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2019 Vol 32 Nos 1 and 2 Vol 32 No 1 and 2 2019 MATHEMATICS COMPETITIONS JOURNAL OF THE WORLD FEDERATION OF NATIONAL MATHEMATICS COMPETITIONS Vol 32 No 1 and 2 2019 MATHEMATICS COMPETITIONS JOURNAL OF THE WORLD FEDERATION OF NATIONAL MATHEMATICS COMPETITIONS MATHEMATICS COMPETITIONS Journal of the World Federation of National Mathematics Competitions (ISSN 1031 - 7503) Published biannually by Australian Maths Trust 170 Haydon Drive Bruce ACT 2617 Australia Articles (in English) are welcome. Please send articles to: Professor Alexander Soifer University of Colorado 1420 Austin Bluffs Parkway Colorado Springs, CO 80918 USA [email protected] Typesetting by the Moscow Center for Continuing Mathematical Education, Moscow, Russia c 2019 Australian Mathematics Trust, AMTT Limited ACN 083 950 341 Mathematics Competitions. Vol. 32. No. 1 and 2. 2019 Contents World Federation of National Mathematics Competitions 3 From the President 5 Editor’s Page 6 Examples of Mathematics and Competitions influencing each other Peter James Taylor 8 Yes We Can! A History of Rolf Nevanlinna’s name on IMU Prize Alexander Soifer 24 Theoretical or foundational considerations for mathematics education as related to competitions Mar´ıa Falk de Losada 31 What kind of math reasoning may help students in multiple-choice competition Borislav Lazarov 39 2 Mathematics Competitions. Vol. 32. No. 1 and 2. 2019 Mathematics Competitions. Vol. 32. No. 1 and 2. 2019 Contents World Federation of National Mathematics Competitions World Federation of National Mathematics Competitions 3 Executive From the President 5 President: Kiril Bankov Editor’s Page 6 University of Sofia Sofia, BULGARIA Examples of Mathematics and Competitions influencing each other [email protected] Peter James Taylor 8 Senior Vice President: Robert Geretschl¨ager Yes We Can! A History of Rolf Nevanlinna’s name on IMU Prize BRG Kepler Alexander Soifer 24 Graz, AUSTRIA [email protected] Theoretical or foundational considerations for mathematics education Vice Presidents : Sergey Dorichenko as related to competitions School 179 Mar´ıa Falk de Losada 31 Moscow, RUSSIA What kind of math reasoning may help students in multiple-choice [email protected] competition Krzysztof Ciesielski Borislav Lazarov 39 Jagiellonian University Krakow, POLAND [email protected] Publications Officer: Alexander Soifer University of Colorado Colorado Springs, USA [email protected] Secretary: David Crawford Leicester Grammar School Leicester, UK [email protected] Immediate Past President Alexander Soifer Chair, Award Committee: University of Colorado Colorado Springs, USA [email protected] Treasurer : Peter Taylor University of Canberra Canberra, AUSTRALIA [email protected] 2 3 Mathematics Competitions. Vol. 32. No. 1 and 2. 2019 Past Presidents: Maria Falk de Losada Universidad Antonio Nari˜no Bogot´a,COLOMBIA [email protected] Petar Kenderov Bulgarian Academy of Sciences Sofia, BULGARIA [email protected] Regional Representatives Africa: Liam Baker University of Stellenbosch Stellenbosch, SOUTH AFRICA [email protected] Asia: M. Suhaimi Ramly Ardent Educational Consultants Sdn. Bhd Kuala Lumpur, MALAYSIA [email protected] Europe: Francisco Bellot Rosado Royal Spanish Math Society Valladolid, SPAIN [email protected] Jaroslav Sˇ v r ˇc e k Palack´yUniversity Olomouc, CZECH REPUBLIC [email protected] North America: Alexander Soifer University of Colorado Colorado Springs, USA [email protected] Oceania : Peter Taylor University of Canberra Canberra, AUSTRALIA [email protected] South America: Maria Falk de Losada Universidad Antonio Nari˜no Bogot´a,COLOMBIA [email protected] For WFNMC Standing Committees please refer to “About WFNMC” section of the WFNMC website http://www.wfnmc.org/. 4 Mathematics Competitions. Vol. 32. No. 1 and 2. 2019 Mathematics Competitions. Vol. 32. No. 1 and 2. 2019 Past Presidents: Maria Falk de Losada Universidad Antonio Nari˜no From the President Bogot´a,COLOMBIA [email protected] Dear readers of Mathematics Competitions journal! Petar Kenderov I wish to start my first message as President of WFNMC by ex- Bulgarian Academy of Sciences pressing thanks to those who attended the Congress in Graz in July Sofia, BULGARIA 2018 and participated in the discussion of the Constitutional Amend- [email protected] ments proposed by Alexander Soifer our Immediate Past President. The Regional Representatives approved Amendments establish up to two 4-year consecutive terms Africa: Liam Baker limits for certain Officers and Members of the Standing Committees, University of Stellenbosch and a full voting membership on the Executive for Past Presidents of Stellenbosch, SOUTH AFRICA the Federation, who are willing to continue their contributions to the [email protected] Executive and