Aimo Sample Papers Sample Doc

Published by A M T 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.

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Australian Intermediate Mathematics Olympiad Committee Dr K McAvaney Mr J Dowsey Dr M Evans Mr B Henry

Assoc Prof H Lausch Mr R Longmuir Adj Prof M Clapper

Deakin University (Chair, 2007–2013) University of Melbourne AMSI, Victoria Victoria (Chair, 1999–2006) Monash University China Australian Mathematics Trust

7 years; 2007–2013 15 years; 1999–2013 15 years; 1999–2013 15 years; 1999–2013 15 years; 1999–2013 2 years; 1999–2000 1 year; 2013

Moderators for AIMO Dr G Carter

Queensland University of Technology ACT Dept of Education Mr J Carty University of Tasmania Dr K Dharmadasa University of Melbourne Dr A Di Pasquale St Michael’s Collegiate School, TAS Mr W Evers University of Western Australia Dr G Gamble SA Department of Education Mr K Hamann Burgmann Anglican School, ACT Mr J Hassall Assoc Prof D Hunt UNSW University of Sydney Dr W Palmer Adelaide Dr M Peake University of Queensland Dr V Scharaschkin Assoc Prof P Schulz University of Western Australia Australian Mathematics Trust Dr A Storozhev WA Department of Education Dr E Stoyanova King David School, VIC Dr P Swedosh University of Queensland Dr N H Williams University of Southern Queensland Dr O Yevdokimov

12 years; 2001–2012 14 years; 1999–2012 10 years; 2004–2013 5 years; 2009–2013 5 years; 1999–2003 8 years; 2006–2013 7 years; 1999–2005 2 years; 2012–2013 7 years; 2007–2013 15 years; 1999–2013 7 years; 2006–2012 3 years; 2011–2013 1 year; 1999 2 years; 2007–2008 6 years; 2000–2005 15 years; 1999–2013 2 years; 1999–2000 4 years; 2010–2013

CONTENTS

• PREFACE

vii

• AUSTRALIAN INTERMEDIATE MATHEMATICS

OLYMPIAD COMMITTE E • QUESTIONS

ix 1

Australian Intermediate Mathematics Olympiad 1999

3


5


7


9

Australian Intermedi ate Mathemat ics Olympiad 2003

11


13


15


17


19


21


23

Australian Intermedi ate Mathemati cs Olympiad 2010

25

Australian Intermedi ate Mathemati cs Olympiad 2011

27


29

Australian Intermedi ate Mathemat ics Olympiad 2013

31

• SOLUTIONS

33

Australian Intermediate Mathematics Olympiad 1999 Solutions

35


40


44


48


53


58


63


70


76


82


88

Australian Intermedi ate Mathemat ics Olympiad 2010 Solutions

96


109


119


131

QUESTIONS

Aimo Sample Papers Sample Doc

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