Published by AMT P U B L I S H I N G Australian Mathematics Trust University of Canberra Locked Bag 1 Canberra GPO ACT 2601 AUSTRALIA Copyright 2014 AMT Publishing Telephone: +61 2 6201 5137 www.amt.edu.au AMTT Limited ACN 083 950 341 National Library of Australia Card Number and ISSN Australian Mathematics Trust Enrichment Series ISSN 1326-0170 Australian Intermediate Mathematics Olympiads 1999-2013 ISBN 978-1-876420-73-4
vii Preface Australia entered a team in the International Mathematical Olympiad (IMO) for the first time in 1981 and has participated in this competition ever since, enjoying significant success and assisting in the development of many fine young mathematicians. In 1983, the Australian Mathematical Olympiad Committee (AMOC) was set up to identify and train students for international competition, as well as to stimulate a general interest in mathematical problem solving. Key components of this identification process were the AMOC Inter-State competitions which led to participation in training and then the Australian Mathematical Olympiad (AMO). In 1986, a junior division of the Inter-State competition was introduced, aimed at students in Years 7 10. The importance of identifying talented students as young as possible was recognised; a small number of outstanding juniors were invited to training camps. This contest was renamed the Telecom Junior Contest in 1990, and in 1993, the Telecom Intermediate Contest. Eventually, and in a revised format, it became the Australian Intermediate Mathematics Olympiad in 1999. At every stage the purpose of this competition has been to provide a stimulating set of challenging questions for young mathematicians, with the hope of identifying talented individuals, who might become involved in state or national training leading to participation in the senior Olympiad program. Whilst the Australian Mathematics Competition attracts many more students and is also used to identify potential, the AIMO is a longer exam (4 hours) and requires some proofs and investigation, essential skills at the Olympiad level. The AIMO is now seen as the culmination of the Mathematics Challenge for Young Australians (MCYA) and there could be no better preparation for the AIMO than to complete the Challenge and Enrichment Stages. Indeed, the AIMO is based on material found in the later stages of the Enrichment Stage (particularly Gauss and Noether). The AIMO papers have been developed by a small committee, chaired initally by Bruce Henry and, from 2007, by Kevin McAvaney. While the AIMO is certainly challenging, we feel that some students who might be able to do well are not encouraged to enter or do not know about the competition. In producing this book of past papers, we hope to bring the contest to a wider audience and to provide some opportunity to practise. Individually, AIMO papers have been available in the AMOC yearbook, Mathematics Contests The Australian Scene, but this is the first time that a collection of papers has been put together. We hope that it will prove a useful and stimulating resource for teachers and students. The papers are presented in very much their original form, though edited to fit on smaller pages. Some diagrams have been redrawn for greater clarity. The student instructions have changed very little over the years, and are provided on the next page. These instructions have been removed from the individual papers in the interests of space. Statistics are provided next to the marks for each question as to the number of students with the correct answer per total number of students. For questions 9 and 10, the mean number of marks obtained is given. The solutions are as originally published in Mathematics Contests The Australian Scene each year, sometimes with several alternatives for each question. I am extremely grateful for the efforts of Bruce Henry and Kevin McAvaney, not only in their many years as successive Chairs of the AIMO Committee, but also in the preparation of this volume of collected papers. I also acknowledge the work of our in-house editor, Bernadette Webster, whose tireless efforts in proofreading and editing have eliminated many errors and greatly improved the final appearance, and Heather Sommariva, our graphic designer, who produced the cover and other aspects of the final design and layout. Mike Clapper Executive Director, Australian Mathematics Trust Adjunct Professor, University of Canberra
viii Student Instructions Time allowed : 4 hours No calculators are to be used. Questions 1 to 8 require only numerical answers, all non-negative integers less than 1000. Questions 9 and 10 require written solutions which may include proofs. The investigation in Question 10 offers bonus marks, used only to determine prize winners where required.
