Introduction to Math Olympiad Problems

Michael A. Radin

• Publisher: Chapman and Hall/CRC
• Year: 2021
• ISBN:
  9781000396737
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics in Education
• Index: Mathematics

Introduction to Math Olympiad Problems aims to introduce high school students to all the necessary topics that frequently emerge in international Math Olympiad competitions. In addition to introducing the topics, the book will also provide several repetitive-type guided problems to help develop vital techniques in solving problems correctly and efficiently. The techniques employed in the book will help prepare students for the topics they will typically face in an Olympiad-style event, but also for future college mathematics courses in Discrete Mathematics, Graph Theory, Differential Equations, Number Theory and Abstract Algebra.

Features:

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• Numerous problems designed to embed good practice in readers, and build underlying reasoning, analysis and problem-solving skills

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• Suitable for advanced high school students preparing for Math Olympiad competitions

CHAPTER 1

Introduction

The aim of this chapter is to get acquainted with the basic fundamental tools that we will use to approach, analyze and solve assorted problems that may require just one or two steps to solve or perhaps require multiple number of steps to solve. We will discover that every problem will reduce to understanding and deciphering of basic fundamentals. Therefore, it is vital to establish and understand the intrinsic knowledge that will guide us to unfolding very challenging problems. We will emerge with the foundation of assorted sequences and patterns.

1.1 Patterns and Sequences

This section’s aims are to recognize various patterns and sequences. We will commence with linear sequences where the difference between two neighboring terms is a constant. For instance, the sequence that lists the consecutive positive integers starting at 1 is graphically portrayed with the corresponding diagram.

Analytically, we express the sequence of positive integers in Figure 1.1 as

$$\begin{array}{l}{\left\{n\right\}}_{n=1}^{\infty }.\hfill \end{array}$$ (1.1)

Figure 1.1 List of positive consecutive integers.

Analogous to Figure 1.1, we will assemble similar diagrams when solving related problems with consecutive integers and other homologous contents. In addition, (1.1) can be expressed as a recursive sequence. Observe that we start at 1 and tr...