Marcus du Sautoy is Professor of Mathematics and Simonyi Professor for the Public Understanding of Science at the University of Oxford.
Introduction
This book contains a selection of problems drawn from the various competitions and other activities organised by the United Kingdom Mathematics Trust (UKMT).
The UKMT was founded in 1996 by bringing together a number of mathematics competitions for school students organised by three different bodies: the British Mathematical Olympiad Committee, the National Committee for Mathematics Contests and the United Kingdom Mathematics Foundation. Today the UKMT organises a range of different competitions and other activities for students of all ages from 10 to 18, with the stated aim ‘to advance the education of children and young people in mathematics’.
This book consists of one short problem for each day of the year, with an interlude every fourteen days that consists of a longer challenge. The difficulty of the problems varies. We have listed at the end of the book the source of each problem. Together with the description of the competition from which it is drawn, this should give the reader some indication of its intended difficulty. The difficulty of a problem is hard to assess and will naturally depend on the reader. Our intention is that the problems should get gradually harder as each week progresses, and from the start of the year to the end. However, you will probably find many exceptions to this general rule.
Many of the problems were originally set as multiple-choice questions but have been rewritten to remove this feature. This may make some of the problems harder than they originally were, although it should be remembered that in all our competitions students are given only a limited amount of time. It is this time pressure, from which the reader of this book is free, that makes our competitions a tough challenge for school students.
In the Olympiad papers students are asked to write out full solutions and not just give the answer. In marking these papers most of the credit is given for coming up with a cogent and clearly expressed argument; a correct numerical answer by itself counts for little. This feature is missing in the Olympiad problems given in this book, as we are not able to mark your answers! For this reason the full flavour of these problems as presented here is missing. You will find it in the UKMT publications, listed below, which cover the Olympiad problems.
You should also note that the UKMT has a ‘no calculators’ rule for all its competitions. This is because our aim is to encourage good mathematical thinking. Calculators are helpful when numerical answers are required, but using them is often a substitute for thinking. You, dear reader, are, of course, free to use calculators and other electronic devices in tackling these problems, but we think they will only rarely be of any help.
How the problems have been chosen
UKMT problems are designed for students who are currently studying mathematics. In selecting problems for this book aimed at a general reader we have mainly avoided problems that require up-to-date knowledge of the current mathematics curricula. For example, although algebra will often be useful in tackling a problem, we have selected very few problems that are explicitly about algebra. We have included some geometry problems, because geometry is such a beautiful subject. However, most of the problems can be solved using no more than a little numerical knowledge, logical thinking and native wit.
For those whose knowledge is a little rusty we have included a Glossary containing some reminders of mathematical terminology and some basic geometrical facts about angles.
Solutions
To include full solutions, with detailed explanations, to all the problems would require a book four times as large as the current volume. We have therefore generally only given short solutions, but we hope that these will let you know whether you solved the problem correctly, and will be of help if you are ever baffled.
Detailed solutions to many of the problems may be found in the UKMT publications listed below. Solutions to many of the problems, especially those from recent years, may also be downloaded (for free!) from the UKMT website: www.ukmt.org.uk
The range of UKMT competitions
We list below the different UKMT competitions from which these problems have been drawn, giving an indication of the students they are aimed at and the amount of time they are given. Which competition a student is qualified for depends on their school year. Because school years are denominated differently in different parts of the UK, for simplicity we have translated them into the standard ages of the students in these school years.
Primary Team Maths Resources (PTMR)
These are a range of different team activities designed for 10- and 11-year-old students with the aim of facilitating secondary schools in running events for their feeder schools. The activities are mostly similar to those in the Team Maths Challenge but also include some logic problems that we have used for some of our interludes.
Junior Mathematical Challenge (JMC)
Intermediate Mathematical Challenge (IMC)
Senior Mathematical Challenge (SMC)
Each of these papers is made up of 25 multiple-choice questions. The JMC is aimed at students aged 12 or 13, the IMC at students aged 14, 15 or 16, and the SMC at students aged 17 or 18. These papers are often taken by students younger than the target range. For the JMC and IMC, students are given 60 minutes, and for the SMC they are given 90 minutes.
Kangaroo Competitions
The International Mathematical Kangaroo (Kangourou sans Frontières) is an international competition founded in France in 1991 in which over 50 countries now take part. The idea came from the Australian Mathematics Competition, and was named the Kangaroo in recognition of this. The UKMT uses Kangaroo questions for four competitions that are open to students who do sufficiently well in the Challenges, as follows:
Junior Kangaroo for students aged 12 or 13;
Grey Kangaroo for students aged 14;
Pink Kangaroo for students aged 15 or 16;
Senior Kangaroo for students aged 17 or 18.
Each paper consists of 25 questions, for which 60 minutes are allowed. The Junior, Grey and Pink Kangaroo papers are made up of multiple-choice questions. In the Senior Kangaroo, students are asked to give numerical answers but explanations are not required.
Olympiad Papers
These papers are designed for students who do extremely well in the Mathematical Challenges. The heart of all these papers is a small number of tough questions for which fully explained answers are required. They are given the name Olympiad because they are part of a pathway that leads to the UKMT team for the annual International Mathematical Olympiad, a competition in which over one hundred countries take part.
The Olympiad paper that follows on from the Junior Mathematical Challenge is the Junior Mathematical Olympiad for students aged 12 or 13. This is a two-hour paper made up of 10 A-section questions (answers only) and 5 B-section questions (full explanations required).
There are three Olympiad papers that follow on from the Intermediate Mathematical Challenge. These are the Cayley Mathematical Olympiad for students aged 14, the Hamilton Mathematical Olympiad for students aged 15, and the Maclaurin Mathematical Olympiad for students aged 16. Each of these is a two-hour paper with six questions for which full explanations are required. (Before 2004 these papers had A and B sections.)
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