A First Step to Mathematical Olympiad Problems

Derek Holton

• Publisher: WSPC
• Year: 2009
• ISBN:
  9789814365253
• Topic: Mathematics
• Subtopic: Mathematics in Education
• Index: Mathematics

See also A SECOND STEP TO MATHEMATICAL OLYMPIAD PROBLEMS

The International Mathematical Olympiad (IMO) is an annual international mathematics competition held for pre-collegiate students. It is also the oldest of the international science olympiads, and competition for places is particularly fierce. This book is an amalgamation of the first 8 of 15 booklets originally produced to guide students intending to contend for placement on their country's IMO team.

The material contained in this book provides an introduction to the main mathematical topics covered in the IMO, which are: Combinatorics, Geometry and Number Theory. In addition, there is a special emphasis on how to approach unseen questions in Mathematics, and model the writing of proofs. Full answers are given to all questions.

Though A First Step to Mathematical Olympiad Problems is written from the perspective of a mathematician, it is written in a way that makes it easily comprehensible to adolescents. This book is also a must-read for coaches and instructors of mathematical competitions.

Contents:

• Jugs and Stamps: How to Solve Problems
• Combinatorics I
• Graph Theory
• Number Theory 1
• Geometry 1
• Proof
• Geometry 2
• Some IMO Problems

Readership: School students keen to learn more of mathematics and specifically mathematics related to the IMO; coaches and instructors of mathematical competitions.

Chapter 1

Jugs and Stamps: How To Solve Problems

1.1. Introduction

In this chapter I look at some number problems associated with jugs, consecutive numbers and stamps. I extend and develop these problems in the way that a research mathematician might. At the same time as this is being done, I develop skills of problem solving and introduce some basic mathematical theory, especially about a basic fact of relatively prime numbers.

Whether you are reading this book as a prelude to IMO training or out of interest and curiosity, you should know from the start that mathematics is all about solving problems. Hence the book concentrates on problem solving. Now a problem is only something that, at first sight, you have no idea how to solve. This doesn't mean that a problem is a problem for everyone. Indeed, after you have solved it, it isn't a problem for you any more either. But what I am trying to do here is to both introduce you to some new mathematics and at the same time show you how to tackle a problem that you have no idea at first how to solve.

This book tackles areas of mathematics that are usually not covered in most regular school syllabuses. Sometimes some background is required before getting started but the goal is to show how mathematics is created and how mathematicians solve problems. In the process I hope you, the reader, get a great deal of pleasure out of the work involved in this book.

I have tried to design the material so that it can be worked through by individuals in the privacy of their own brains. But mathematics, like other human pursuits is more fun when engaged in by a group. So let me encourage you to rope in a friend or two to work with you. Friends are also good for talking to about mathematics even if they know nothing about the subject. It's amazing how answers to problems appear when you say your problem out loud.

Now I expect the geniuses amongst you will be able to work through all this book from cover to cover without a break. The mere mortals, however, will most likely read, get stuck somewhere, put the book down (or throw it away) and hopefully go back to it later. Sometimes you'll skim over a difficulty and go back later (maybe much later). But I hope that you will all get some enjoyment out of solving the problems here.

1.2. A Drinking Problem

No problem solving can be done without a problem, so here is the first of many.

Problem 1. Given a 3 litre jug and a 5 litre jug can I measure exactly 7 litres of water?

Discussion. You've probably seen this question or one like it before but even if you haven't you can most likely solve it very quickly. Being older and more senile than most of you, bear with me while I slog through it.

I can't see how to get 7. So I'll doodle a while. Hmm. I can make 3, 6 or 9 litres just using the 3 litre jug and 5, 10 or 15 litres with the 5 litre jug. It's obvious, from those calculations that I'm going to have to use both jugs.

Well, it's also pretty clear that 7 ≠ 3a + 56 if I keep a and b positive or zero. So I can't get 7 by just adding water from the two jugs in some combination. So what if I pour water from one jug into another?

Let's fill up the 3 litre jug, then pour the water into the 5 litre jug. I can then fill up the 3 litre jug and pour into the bigger jug again until it's full. That leaves 1 litre in the 3 litre jug. Now if I drink the 5 litres of water from the larger jug I could pour 1 litre of water into some container.

So it's easy. Repeat the performance seven times and we've got a container with 7 litres of water!

Exercises

1. Drink 35 litres of water.

2. Find a more efficient way of producing 7 litres.

What does it mean by “more efficient”? Does it mean you'll have to drink less or you'll use less water or what?

1.3. About Solving Problems

Now we've seen a problem and worked out a solution, however rough, let's look at the whole business of problem solving. There is no way that at the first reading I can expect you to grasp all the infinite subtleties of the following discussion. So read it a couple of times and move on. But do come back to it from time to time. Hopefully you'll make more sense of it all as time goes on.

Welcome to the Holton analysis of solving problems.

(a) First take one problem. Problem solving differs in only one or two respects to mathematical research. The difference is simply that most problems are precisely stated and there is a definite answer (which is known to someone else at the outset). All the steps in between problem and solution are common to both problem solving and research. The extra skill of a research mathematician is learning to pose problems precisely. Of course he/she has more mathematical techniques to hand too.

(b) Read and understand. It is often necessary to read a problem through several times. You will probably initially need to read it through two or three times just to get a feel for what's needed. Almost certainly you will need to remind yourself of some details in mid solution. You will definitely need to read it again at the end to make sure you have answered the problem that was actually posed and not something similar that you invented along the way because you could solve the something similar.

(c) Important words. What are the key words in a problem? This is often a difficult question to answer, especially on the first reading. However, here is one useful tip. Change a word or a phrase in the problem. If this changes the problem then the word or phrase is important. Usually numbers are important. In the problem of the last section, “jug” is only partially important. Clearly if “jug” was changed to “vase” everywhere, the problem is essentially not changed. However “3” can't be changed to “7” without affecting the problem.

Now you've come this far restate the problem in your own words.

(d) Panic! At this stage it's often totally unclear as to what to do next. So, doodle, try some examples, think “have I seen a problem like this before?”. Don't be afraid to think “I'll never solve this (expletives deleted) problem”. Hopefully you'll get inspiration somewhere. Try another problem. Keep coming back to the one you're stuck on and keep giving it another go. If, after a week, you're still without inspiration, then talk to a friend. Even mothers (who may know nothing about the problem) are marvellous sounding boards. Often the mere act of explaining your difficulties produces an idea or two. However, if you've hit a real toughie, then get in touch with your teacher — that's why they exist. Even then don't ask for a solution. Explain your difficulty and ask for a hint.

(e) System. At the doodling stage and later, it's important to bring some system into your work. Tables, charts, graphs, diagrams are all valuable tools. Never throw any of this initial material away. Just as soon as you get rid of it you're bound to want to use it.

Oh, and if you're using a diagram make sure it's a big one. Pokey little diagrams are often worse than no diagram.

And also make sure your diagram covers all possibilities. Sometimes a diagram can lead you to consider only part of a problem.

(f) Patterns. Among your doodles, tables and so forth look for patterns. The exploitation of pattern is fundamental to mathematics and is one of its basic powers.

(g) Guess. Yes, guess! Don't be afraid to gu...