Page 1 About the Editor and Contributors Kiril Bankov, Bulgaria

About the Editor and Contributors

Kiril Bankov, Bulgaria, a Ph.D. in mathematics, prepares future mathematics teachers as a Professor of Mathematics Education at the University of Soﬁa and the Bulgarian Academy of Sciences. He has been involved in mathematics competitions in Bulgaria for more than 20 years as an author of contest problems and as a member of juries. He has written many articles, made presentations, and is a co-author of books on mathematics competitions, problem solving, work with mathematically gifted students, etc. Two of his papers in “Mathematical Gazette” won “Article of the Year” citation (1995 and 1999). Bankov has an extensive experience in international large-scale studies in mathematics education. For more than a decade he has been a member of the International Expert Committee (SMIRC) for TIMSS study. He also worked as a mathematics coordinator for the International Study Center of the Teacher Education and Development Study—Mathe- matics (TEDS-M) at Michigan State University (USA). In 2014 he was the Chairperson of the European Baccalaureate Examining Board. Bankov was the Secretary of World Federation of National Mathematics Competitions (WFNMC) 2008–2012, and serves presently as the Senior Vice President. Francisco Bellot–Rosado, Spain, served as Chair of the Mathematics Department of High Schools “Marqués de la Ensenada”, Logroño (1966–70) and “Emilio Ferrari”, Valladolid (1970–2006), and Associate Professor on the Faculty of Sciences of the University of Valladolid (several times during 1971–1999). He was Team Leader or Deputy Leader of the Spanish Team in the International Mathematical Olympiad (IMO) and Iberoamerican Math Olympiad (OIM), 1988–1997, Europe Representative in WFNMC 1996– present; and Member of the Joint Committee of Governors of the RSME in various periods 1997–present. He served as the Editor of the Journal SIPROMA (1996–98), published by the Organization of Iberoamerican States (OEI). He founded the Mediterranean Mathematics Competition (Peter O’Halloran Memorial) in 1998; and starting in 1984 organized in Spain the Mathematical Kangaroo Contest. In 2000, Bellot-Rosado was presented the

Paul Erdős Award by the WFNMC. Starting in 2002, he served as the Editor of the digital journal Revista Escolar de la OIM. H served as the Leader of the Spanish team at the 5th European Girls Mathematics Olympiad (Busteni, Romania, 2016). Luis F. Cáceres-Duque, Puerto Rico, a Ph.D. in Logic from The University of Iowa, is a Professor at the University of Puerto Rico, Maya- guez Campus. He is well recognized in the educational system of Puerto Rico, and has worked for the last ﬁfteen years in several educational projects including professional development of mathematics and science teachers. Cáceres is well-known in the ﬁeld of international mathematical competitions. He is the Secretary of the Association Kangaurou Sans Frontieres. He is also the Founder and Director of The Mathematical Olympiad Program in Puerto Rico. In 2016, Cáceres was presented the Paul Erdős Award by the World Federation for National Mathematical Competitions. He has several papers and books in Logic, Ring theory, math Olympiads and secondary school mathematics. In 2014, he published a joint paper “So Much Hap- pening, So Much to Do,” Notices of the AMS. He recently published the book “Math Olympiads in Puerto Rico: 2015–2016.” Cáceres collaborates with the organization of math Olympiads in different countries and he is invited frequently to different countries to offer courses and conferences. His interests include geometry, math Olympiads problems, math talented students and professional development of math teachers. Sergey Dorichenko, Russia, a 1995 M.A. in Mathematics from Moscow State University (MGU), has taught in the Mathematical Circles program of MGU. As a result, he published the book “A Moscow Math Circle: Week-by-Week Problem Sets,” Math Circles Library, MSRI, AMS, 2012. Multiple times he was Deputy Chair of the Organizing Committee and Head Judge of the Moscow Mathematical Olympiad; for several years he worked as a judge and a member of the Organizing Committee of the Lomonosov Tournament. Dorichenko served as a Teacher of Mathematics in Moscow schools 57 (1992–2011) and 179 (2002–present). Deputy Director of the mathematical section of Moscow school №179 (from 2007). Works at the Center of Pedagogical Mastery in Moscow. He also served as the Chair of the Central Jury and Problems Committee of the International Mathematics Tournament of Towns; and the Chair of the Jury of the Summer Conference of the Tournament of Towns. He is the Head of the Department of Mathematics a member of the Editorial Board of the magazine “Kvant” (2008–2016). In 2012 he founded the magazine «Kvantik» and has since served as its Editor-in-Chief. Dorichenko is presently the Secretary of the World Federation of National Mathematical About the Editor and Contributors 375

