A Central European Olympiad

The Mathematical Duel

Robert Geretschläger, Józef Kalinowski;Jaroslav Švrček

• Publisher: WSPC
• Year: 2017
• ISBN:
  9789813223929
• Topic: Pedagogía
• Subtopic: Enseñanza de matemáticas

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This book contains the most interesting problems from the first 24 years of the "Mathematical Duel", an annual international mathematics competition between the students of four schools: the Gymnázium Mikuláše Koperníka in Bílovec, Czech Republic, the Akademicki Zespół Szkół Ogólnokształcących in Chorzów, Poland, the Bundesrealgymnasium Kepler in Graz, Austria and the Gymnázium Jakuba Škody in Přerov, Czech Republic.

The problems are presented by topic, grouped under the headings Geometry, Combinatorics, Number Theory and Algebra, which is typical for olympiad-style competitions.

Above all, it is of interest to students preparing for mathematics competitions as well as teachers looking for material to prepare their students, as well as mathematically interested enthusiasts from all walks of life looking for an intellectual challenge.

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--> Contents:

• Introduction
• Number Theory
• Algebra
• Combinatorics
• Geometry
• 4! Years of Problems

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--> Readership: General public, students and teachers preparing for olympiad-style mathematical competitions -->
Keywords:Mathematics Competition;Problem SolvingReview: Key Features:

• The wide selection of problems makes it especially interesting for students and teachers preparing for olympiad-style mathematical competitions
• The participants in this particular competition range in age from 13 to 18, and the problems are created with this wide range in mind
• Any interested reader is bound to find something interesting to suit their own level of experience

Chapter 1

Introduction

A Special Kind of Mathematics Competition

One of the wonderful aspects of international mathematics competitions is the opportunity they offer to bring together people from different geographical backgrounds, but sharing a strong common interest. As we all know, a strong interest in mathematics is not all that common, and young people fascinated by logical thought and abstract puzzles may not always find it easy to find like-minded peers in their own classrooms. As anyone who has ever been involved with any type of international competition can attest to, such a meeting of competitors gives the participants an opportunity to forge friendships across borders that can last a lifetime. Even those taking part who are not quite so fortunate, can at least hope for a chance to gain some understanding of other cultures and other viewpoints. This is the story of an international mathematics competition that has been going strong for nearly a quarter century, and shows no sign of burning out.

Since 1993, the mathematics competition “Mathematical Duel” has been held annually between students from two central European schools, the Gymnázium Mikuláše Koperníka (GMK) in Bílovec, Czech Republic, and the I Liceum Ogółnokształc

ce im. Juliusza Słowackiego (now the Akademicki Zespół Szkół Ogólnokształc

cych) in Chorzów, Poland. Since 1997, they have been joined by the Bundesrealgymnasium (BRG) Kepler in Graz, Austria, and since 2008 by Gymnázium Jakuba Škody (GJŠ) in Přerov, Czech Republic. The original inspiration for this activity came from several international competitions that were already well established at the time, such as the Baltic Way or the Austrian-Polish Mathematics Competition, but the Duel was to develop into something quite different.

Structure of the Mathematical Duel

Anyone familiar with olympiad-style mathematics will immediately recognise the typical type of problems posed at the Duel. As is the case in many such competitions the world over, participants aim to write mathematical proofs for problems stemming from various topics in pure mathematics and appropriate to their age- and competence-levels.

The competition is divided into three categories. Similar to the Czech olympiad system, division A is for grades 11–12, B for grades 9–10 and C for grades 8 or younger. Typically, four students from each school in each of the divisions come together for a competition (on occasion, a school will not be able to field the complete contingent of four for a group because of sudden illness or for some other reason).

The students participate in an individual competition comprising four problems to be solved in 150 minutes and in a team competition comprising three problems to be solved in 100 minutes. The two competitions are completely independent of one another and yield separate results. While the individual competition is written in supervised silence, the team competition sees the four-member team from each school in each category placed together in a room with no supervision. The students spend their time devising a common group answer to each problem, and only one answer sheet is accepted from each group at the end. The dynamics of the group competition are quite different from the individual work common to most competitions, and are quite fascinating to observe.

