Features of a high school olympiad problem

TL;DR

This paper explores a high school olympiad problem involving sequences and rational numbers, illustrating key mathematical concepts like countability, continued fractions, and 2-adic integers, with detailed explanations suitable for educators.

Contribution

It provides an in-depth analysis of a specific olympiad problem, connecting it to broader mathematical ideas and extending known concepts to 2-adic integers.

Findings

Demonstrates the countability of rational numbers via a specific sequence.

Shows the correspondence between finite continued fractions and rational numbers.

Extends the sequence index to 2-adic integers, linking positive reals and 2-adic integers.

Abstract

This paper is a supplement to a talk for mathematics teachers given at the 2016 LSU Mathematics Contest for High School Students. The paper covers more details and aspects than could be covered in the talk. We start with an interesting problem from the 2009 Iberoamerican Math Olympiad concerning a particular sequence. We include a solution to the problem, but also relate it to several areas of mathematics. This problem demonstrates the countability of the rational numbers with a direct one-to-one correspondence. The problem also shows the one-to-one correspondence of finite continued fractions and rational numbers. The subsequence of odd indexed terms was constructed by Johannes Kepler and is discussed. We also show that the indices have an extension to the 2-adic integers giving a one-to-one correspondence between the positive real numbers and the 2-adic integers. Everything but the extension to the 2-adic integers is known to have appeared elsewhere.