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Recurrences at the International Mathematical Olympiad

EasyChair Preprint no. 2758

10 pagesDate: February 22, 2020

Abstract

Some international olympics exercises require a multi-step solution that integrates various mathematical skills. In this article we discuss in detail three problems proposed for the International Mathematical Olympiad (IMO) where the focus is on dealing with recurrences and each of them can be associated with a sequence of real numbers. In the rst problem, recurrence is a second order equality, linear and inhomogeneous, and it is asked to show the validity of a given property. In the second, the law of recurrence is dened using a second-order linear and homogeneous inequality and it must be shown that another inequality is valid for the terms of the corresponding sequence. In the third problem, recurrence is of the rst order, but not linear, and it is necessary to nd a closed (explicit) formula for the terms of the related sequence. The exercises allows to train the use of various techniques such as sums and telescopic products, the sum of an arithmetic and geometric progression and demonstration by contradiction.

Keyphrases: high school education, International Mathematical Olympiad, Recurrences, sequences, university teaching