The venue for the competition rotates annually between the participating cities Bílovec, Chorzów, Graz and Přerov. In early years, the participants were given the questions in their own languages, but English has been used as a common neutral language since Duel VIII in the year 2000. The students still write their answers in their own languages, and any difficulties in understanding the problems are solved on the spot as the problems are handed out. This method of dealing with the language issue certainly makes for some interesting situations, but the jury has always been able to iron out any difficulties as they have arisen.

The competition was traditionally organised over the course of four days, the first and last of which are spent travelling. (This has changed in the last two years, since the Duel became a part of an EU-sponsored Erasmus Plus project. More will be said about this further on.) The farthest any two of the schools involved are apart is about 800 km, and travel is therefore always possible by bus or train. On one day, there is always an excursion for the participants. This can include some hiking, or a visit to a museum. Also, other activities like sports or puzzle events are organised. Of course, there are also some small prizes for the students with the best results, as well as t-shirts or other souvenirs of the competition, and diplomas for all the participants.

A Special Duel and Guest Schools

In the year 2005, Duel XIII was slated to take place in Graz. Coincidentally, this was the year that BRG Kepler was planning its second Europe Days, a school project meant to foster understanding between the peoples of Europe. It was therefore decided to have a special “European” Duel. Several other schools, with which a long-standing cooperation already existed, were therefore invited to participate in the Mathematical Duel as guests. This was the start of an idea which has become a tradition at the Mathematical Duel, namely for the hosting school to invite partner schools to participate as guests, and this tradition has helped to increase the international flavour of the competition a great deal. The following schools have participated in the Mathematical Duel as guests since 2005:

2005 — Duel XIII — Graz: Diósgyőri Gimnázium (Kilián György Gimnázium), Miskolc, Hungary; GJŠ Přerov;

2006 — Duel XIV — Bílovec: GJŠ Přerov;

2008 — Duel XVII — Přerov: Gymnázium Olomouc — Hejčín, Czech Republic; Slovanské Gymnázium Olomouc, Czech Republic;

2009 — Duel XVII — Graz: mixed team, various schools in Graz, Austria;

2010 — Duel XVIII — Chorzów: Colegiul Naţional “Ion Luca Caragiale”, Ploieşti, Romania;

2011 — Duel XIX — Přerov: Liceo Scientifico Statale “Antonio Labriola”, Roma-Ostia, Italy;

2012 — Duel XX — Bílovec: Liceo Scientifico Statale “Antonio Labriola”, Roma-Ostia, Italy; Colegiul Naţional “Ion Luca Caragiale”, Ploieşti, Romania; Sofiyska Matematicheska Gimnaziya Paisiy Hilendarski, Sofia, Bulgaria;

2013 — Duel XXI — Graz: mixed team, various schools in Graz, Austria;

2014 — Duel XXII — Přerov: Sofiyska Matematicheska Gimnaziya Paisiy Hilendarski, Sofia, Bulgaria;

2015 — Duel XXIII — Bielsko-Biała: V Liceum Ogólnokształc

ce, Bielsko-Biała, Poland.

Reaching Out Beyond the Core — The Mathematical Duel Outside the Participating Schools

Since several schools have participated in the Mathematical Duel as guests along with the regulars, awareness of the competition outside the confines of the partner schools has developed quite naturally. Along with this, however, a conscious effort has also been made to make the problems from the competition available to a wider audience. Articles about the competition were published in various regional publications as well as in the international journal Mathematics Competitions, the periodical of the World Federation of National Mathematics Competitions. (A Local International Mathematics Competition (Special Edition), Robert Geretschläger and Jaroslav Švrček, Mathematics Competitions, Vol. 18, No. 1, 2005, pp. 39–51.)

Starting in 2008 (with an interruption in 2009), booklets were published containing the complete problems and solutions of the competition, along with the complete results. These booklets were distributed among mathematics educators in the participating countries.

This lead to the use of the problems of the Duel for another, similar competition, annually pitting the students of the BG/BRG Leibnitz (Austria) against those from the Österreichische Schule (Austrian High School) in Budapest (Hungary).

This development has recently found an unexpected climax, with the Mathematical Duel finding Europe-wide recognition.

The Mathematical Duel as an Erasmus Plus Project

Erasmus Plus is a program combining the various funding schemes the European Union finances for education, training, youth and sport. It was launched in its current form as a continuation of numerous predecessor programs in January 2014, and the idea to apply for an Erasmus Plus grant for a project that would include the Duel was born as soon as the creation of the program was announced.

