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1 Published byA M T PU B L I S H I N GAustralian Mathematics TrustUniversity of Canberra Locked Bag 1 Canberra GPO ACT 2601 AUSTRALIAC opyright 2014 AMT PublishingTelephone: +61 2 6201 Limited ACN 083 950 341 National Library of Australia Card Number and ISSNA ustralian Mathematics Trust Enrichment Series ISSN 1326-0170 Australian Intermediate Mathematics Olympiads 1999-2013 ISBN 978-1-876420-73-4ii AIMO book1999-2013b5 2014/6/18 16:33 page vii #2iiiiiiviiPrefaceAustralia entered a team in the International Mathematical Olympiad (IMO) for the firsttime in 1981 and has participated in this competition ever since, enjoying significant successand assisting in the development of many fine young mathematicians. In 1983, the AustralianMathematical Olympiad Committee (AMOC) was set up to identify and train students forinternational competition, as well as to stimulate a general interest in mathematical problemsolving.

2 Key components of this identification process were the AMOC Inter-State competi-tions 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 atstudents in Years 7 10. The importance of identifying talented students as young as possiblewas recognised; a small number of outstanding juniors were invited to training camps. Thiscontest was renamed the Telecom Junior Contest in 1990, and in 1993, the Telecom Interme-diate Contest. Eventually, and in a revised format, it became the Australian IntermediateMathematics Olympiad in every stage the purpose of this competition has been to provide a stimulating set ofchallenging questions for young mathematicians, with the hope of identifying talented in-dividuals, who might become involved in state or national training leading to participationin the senior Olympiad program.

3 Whilst the Australian Mathematics Competition attractsmany more students and is also used to identify potential, the AIMO is a longer exam (4hours) and requires some proofs and investigation, essential skills at the Olympiad AIMO is now seen as the culmination of the Mathematics Challenge for Young Aus-tralians (MCYA) and there could be no better preparation for the AIMO than to completethe Challenge and Enrichment Stages. Indeed, the AIMO is based on material found in thelater stages of the Enrichment Stage (particularly Gauss and Noether). The AIMO papershave 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 studentswho might be able to do well are not encouraged to enter or do not know about the com-petition.

4 In producing this book of past papers, we hope to bring the contest to a wideraudience and to provide some opportunity to practise. Individually, AIMO papers have beenavailable in the AMOC yearbook,Mathematics Contests The Australian Scene, but this isthe first time that a collection of papers has been put together. We hope that it will provea useful and stimulating resource for teachers and papers are presented in very much their original form, though edited to fit on smallerpages. Some diagrams have been redrawn for greater clarity. The student instructions havechanged very little over the years, and are provided on the next page. These instructionshave been removed from the individual papers in the interests of are provided next to the marks for each question as to the number of students withthe correct answer per total number of students.

5 For questions 9 and 10, the mean number ofmarks obtained is given. The solutions are as originally Published inMathematics Contests The Australian Sceneeach year, sometimes with several alternatives for each am extremely grateful for the efforts of Bruce Henry and Kevin McAvaney, not only in theirmany years as successive Chairs of the AIMO Committee, but also in the preparation of thisvolume of collected papers. I also acknowledge the work of our in-house editor, BernadetteWebster, whose tireless efforts in proofreading and editing have eliminated many errors andgreatly improved the final appearance, and Heather Sommariva, our graphic designer, whoproduced the cover and other aspects of the final design and ClapperExecutive Director, Australian Mathematics TrustAdjunct Professor, University of Canberraii AIMO book1999-2013b5 2014/6/18 16:33 page viii #3iiiiiiviiiStudent InstructionsTime allowed.

6 4 hoursNo calculators are to be 1 to 8 require only numerical answers, all non-negative integers less than 9 and 10 require written solutions which may include investigation in Question 10 offers bonus marks, used only to determine prize winnerswhere AIMO book1999-2013b5 2014/6/18 16:33 page ix #4iiiiiiixAustralian Intermediate Mathematics Olympiad CommitteeDr K McAvaneyDeakin University(Chair, 2007 2013)7 years; 2007 2013Mr J DowseyUniversity of Melbourne15 years; 1999 2013Dr M EvansAMSI, Victoria15 years; 1999 2013Mr B HenryVictoria(Chair, 1999 2006)15 years; 1999 2013 Assoc Prof H LauschMonash University15 years; 1999 2013Mr R LongmuirChina2 years; 1999 2000 Adj Prof M ClapperAustralian Mathematics Trust1 year; 2013 Moderators for AIMODr G CarterQueensland University ofTechnology12 years; 2001 2012Mr J CartyACT Dept of Education14 years; 1999 2012Dr K DharmadasaUniversity of Tasmania10 years; 2004 2013Dr A Di PasqualeUniversity of Melbourne5 years; 2009 2013Mr W EversSt Michael s Collegiate School, TAS5 years; 1999 2003Dr G GambleUniversity of Western Australia8 years; 2006 2013Mr K HamannSA Department of Education7 years; 1999 2005Mr J HassallBurgmann Anglican School, ACT2 years; 2012 2013 Assoc Prof D HuntUNSW7 years; 2007 2013Dr W PalmerUniversity of Sydney15 years; 1999 2013Dr M PeakeAdelaide7 years; 2006 2012Dr V ScharaschkinUniversity of Queensland3 years; 2011 2013 Assoc Prof P SchulzUniversity of Western Australia1 year.

7 1999Dr A StorozhevAustralian Mathematics Trust2 years; 2007 2008Dr E StoyanovaWA Department of Education6 years; 2000 2005Dr P SwedoshKing David School, VIC15 years; 1999 2013Dr N H WilliamsUniversity of Queensland2 years; 1999 2000Dr O YevdokimovUniversity of Southern Queensland4 years; 2010 2013C ONTENTS P REFACE vii A USTRALIAN INTERMEDIATE MATHEMATICS O LYMPIAD COMMITTEE 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

8 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 11 9 Australian Intermediate Mathematics Olympiad 2013 Solutions 131 Questions