aimo sample papers sample doc
DESCRIPTION
Sample of the book produced by the AMTTRANSCRIPT
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Published by
AMT PUBL ISH ING
Australian Mathematics TrustUniversity of Canberra Locked Bag 1
Canberra GPO ACT 2601AUSTRALIA
Copyright 2014 AMT Publishing
Telephone: +61 2 6201 5137www.amt.edu.au
AMTT Limited ACN 083 950 341
National Library of Australia Card Number and ISSNAustralian Mathematics Trust Enrichment Series ISSN 1326-0170
Australian Intermediate Mathematics Olympiads 1999-2013ISBN 978-1-876420-73-4
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Preface
Australia 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. 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 710. 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 1999.
At 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. 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 level.
The 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. 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 students.
The 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 space.
Statistics are provided next to the marks for each question as to the number of students withthe correct answer per total number of students. For questions 9 and 10, the mean number ofmarks 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 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 layout.
Mike Clapper
Executive Director, Australian Mathematics TrustAdjunct Professor, University of Canberra
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Student Instructions
Time allowed : 4 hoursNo 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 winnerswhere required.
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Australian Intermediate Mathematics Olympiad Committee
Dr K McAvaney Deakin University(Chair, 20072013)
7 years; 20072013
Mr J Dowsey University of Melbourne 15 years; 19992013
Dr M Evans AMSI, Victoria 15 years; 19992013
Mr B Henry Victoria(Chair, 19992006)
15 years; 19992013
Assoc Prof H Lausch Monash University 15 years; 19992013
Mr R Longmuir China 2 years; 19992000
Adj Prof M Clapper Australian Mathematics Trust 1 year; 2013
Moderators for AIMO
Dr G Carter Queensland University ofTechnology
12 years; 20012012
Mr J Carty ACT Dept of Education 14 years; 19992012
Dr K Dharmadasa University of Tasmania 10 years; 20042013
Dr A Di Pasquale University of Melbourne 5 years; 20092013
Mr W Evers St Michaels Collegiate School, TAS 5 years; 19992003
Dr G Gamble University of Western Australia 8 years; 20062013
Mr K Hamann SA Department of Education 7 years; 19992005
Mr J Hassall Burgmann Anglican School, ACT 2 years; 20122013
Assoc Prof D Hunt UNSW 7 years; 20072013
Dr W Palmer University of Sydney 15 years; 19992013
Dr M Peake Adelaide 7 years; 20062012
Dr V Scharaschkin University of Queensland 3 years; 20112013
Assoc Prof P Schulz University of Western Australia 1 year; 1999
Dr A Storozhev Australian Mathematics Trust 2 years; 20072008
Dr E Stoyanova WA Department of Education 6 years; 20002005
Dr P Swedosh King David School, VIC 15 years; 19992013
Dr N H Williams University of Queensland 2 years; 19992000
Dr O Yevdokimov University of Southern Queensland 4 years; 20102013
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CONTENTS
PREFACE vii
AUSTRALIAN INTERMEDIATE MATHEMATICS OLYMPIAD 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 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
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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
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Questions
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AIMO-Book-Text-12AIMO-Book-Text-13AIMO-Book-Text-14AIMO-Book-Text-15AIMO-Book-Text-16AIMO-Book-Text-17AIMO-Book-Text-18AIMO-Book-Text-19AIMO-Book-Text-20