Competitions. He authored several books and multiple articles, published in magazines «Kvant», «Kvantik», «Mathematical Enlightenment», and «Mathematical Education». Mary Falk de Losada (María), Colombia, is a Professor at the Univer- sidad Antonio Nariño in Bogotá. She did her graduate work in mathematics and math education at Harvard University and the University of Illinois at Chicago. She is cofounder with Ricardo Losada and Rafael Mariño of the Colombian Mathematics Olympiads, the Iberoamericans Mathematics Olympiad, and with many others of the World Federation of Mathematics Competitions (WFNMC). She has received the David Hilbert Award from WFNMC and the José Celestino Mutis Prize from the Colombian Mathe- matics Society for her work in mathematics Olympiads in Colombia and Latin America. She was elected member at large of the ICMI Executive Committee and has worked in various capacities on the organization of ICMEs since 1996, as well as currently being a member of the International Mathematics Union Circle. In 2003, she served as the President of the International Jury of International Mathematical Olympiad in Colombia. María served as the President of WFNMC, and at this time is the Immediate Past President, serving on the Executive Committee and chairing the WFNMC awards committee that determines the recipients of the Paul Erdős Award. She has published several books in Colombia for future math teachers as well as many of the yearly collections of problems from the Colombian Math Olympiads. Retired as professor of the National University of Colombia, she continues to divide her time between the Colombian Math Olympiads and a doctoral program in mathematics education at the Universidad Antonio Nariño in Bogotá. Married for over fifty years to Ricardo Losada, they have five children—all in fields related to mathematics —and nine grandchildren. Robert Geretschläger, Austria Robert (full name: OStR Prof. Mag.rer.nat. Dr.phil. Robert Rudolf Alfred Geretschläger; quite a mouthful) has been active as a mathematics teacher, author and competitions coordinator for over 35 years. Along with his full-time teaching duties at Bundesrealgym- nasium Keplerstrasse in Graz, Austria, and his work at the Karl-Franzens University in Graz, he has also worked on curriculum development in Austria and co-authored a series of high school mathematics textbooks. He is the author of “Geometric Origami” and co-author of such books as “The Circle” along with numerous compilations of competition problems. His main interests lie in the preparation and organization of mathematics competitions. He has been responsible for the introduction and organization of several competitions in Austria, such as the Mathematical Kangaroo, the 376 About the Editor and Contributors

International Tournament of the Towns, and the Mediterranean Mathematics Competition, and has been actively involved in the Austrian Mathematical Olympiad for over 30 years. Since the year 2000. He has represented Austria at the International Mathematical Olympiad, ﬁrst as Deputy Leader of the Austrian team, and starting in 2006 as a member of the problem selection committee, and the Austrian Team Leader. He is also a member of the international boards of the Kangourou sans Frontières (KSF) and presently a Vice President of the World Federation of National Mathematics Competitions (WFNMC). Matúš Harminc, Slovakia, a 1986 Ph.D. in Group Theory and Algebra, is an Associate Professor at the Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University, Košice. After graduation, he worked at the Mathematical Institute of the Slovak Academy of Sciences. From 1975 to 1986 he taught on several Slovak faculties. He gives courses on Number Theory, Mathematical Methods of Problem Solving, and the Introduction to Mathematics for pre-service teachers. His main research areas are teaching methods and the solving of math word problems. Martina Jesenská, Slovakia, works at the Giles Academy, Boston, Lin- colnshire, England. In 2015, she graduated from Pavol Jozef Šafárik University in Košice with the master's degree in Teaching Mathematics. She is presently on her second year of teaching Mathematics. Romualdas Kašuba, Lithuania, is Ph.D. from the University of Greif- swald, Germany, teaches Mathematics, Communications Skills, and Ethics at Vilnius University. He has a long history with various Mathematical Olympiads (MO): being a jury member of the Lithuanian MO since 1979, the Deputy Leader of the Lithuanian IMO team since 1996, and the leader of the Lithuanian team at the Baltic Way team-contest since 1995. In 1999, he initiated the Lithuanian Mathematical Olympiad for youngsters and has been involved in Lithuanian Kangaroo movement from the same year, as well as being the leader of the Lithuanian MEMO team since 2009. Besides being the author of several books in Lithuanian, he has composed several booklets in English (“What to do when you do not know what to do,” Parts I and II, as well as “Once upon a time I saw the puzzle,” Parts I, II, and III). Furthermore, he rewrote and expanded a book with a similar title in Russian, which was published in Moscow in 2012 (2nd ed. 2014). During 2008–2016 he represented Lithuania at ICMI, 2010–2014 was the Board member of MCG (International Group for Mathematical Creativity and Giftedness), and is the member of the Editorial Board of MCG Newsletter. About the Editor and Contributors 377

