Mathematics Contests the Australian Scene 2018 Combined MCYA and AMOC
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Mathematics Contests The Australian Scene 2018 Combined MCYA and AMOC A DI PASQUALE, N DO, KL MCAVANEY AND T SHAW Published by AMT PUBLISHING Australian Maths Trust 170 Haydon Drive Bruce ACT 2617 AUSTRALIA Telephone: +61 2 6201 5136 www.amt.edu.au Copyright © 2019 Australian Mathematics Trust AMTT Limited ACN 083 950 341 National Library of Australia Card Number and ISSN 1323-6490 SUPPORT FOR THE AUSTRALIAN MATHEMATICAL OLYMPIAD COMMITTEE TRAINING PROGRAM The Australian Mathematical Olympiad Committee Training Program is an activity of the Australian Mathematical Olympiad Committee, a department of the Australian Maths Trust. Sponsors The Olympiad programs are funded through the Australian Government’s National Innovation and Science Agenda. The AMT’s EGMO initiative received grant funding from the Australian Government through the Department of Industry, Innovation and Science under the Inspiring Australia – Science Engagement Programme. The Australian Mathematical Olympiad Committee (AMOC) also acknowledges the significant financial support it has received from the Australian Government towards the training of our Olympiad candidates and the participation of our team at the European Girls’ Mathematical Olympiad (EGMO), and the International Mathematical Olympiad (IMO). The views expressed here are those of the authors and do not necessarily represent the views of the government. The Olympiad programs are also supported by the Trust’s National Sponsor of the Australian Informatics and Mathematical Olympiads, Optiver. Optiver is a market maker, offering trading opportunities on major global financial markets using their own capital at their own risk. They hire top talent at graduate and undergraduate levels with science, technology, engineering and mathematics (STEM) qualifications from the world’s leading universities. Optiver employs over 900 people across offices in the United States, Europe and Asia Pacific— including past Olympians and students that have been involved in AMT competitions and programs. Special thanks With special thanks to the Australian Mathematical Society, the Australian Association of Mathematics Teachers and all those schools, societies, families and friends who have contributed to the expense of sending the 2018 EGMO team to Florence and the 2018 IMO team to Cluj–Napoca. Mathematics Contests The Australian Scene 2018 | Support for the AMOC Training Program | i ACKNOWLEDGEMENTS The Australian Mathematical Olympiad Committee (AMOC) sincerely thanks all sponsors, teachers, mathematicians and others who have contributed in one way or another to the continued success of its activities. The editors sincerely thank those who have assisted in the compilation of this book, in particular the students who have provided solutions to the 2018 EGMO and the 2018 IMO. Thanks also to members of AMOC and Challenge Problems Committee, Chief Mathematician Mike Clapper, Chief Executive Officer Nathan Ford, staff of the Australian Maths Trust and others who are acknowledged elsewhere in the book. Mathematics Contests The Australian Scene 2018 | Acknowledgements | ii FROM THE AMT CHIEF EXECUTIVE OFFICER In 2018, the Australian team continued our strong performance at the IMO, ranking 11th in the competition, with 2 Gold medals, 3 Silvers and a Bronze. Australia has placed 15th or higher at the IMO 4 times in the last six years. Our inaugural European Girls’ Mathematical Olympiad team also performed well, placing 20th out of 52 countries with a Silver medal, two Bronze medals and an Honourable Mention. This was a wonderful result for our first attempt, and we are looking forward to continuing our participation in this great competition. These results are a testament to the incredible work of our Australian Mathematical Olympiad Committee (AMOC), our Director of Training, our Team Leaders and Deputies, our AMOC State Directors, the Senior Problems Committee and the dozens of other volunteers and staff who work to support our training program. On behalf of the Australian Maths Trust, I would like to thank in particular: Emeritus Professor Cheryl Praeger AM, AMOC, Chair Dr Angelo Di Pasquale, AMOC Director of Training and International Mathematical Olympiad (IMO), Team Leader Mr Andrew Elvey Price, IMO Deputy Team Leader Miss Thanom Shaw, EGMO Team Leader Miss Michelle Chen, EGMO Deputy Team Leader Mr Mike Clapper, AMT Chief Mathematician Dr Norman Do, Chair, AMOC Senior Problems Committee Dr Kevin McAvaney, Chair, MCYA Challenge Committee Members of the AMT Board and AMOC Committee AMOC tutors, mentors, volunteers and ex-Olympians Nathan Ford April 2019 Mathematics Contests The Australian Scene 2018 | From the AMT Chief Executive Officer | iii CONTENTS Support for the Australian Mathematical Olympiad Committee Training Program .................................... i Acknowledgements ........................................................................................................................................ii From the AMT Chief Executive Officer .........................................................................................................iii Background Notes on the EGMO, IMO and AMOC ...................................................................................... 1 Summary of Australia’s achievements at EGMO ...................................................................................... 2 Summary of Australia’s achievements at previous IMOs ......................................................................... 2 Mathematics Challenge for Young Australians ............................................................................................. 4 Membership of MCYA Committees ............................................................................................................... 6 Membership of AMOC .................................................................................................................................... 8 AMOC Timetable for Selection of the Teams to the 2018 EGMO and 2018 IMO ........................................ 9 Activities of AMOC Senior Problems Committee ...................................................................................... 10 Challenge Problems – Middle Primary ........................................................................................................ 11 Challenge Problems – Upper Primary ......................................................................................................... 14 Challenge Problems – Junior ....................................................................................................................... 17 Challenge Problems – Intermediate ............................................................................................................ 22 Challenge Solutions – Middle Primary......................................................................................................... 26 Challenge Solutions – Upper Primary ......................................................................................................... 30 Challenge Solutions – Junior ....................................................................................................................... 34 Challenge Solutions – Intermediate ............................................................................................................ 42 Challenge Statistics – Middle Primary ......................................................................................................... 53 Challenge Statistics – Upper Primary .......................................................................................................... 54 Challenge Statistics – Junior ........................................................................................................................ 55 Challenge Statistics – Intermediate ............................................................................................................. 56 Australian Intermediate Mathematics Olympiad ........................................................................................ 57 Australian Intermediate Mathematics Olympiad Solutions ........................................................................ 58 Australian Intermediate Mathematics Olympiad Statistics ........................................................................ 69 AMOC Senior Contest .................................................................................................................................. 70 AMOC Senior Contest Solutions ................................................................................................................. 71 AMOC Senior Contest Results ..................................................................................................................... 78 AMOC Senior Contest Statistics .................................................................................................................. 80 AMOC School of Excellence ........................................................................................................................ 81 Participants at the 2017 AMOC School of Excellence ............................................................................ 82 Australian Mathematical Olympiad ............................................................................................................. 83 Australian Mathematical Olympiad Solutions ............................................................................................. 85 Australian Mathematical