the Federation. Asia: M. Suhaimi Ramly In Graz we elected the Executive and the Standing Committees of Ardent Educational Consultants Sdn. Bhd the Federation. I am glad to see new names there and I hope we all will Kuala Lumpur, MALAYSIA [email protected] do our best for the future prosperity of the Federation. Mathematics Competitions Europe: Francisco Bellot Rosado As already reported, the WFNMC journal Royal Spanish Math Society has a new editor. It is my pleasure to thank the past editor Jaroslav ˇ Valladolid, SPAIN Svrˇcek for his longstanding dedicated work and contributions to make [email protected] the journal so impressive. At the same time I congratulate the new editor Jaroslav Sˇ v r ˇc e k Alexander Soifer and the assistant editor Sergey Dorichenko. I am sure Palack´yUniversity they will continue the increase of the role and influence of Mathematics Olomouc, CZECH REPUBLIC Competitions as a unique journal in this area. I appeal to you, dear [email protected] colleagues, with a call to help the new editors by submitting interesting North America: Alexander Soifer materials to the journal. University of Colorado I realize that the sound foundations of my presidency are built by the Colorado Springs, USA past presidents of the Federation: Peter O’Halloran, Blagovest Sendov, [email protected] Ron Dunkley, Peter Taylor, Petar Kenderov, Maria Falk de Losada, and Oceania : Peter Taylor Alexander Soifer. I hope to continue the directions and the traditions in University of Canberra the development of the Federation. Canberra, AUSTRALIA Finally, let me cite a thought of Alexander Soifer that I like a lot: [email protected] “The future of the Federation is in our hands”. By saying this I want to South America: Maria Falk de Losada stress that I rely on you all to work together for the aims of WFNMC. Universidad Antonio Nari˜no Bogot´a,COLOMBIA With warm regards, [email protected] For WFNMC Standing Committees please refer to “About WFNMC” Kiril Bankov, President of WFNMC section of the WFNMC website http://www.wfnmc.org/. May 2019 4 5 Mathematics Competitions. Vol. 32. No. 1 and 2. 2019 Editor’s Page Dear Competitions enthusiasts, readers of our Mathematics Competitions journal! Sergei Dorichenko and I are inviting you to submit to our Mathemat- ics Competitions your creative essays on a variety of topics related to creating original problems, working with students and teachers, organiz- ing and running mathematics competitions, historical and philosophical views on mathematics and closely related fields, and even your original literary works related to mathematics. Just be original, creative, and inspirational. Share your ideas, prob- lems, conjectures, and solutions with all your colleagues by publishing them here. We have formalized the submission format to establish uniformity in our journal. Submission Format Please, submit your essay to the Editor Alexander Soifer. Format: should be LaTeX, TeX, or Microsoft Word, accompanied by another copy in pdf. Illustrations: must be inserted at about the correct place of the text of your submission in one of the following formats: jpeg,pdf,tiff, eps, or mp. Your illustration will not be redrawn. Resolution of your illustrations must be at least 300 dpi, or, preferably, done as vector illustrations. If a text is needed in illustrations, use a font from the Times New Roman family in 11 pt. Start: with the title in BOLD 14 pt, followed on the next line by the author(s)’ name(s) in italic 12 pt. Main Text: Use a font from the Times New Roman family in 11 pt. End: with your name-address-email and your website (if applicable). Include: your high resolution small photo and a concise professional summary of your works and titles. Please, submit your manuscripts to me at [email protected]. 6 Mathematics Competitions. Vol. 32. No. 1 and 2. 2019 Mathematics Competitions. Vol. 32. No. 1 and 2. 2019 The success of the World Federation of National Mathematics Com- Editor’s Page petitions (WFNMC), including its journal Mathematics Competitions, is in your hands — and in your minds! Best wishes, Dear Competitions enthusiasts, readers of our Mathematics Competitions journal! Alexander Soifer, Editor, Mathematics Competitions Sergei Dorichenko and I are inviting you to submit to our Mathemat- Immediate Past President, WFNMC ics Competitions your creative essays on a variety of topics related to creating original problems, working with students and teachers, organiz- ing and running mathematics competitions, historical and philosophical views on mathematics and closely related fields, and even your original literary works related to mathematics. Just be original, creative, and inspirational. Share your ideas, prob- lems, conjectures, and solutions with all
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