Nikolai Konstantinov, Russia, a 1963 Ph.D. in mathematics and physics, worked at the Physics Department at Moscow State University, the Institute of Theoretical and Experimental Physics, the Institute of Patent Information, Institute of Economics of the USSR Science Academy, Moscow Institute of Open Education, and at several Moscow schools: 7, 57, 91, and 179. He is currently a scientific supervisor at the Moscow school 179. In 1968 he coauthored a computer program that animated and modeled the movement of a cat (the program operated by solving differential equations). In 1979 founded the multidisciplinary Lomonosov Tournament for secondary students and has been its chair ever since. During 1967–1980, he was a member of the jury of the Soviet Union Mathematical Olympiad. In 1980 he founded the International Mathematics Tournament of Towns and has since been its president. In 1989 he organized the International Summer Con- ference of the Tournament of Towns. He founded the Independent Moscow University IMU) and is currently a trustee of IMU and of the Moscow Center of Continuous Mathematical Education. Konstantinov has been a member of the advisory board of “Kvant” magazine, and of the editorial boards of Russian magazines “Mathematical education” and “Mathematical Enlightenment”, Enrichment series (Australian International Centre for Mathematics Enrichment). Konstantinov’s publications include two articles on set-theoretic geometry, an article on computer science, over ten articles on mathematical economics, and a series of articles on mathematical education (including specific courses of calculus taught at mathematical schools). Konstantinov is a 1992 recipient of the Paul Erdős Award from the World Federation of National Mathematics Competitions (WFNMC) “for his significant contribution in developing the Tournament of the Towns’ Contest in Russia”; a 2002 Award of the Moscow City Council “for services to the city.” In WFNMC he is a representative of Europe. José Heber Nieto Said, Venezuela, is a Professor at Universidad del Zulia. Born in Uruguay, he earned his M.Sc. degree from Universidad de Buenos Aires in Argentine, and Dr.H.C. from Universidad del Zulia, Venezuela. He has written several papers and books on mathematics, especially on combinatorics, problem solving and mathematical Olympiads. His research interests include algebraic and analytic combinatorics. He has been involved with mathematical Olympiads for the past twenty years: he has been a coach, Leader or Deputy Leader of the Venezuelan Team and participates actively in the organization of Olympiads in Venezuela and the Caribbean region. He is particularly fond of problem solving and regularly submits problems and solutions to math competitions and to the problems sections of several journals and on-line forums. 378 About the Editor and Contributors

Rafael Sánchez Lamoneda, Colombia, a 1986 Ph.D. in Mathematics from Brandeis University, USA, is an Assistant Professor at the Universidad Antonio Nariño, Bogota, Colombia. He was born in Venezuela. He has written several papers and books on Algebra and Mathematical Olympiads problems. He was Chair of the Venezuelan Institute for Scientiﬁc Research’s Mathematical Department. In 2010 he won the Paul Erdős Award in 2010, given by the World Federation for National Mathematical Competitions, and in 1993 Best Mathematical Paper Prize from the Venezuelan National Council for Scientiﬁc Research. He is member of the Venezuelan Mathematical Olympiads Commission since 1978, Coach, Leader and Deputy Leader of Venezuelan teams to international mathematical competitions, including the International Mathematical Olympiad (IMO). From 2012 to 2016 he was an elected member for the International Mathematical Olympiad Advisory Board, IMOAB, and since 2012 Presi- dent of the IMO Ethics Committee. He is advisor of the Venezuelan Academy of Physical, Mathematical and Natural Sciences. He was President of the Venezuelan Mathematical Association and actual president and founder of the Venezuelan Association for Mathematical Competitions. His research interests include Characteristic-Free Representation Theory of Gl (n), Schur and Weyl Modules, Homological Algebra and Mathematical Olympiads problems. Ingrid Semanišinová, Slovakia, is a 2007 Ph.D. in Theory of Mathematics Education, is an Assistant Professor on the Faculty of Science, Pavol Jozef Šafárik University, Košice, and teaches Didactics of Mathematics, History of Mathematics, Problem Solving in Mathematics Education and Mathe- matics for Geographers. Her research topics are aimed at pre-service teacher preparation, activities for students and developing creativity in the classroom. She has six years’ experience with teaching sixth to ninth grade mathematics at a local middle school. She is a member of the regional committee of Mathematical Olympiad. Vinayak M. Sholapurkar, India, is the Head, Center for Postgraduate Studies in Mathematics, S. P. College, Pune, India. He completed his Ph. D. in Operator Theory from Savitribai Phule Pune University. He has an extensive experience of teaching graduate level mathematics courses and has been involved in training Math Olympiad students at the regional and National levels, as well as in the International Mathematical Olympiad training camps. He was the National Coordinator for Mathematical Olym- piads in India from 2013 to 2016. Dr Sholapurkar served as an Observer with Indian Team in IMO, Cape Town, 2014; and led the Indian Team in EGMO, Busteni, Romania, 2016. He has been working as the National About the Editor and Contributors 379

