Aimo Sample Papers Sample Doc

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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

CONTENTS

PREFACE vii

AUSTRALIAN INTERMEDIATE MATHEMATICS OLYMPIAD COMMITTEE ix

QUESTIONS 1

Australian Intermediate Mathematics Olympiad 1999 3















SOLUTIONS 33

Australian Intermediate Mathematics Olympiad 1999 Solutions 35









Questions

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

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Aimo Sample Papers Sample Doc