In order to qualify as an Erasmus Plus program, the scope of the activity needed to be widened quite a bit. Most importantly, the Duel needed to be included in a research framework, and this meant widening the group of institutions involved to include some tertiary institutions. For this reason, a university from each of the partner countries was included in the project, now named “Mathematical Duel plus”. These are The Karl-Franzens University in Graz, Austria, the Palacký University in Olomouc, Czech Republic, and the University of Silesia in Katowice, Poland. A wide research program, looking into the implications of mathematics competitions in education from various viewpoints was introduced, and several academic papers have already appeared. Along with many other documents relating to the project, these can be found at the project website at

http://www.mathematicalduel.eu/.

Since the rules of the Erasmus program require international student activities to be at least five days in length, a fifth day was added to the usual program of the mathematical duel, in which a scientific program with mathematical talks and other activities is organised for the students.

Due to the widened scope of the competition, the competition has been shifted to slightly larger venues in the years of the Erasmus Plus project. In 2015, the contest was held at a hotel catering specially to sports teams in Bialsko-Biała (which is not too far from Chorzów), and in 2016 it was held at a hotel in Ostrava (a larger city near Bílovec).

The inclusion of the Mathematical Duel in this Erasmus Plus project (as this is being written, two of the three years of the initial project have passed) can certainly be considered a success by any measure. A great deal of material is being produced for use in schools all over the European Union, the participating students are being given the opportunity to experience much more mathematics and more cultural crossover than would otherwise be the case, and the website makes the material accessible to anyone interested in immersing themselves in the project.

Problem Development

Finally, before we get to the problems themselves, a few words concerning their genesis seem in order.

Finding appropriate problems for the Mathematical Duel is not an easy task. As is the case for so many similar competitions, every effort is made to develop original material that has not appeared in the published problem sets from all the various competitions around the world. Not only that, the problems should be interesting, age-appropriate and similar in style to the type of problem the students are prepared for. How well we have succeeded in this is up to each reader of this collection to decide for him- or herself.

The majority of the problems used in the competition over the years were suggested by the three authors of this book, but many were suggested by our colleagues. Unfortunately, in the early years of the competition, there was not really any thought given to collecting information like the names of the proposers of specific problems for posterity. For this reason, we must apologise for possibly not being able to thank some early collaborator for his or her contributions. Since 2011, however, this information has been collected systematically, and together with the names of problem proposers we were able to reconstruct from various other sources, we gratefully acknowledge the following problem authors for their suggestions:

Pavel Calábek (A-T-1-99, A-T-1-04, A-T-3-07, A-T-3-08, A-I-1-09, A-T-3-11, C-I-4-11, A-I-3-12, A-T-1-12, A-T-3-12, A-I-2-13, A-T-2-13, A-I-2-14, A-T-2-14, A-T-1-15, B-T-3-15, C-T-1-15, A-I-4-16, A-T-3-16, B-I-4-16)

Emil Calda (A-T-I-02)

Radek Horenský (A-I-4-10)

Gottfried Perz (A-I-4-15, B-I-2-15)

Stanislav Trávníček (B-I-1-14, C-I-4-14)

Jacek Uryga (A-I-3-11, A-T-1-11, B-T-2-11, A-I-1-12, B-I-2-12, A-I-1-13, B-I-3-13, C-I-2-13, A-I-1-14, A-I-3-14, B-I-3-14, A-I-2-15, A-T-2-15, C-T-2-15, A-T-1-16, A-T-2-16, B-I-3-16, B-T-3-16)

Erich Windischbacher (C-T-2-12, A-I-3-13)

Note that each problem is referred to in the form (category–individual/team–number–year). This means that (B-I-2-03) refers to problem number 2 of the individual competition in the B category of the year 2003. Furthermore, the number of the competition results by adding 8 to the digits of the year. The year 2003 was therefore the year of the 03+8=11th Duel.

In each subsection, a brief introduction is followed by a list of problems. Interested readers are invited to try their hand at finding solutions before heading on to the solutions that follow immediately afterwards. Of course, if you prefer to head straight to the solutions, you may just want to skip right ahead. Either way, we hope you will enjoy this collection of problems from our Mathematical Duel.

Chapter 2

Number Theory

2.1.Numbers with Interesting Digits

A type of problem used quite often in the Mathematical Duel concerns problems that specifica...