ix Australian Intermediate Mathematics Olympiad Committee Dr K McAvaney Deakin University 7 years; 2007 2013 (Chair, 2007 2013) Mr J Dowsey University of Melbourne 15 years; 1999 2013 Dr M Evans AMSI, Victoria 15 years; 1999 2013 Mr B Henry Victoria 15 years; 1999 2013 (Chair, 1999 2006) Assoc Prof H Lausch Monash University 15 years; 1999 2013 Mr R Longmuir China 2 years; 1999 2000 Adj Prof M Clapper Australian Mathematics Trust 1 year; 2013 Moderators for AIMO Dr G Carter Queensland University of 12 years; 2001 2012 Technology Mr J Carty ACT Dept of Education 14 years; 1999 2012 Dr K Dharmadasa University of Tasmania 10 years; 2004 2013 Dr A Di Pasquale University of Melbourne 5 years; 2009 2013 Mr W Evers St Michael s Collegiate School, TAS 5 years; 1999 2003 Dr G Gamble University of Western Australia 8 years; 2006 2013 Mr K Hamann SA Department of Education 7 years; 1999 2005 Mr J Hassall Burgmann Anglican School, ACT 2 years; 2012 2013 Assoc Prof D Hunt UNSW 7 years; 2007 2013 Dr W Palmer University of Sydney 15 years; 1999 2013 Dr M Peake Adelaide 7 years; 2006 2012 Dr V Scharaschkin University of Queensland 3 years; 2011 2013 Assoc Prof P Schulz University of Western Australia 1 year; 1999 Dr A Storozhev Australian Mathematics Trust 2 years; 2007 2008 Dr E Stoyanova WA Department of Education 6 years; 2000 2005 Dr P Swedosh King David School, VIC 15 years; 1999 2013 Dr N H Williams University of Queensland 2 years; 1999 2000 Dr O Yevdokimov University of Southern Queensland 4 years; 2010 2013
CONTENTS PREFACE AUSTRALIAN INTERMEDIATE MATHEMATICS OLYMPIAD COMMITTEE vii ix QUESTIONS 1 Australian Intermediate Mathematics Olympiad 1999 3 Australian Intermediate Mathematics Olympiad 2000 5 Australian Intermediate Mathematics Olympiad 2001 7 Australian Intermediate Mathematics Olympiad 2002 9 Australian Intermediate Mathematics Olympiad 2003 11 Australian Intermediate Mathematics Olympiad 2004 13 Australian Intermediate Mathematics Olympiad 2005 15 Australian Intermediate Mathematics Olympiad 2006 17 Australian Intermediate Mathematics Olympiad 2007 19 Australian Intermediate Mathematics Olympiad 2008 21 Australian Intermediate Mathematics Olympiad 2009 23 Australian Intermediate Mathematics Olympiad 2010 25 Australian Intermediate Mathematics Olympiad 2011 27 Australian Intermediate Mathematics Olympiad 2012 29 Australian Intermediate Mathematics Olympiad 2013 31 SOLUTIONS 33 Australian Intermediate Mathematics Olympiad 1999 Solutions 35 Australian Intermediate Mathematics Olympiad 2000 Solutions 40 Australian Intermediate Mathematics Olympiad 2001 Solutions 44 Australian Intermediate Mathematics Olympiad 2002 Solutions 48 Australian Intermediate Mathematics Olympiad 2003 Solutions 53 Australian Intermediate Mathematics Olympiad 2004 Solutions 58 Australian Intermediate Mathematics Olympiad 2005 Solutions 63 Australian Intermediate Mathematics Olympiad 2006 Solutions 70 Australian Intermediate Mathematics Olympiad 2007 Solutions 76
Australian Intermediate Mathematics Olympiad 2008 Solutions 82 Australian Intermediate Mathematics Olympiad 2009 Solutions 88 Australian Intermediate Mathematics Olympiad 2010 Solutions 96 Australian Intermediate Mathematics Olympiad 2011 Solutions 109 Australian Intermediate Mathematics Olympiad 2012 Solutions 119 Australian Intermediate Mathematics Olympiad 2013 Solutions 131
Questions