Holton - A First Step to Mathematical Olympiad Problems: Mathematical Olympiad Series Vol. 1

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.