Coordinator for Madhava Mathematics Competition for Undergraduate students in India since its inception in 2009. He delivered talks at several National and International conferences and workshops and is a co-author of a popular book “An Excursion in Mathematics” meant for the preparation for Math Olympiads. His publications reflect his keen interest in the area of Operator Theory, his field of specialization as also, Functional Analysis. Some of these papers are: Rigidity theorems for spherical hyperexpansions (with Sameer Chavan), Complex Analysis and Operator Theory, Vol. 7(5), (2013), 1545-1568; Completely monotone functions of finite order and Agler’s conditions (with Sameer Chavan), Studia Mathematica, Vol. 226(3), (2015), 229-258; Completely hyperexpansive tuples of finite order (with Sameer Chavan), J. Math. Ana. and Appl. Vol. 447(2017), 1009-1026. Kar-Ping Shum, China, earned his Ph.D. in Mathematics from the University of Alberta in 1971 and was awarded Honorary Doctor of Mathematics by Gomel University, Belarus in 2002. He is the Founding Chairman of Hong Kong IMO committee (1986–present) and has been the leader and inspiration for the organization of the IMO in Hong Kong in 1994 and again in 2016. He has been for years affiliated with the Chinese University of Hong Kong, where he served as Professor of Mathematics (2000–2002), Research Chair Professor (2003–2007), Honorary Professor (2008–2011). He is Honorary Director of the Institute of Mathematics and Professor of Mathematics, Yunnan University at Kunming (2011–present). He was also Honorary Professor, National Research Council, The Chinese Academy of Sciences in Beijing, 1990–1997; President of Hong Kong Mathematics Society, 1985- 1989; President of the Southeast Asian Math- ematical society (1990–1991 and 2001–2002); Chair of the Federation of the Hong Kong Institutions staffs associations (1996–present). He is Editor-in-Chief of Asian European Journal of Mathematics (2007–present); Theory of Semigroup and Its Applications (2008–present); Southeast Asian Bulletin of Mathematics (1989–present). Editor of a number of Journals in Taiwan, Mainland China, Vietnam, Thailand, Malaysia, Iran, Korea, Malaysia, USA, Hungary, Italy, India and Pakistan, etc. Alexander Soifer, USA (the editor and an author), a 1973 Ph.D. from Moscow, is a Professor of Mathematics, Film Studies, and Art History at the University of Colorado (1979–present). In 1978 he left Russia as a refugee in search of freedom, which he found on the shores of the New World. In 1988, Soifer was charmed by Paul Erdős into switching from Abelian group theory to Euclidean Ramsey theory and ‘Erdősian’ discrete geometry. Soifer’s works (some 400 papers and 12 books) include The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators; The 380 About the Editor and Contributors

