum of the alternating harmonic series. A generalization of this problem is found. The third one was proposed for the IMO of 1975 and deepens the understanding of the harmonic series. Taking away an infinite subsequence does the harmonic series remains divergent?

Keyphrases: International Mathematical Olympiad, Pre-university Teaching, harmonic series, sequences, university teaching

@booklet{EasyChair:1519,
  author    = {Juan López Linares and Alexys Bruno-Alfonso and Grazielle Feliciani Barbosa},
  title     = {Three Problems on Harmonic Series in the International Mathematical Olympiad},
  howpublished = {EasyChair Preprint 1519},
  year      = {EasyChair, 2019}}