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699 OLYMPIC MATHEMATICAL CHALLENGES

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Preface

 

Challenging mathematical problems are intellectually demanding proposals that usually require training to be solved. They appear in Mathematical Olympiads and Contests around the world. Behind any proposal of the six that usually form the paper of any competition  there is a key idea to solve it. A basic skill is to gain experience through training. It consist in the application of the usual and most appropriate techniques to solve mathematical problems. These are the usual tools for mathematical proof such as direct reasoning, contradiction, mathematical induction, inclusion-exclusion, double counting,invariants, extremal principle, bijections and visual or proofs without words among others mathematical strategies.The problems given to the contestants in Math Competitions are challenges carefully chosen for the members of the Jury ofthe contest from a Short List presented to them by the so called Problem Selection Committee (PSC). The wording ofthe Problems chosen are carefully revised in order that the statements be absolutely clear and free of ambiguities. It is desirable that the contestants be able to write the answer of the problem posed in a clear and elegant mathematical style explaining in detail the steps of their solutions from the beginning to the end or conclusion.In this book the authors present a collection of math challenges verysuitable for future contestants who want to prepare to participate in math competitions of secondary school or undergraduate levels, or for students and teachers that enjoy solving mathematical problems. The problems with their full solutions are also given in the second part of the book. They contain a lot of basic ideas that are the most frequently used to face mathematical problems. The main topic worked in this book is inequalities that are scattered in Algebra, Geometry, Number theory, Combinatorics, and Mathematical Analysis. The first three are the yelds that usually appears in the math contests addressed to high school levels around the world and the last one is most appropriate for training students of undergra-duate level who want to participate in university math competitions.

Some of the problems are easy, needing no more than common sense and clear reasoning. Other require some results of Elementary Mat-

hematics that can be found in any textbook of math high school or calculus course. I suggest no pay attention to the solutions until your

battle with a problem has resulted in a victory getting a nice solution or a disappointing defeat. Be aware that a problem may be solved in diferent ways distinct of the one given in the book. Finally, I want to say that it is worthy to have this book in the library because the materials contained in it are most interesting and useful for training students for Math Olympiads.

 

Jose Luis Diaz-Barrero Barcelona 2017

BarcelonaTech, Barcelona, Spain.

 

 ABOUT THE AUTHORS

 

Mihaly Bencze, born on 20 November 1954 in Sacele-Negyfalu, Brasov, is principal and teacher of mathematics at "Ady Endre" high school in Bucharest, Romania. He received a Ph.D. in mathematics in 2010 and has written 6 literary and 16 mathematical books. He is Editor in Chief of Gamma (1978 -1989), Octogon Mathematical Magazine (1993 - ), Erdelyi Matematikal Lapok(1996  ) and Brassoi Fuzetek (1985  ), Bukaresti Magyar Elet (2013  ), and member of many scientifc journals' editorial board. From 2013 he is a member of the American Romanian Academy of Arts and Sciences. He is the founder of the "Jozef Wildt" International Mathematical Competition (1990 - ), the Transsylvanian Hungarian Mathematical Competition (1990 - ), the "John Von Neumann" Mathematical Competition, the "Jozsef Bencko" Mathematical Competition and the International Hungarian Mathematical Competition  (1992). He has published more than 23.000 mathematical problems in 43 scientific journals, and more than 800 papers in journals all over the world.

Mihaly Bencze was rewarded with the title "European Citizen 2017" by the European Parliament.

 Daniel Sitaru, born on 9 August 1963 in Craiova, Romania, is teacher at National Economic College "Theodor Costescu" in Drobeta Turnu - Severin. He published 33 mathematical books, last two of these, "Math Phenomenon" and "Algebraic Phenomenon", were very appreciated worldwide. He is founding editor of "Romanian Mathematical Magazine", an Interactive Mathematical Journal with 3.200.000 visitors in 2017 (www.ssmrmh.ro). Many problems from  his books were published in famous journals such as "American Mathematical Monthly", "Crux Mathematicorum", "Math Problems Journal", "The Pentagon Journal", "La Gaceta de la RSME", "SSMA Magazine". He also published an impressive number of original problems in all mathematical journals from Romania

(GMB, Cardinal, Elipsa, Argument, Recreat_ii Matematice). His articles from "Crux Mathematicorum" and "The Pentagon Journal" were also very appreciated.

 

 

Bibliography

 

  1. Bencze Mihály: Erdélyi és Nemzetközi Magyar Matematikai Versenyek (1984-1997), Editura Fulgur, 1997, Braşov, Romania.

ISBN 973-96000-9-3, 452 pagini

  1. Bencze Mihály: Erdélyi és Nemzetközi Magyar Matematikai Versenyek (1997-2002), Editura Fulgur, 2002, Braşov, Romania.

ISBN 973-98342-3-x, 455 pagini

  1. Bencze Mihály, Yuhua Fu and Linfan Mao: Scientific Elements (International Book Series), Vol. I, 2007

ISBN: 978-1-59973-041-7, ILQ, USA, 195 pagini.