Scholar and the State: In Search of Van der Waerden; Mathematics as Problem Solving; The Colorado Mathematical Olympiad: The First Two Decades and Further Explorations; The Colorado Mathematical Olympiad: The Third Decade and Further Explorations; Geometric Etudes in Combi- natorial Mathematics; How Does One Cut a Triangle?—all published by Springer. He co-wrote and edited the monograph Ramsey Theory Yesterday, Today, and Tomorrow, Birkhäuser, 2011. Soifer contributed chapters to Chromatic Graph Theory, Cambridge U Press, 2015; and Open Problems in Mathematics, edited by John F. Nash. Jr.; Springer, Switzerland, 2016. He is a recipient of the Paul Erdős Award (2006) and President of the World Federation of National Mathematics Competitions (2012–present), and has been on its Executive Committee since 1996, serving previously as Senior Vice President and Secretary. Soifer is the publisher and editor of the research quarterly Geombinatorics, 1991–present. He created The Colorado Mathematical Olympiad and has run it for over 33 years, 1984–present. Soifer was first to serve on the judges of both the Soviet Union (1970–1973) and the United States (1996–2005) Mathematical Olympiads. Peter Taylor, Australia, a 1972 Ph.D. in Applied Mathematics from the University of Adelaide, was an academic at the University of Canberra from 1972 to 2012, including Executive Director of the Australian Mathematics Trust from 1994 to 2012. He is now an Emeritus Professor at the University of Canberra. His many publications include Challenging Mathematics in and beyond the Classroom, Springer 2009, being the Final Report of ICME Study 16 he co-chaired, book co-edited with Ed Barbeau. In 2015 he was made an Officer (AO) of the Order of Australia, in 1994 presented the Paul Erdős Award from the World Federation of National Mathematics Com- petitions (WFNMC), and in 1994 BH Neumann Award (Australian Math- ematics Trust). He has been on the Executive of WFNMC since 1996 (President 2000 to 2004) and has been on the Executive of the International Group for Mathematical Creativity and Giftedness since 2012. Iliana Tsvetkova, Bulgaria, is a teacher of Mathematics at Sofia High School of Mathematics. In this position she has gained a vast experience in working with mathematically talented students. An important part of Iliana’s work is the preparation of students for National and International mathematics competitions. Her students won many prizes including gold, silver and bronze medals at 2004 and 2012 IMO. Iliana herself has been a Team Leader of Bulgariathein many international mathematics competitions: Po Leung Kuk Primary Mathematics World Contest (PMWC), Elementary Mathematics International Competition (EMIC), International World Youth Mathematics Competition (IWYMIC), World Youth Mathematics Intercity About the Editor and Contributors 381

Competition (WYMIC), World Mathematics Team Championship WMTC, Tuymaada Olympiad, Jautikov Olympiad. She has shared her experience working with mathematically outstanding students in a number of articles some of them written for “Mathematics Competitions”, the journal of the WFNMC, and the National Council of Teachers of Mathematics (NCTM). Iliana Tsvetkova was awarded as “Teacher of the Year” in 2005 and 2013; this is the most prestigious prize for teachers of mathematics in Bulgaria. Author Index

B K Bankov, Kiril, 101 Kašuba, Romualdas, 267 Bellot-Rosado, Francisco, 27 Konstantinov, Nikolay, 205

C N Cáceres-Duque, L.F., 123 Nieto Said, J.H., 123

D S de Losada, María Falk, 325 Sánchez Lamoneda, R.J., 123 Dorichenko, Sergey, 205 Semanišinová, Ingrid, 171 Sholapurkar, V.M., 239 G Shum, K.P., 55 Geretschläger, Robert, 145 Soifer, Alexander, ix, 3, 347

H T Harminc, Matúš, 171 Taylor, Peter James, 299 Tsvetkova, Iliana, 187 J Jesenská, Martina, 171

A G Admissible operation, 101–103, Geometrical situation, 30 106–114, 120 Gifted students, 188, 209, 304 Applications of mathematics, 148, Goals of instruction, 3, 6, 7 149, 154, 163 Goals of life, 16, 17

C H Caribbean, 123, 142 History of mathematics, 148, 159, Central America, 123, 142 165 Colorado Mathematical Olympiad, Hungarian school, 331, 332, 336 3, 6–8, 10, 13, 14, 18 Combinatorial situation, 102, 105, I – 106, 108 110, 112, 114, 115 International Mathematical Combinatorics, 124, 141 Olympiad (IMO), 336, 337, 341, – Competition, 171 175, 180, 342, 345, 346 182–184 Contemporary mathematics, 348 M Madhava Mathematics Competition, D 243, 248–252, 262, 263 Development, 190, 194, 200, 201 Mathematical competitions, 329, Discovering giftedness, 187, 188 336, 341, 344, 347, 348 Mathematical Olympiads, 244 E Mathematical problem, 29 Epistemology, 331 Mathematical talent, 187, 191, 196, 201 F Mathematics competitions, – Flexibility, 171, 172, 183, 184 145 147, 150, 152, 156, 159, 169, 188, 190, 194–196, 199–201

Methodology, 331 Problem solving, 3, 5–7, 16, Multiple Solution Tasks (MSTs), 330–332, 334, 336, 337, 339, 172 342, 343 Problem solving strategies, 172, 185 O Olympiad, 123, 124, 134, 141, 142 R Optimal arrangements of numbers, Recreational mathematics, 148, 149, 116–118 154, 156, 159, 160, 163

P S Problem, 123, 124, 126, 129, Secondary schools, 147, 154 133–136, 138, 141, 142 Solution, 124, 128–130, 132–140 Problems of mathematics competitions, 102, 115, 121