  1. Bencze Mihály and Šefket Arslanagič: A Mathematical Problem Book, COBISS, BH-ID: 1637194, Sarajevo, 2008, Bosnia şi Herţegovina.

ISBN: 978-9958-9073-0-2, 224 pagini

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ISBN 978-973-88255-4-3, 617 pagini

  1. Bencze Mihály şi Virginia Marinescu: The International Conference- Mathematical Education int the Current Europen Context, Editura Studis, 2011, Iaşi, ISBN 978-606-8242- 98-9, 458 pagini.

       7 Shan-He Wu and Mihály Bencze: Selected Chapters of Mathematical Analysis, STUDIS Publishing House, Iasi, Romania, 2011,

 ISBN 978-606-624-061-1, 366 pagini

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  1. Mihály Bencze: New inequalities based on convexity, LAP LAMBERT Academic Publishing, Germany, 2012, ISBN 978-3-659-18544-1, 188 pagini

 

  1. Ovidiu T. Pop, Nicusor Minculete, Mihály Bencze: An introduction to quadrilateral geometry, Editura Didactica si Pedagocica, R.A., 2013, ISBN 978-973-30-3324-0, 300 pagini

 

  1. Shan-He Wu and Mihály Bencze: Selected Problems and Theorems of Analytic Inequalities, STUDIS Publishing House, Iasi, Romania, 2012, ISBN 978-606-624-124-3, 571 pagini

  

  1. Bencze Mihály si Virginia Marinescu: The International Conference-Mathematical Education in the Current European Context (3), STUDIS Publishing House, 2013, Iasi,Romania, 2013, ISBN 978-606-475-6, 383 paginiBencze Mihály and Virginia Marinescu: The International Conference-Mathematical Education in the Current European Context (4), Editura Studis, 2014, Iasi, Romania, ISSN 2360-5324, ISSN-L 2360-5324, 442 pagini

 

  1. Jose Luis Diaz Barrero and Mihály Bencze: Mathematical Contest Notebook, Studis Publishing House, Iasi, Romania, 2015, ISBN 978-606-624-740-5, 213 pagini

 

 

  1. Mihály Bencze and Ovidiu Pop: Selected Problems for Mathematical Contests, Studis Publishing House, Iasi, Romania, 2015, ISBN 978-606-624-861-7, 291 pagini

 

  1. Bencze Mihály and Virginia Marinescu: The International Conference-Mathematical Education in the Current European Context (5), Editura Studis, 2015, Iasi, 2015, ISSN 2360-5324, ISSN-L 2360-5324, 308 pagini

  

  1. Bencze Mihály and Virginia Marinescu: The International Conference-Mathematical Education in the Current European Context (6), Studis Kiadó, 2015, Iasi, 2015, ISSN 2360-5324, ISSN-L 2360-5324, 296 pagini

 

  1. Mihály Bencze and Marius Dragan: 600 Selected Problems for Mathematical Contest, EdituraStudis, 2017, Iasi, ISBN 978-606-775-616-6, 476 pagini

  

  1. Gamma (Mathematical Magazine) 1978-1989

 

  1. Octogon Mathematical Magazine 1993-2017

  

  1. Popescu M., Sitaru D.: ““Traian Lalescu” Constest. Geometry problems”, Lithography University of Craiova Publishing, Craiova, 1985

 

  1. Daniel Sitaru, Claudia Nănuţi: “National contest of applied mathematics - “Adolf Haimovici”- the county stage”, Ecko – Print Publishing, Drobeta Turnu Severin, 2011

 ISBN 978-606-8332-11-6

  

  1. Daniel Sitaru, Claudia Nănuţi: : “National contest of applied mathematics - “Adolf Haimovici”- the national stage”, Ecko – Print Publishing, Drobeta Turnu Severin, 2011

ISBN 978-606-8332-12-3

 

  1. Daniel Sitaru, Claudia Nănuți: “Contest problems”, Ecko – Print Publishing, Drobeta Turnu Severin, 2011

ISBN 978-606-8332-22-2

 

  1. Daniel Sitaru, Claudia Nănuţi: “Baccalaureate – Problems – Solutions –Topics -Scales”, Ecko – Print Publishing, Drobeta Turnu Severin, 2011,  ISBN 978-606-8332-18-5

  

  1. Daniel Sitaru: „Affine and euclidiane geometry problems”, Ecko – Print Publishing, Drobeta Turnu Severin, 2012

ISBN 978-606-8332-29-1

  

  1. Daniel Sitaru, Claudia Nănuţi, “Baccalaureate – Problems – Tests – Topics – 2010 – 2013”, Ecko – Print Publishing, Drobeta Turnu Severin, 2012    ISBN 978-606-8332-28-4

 

  1. Daniel Sitaru, “Hipercomplex and quaternionic geometry”, Ecko – Print Publishing, Drobeta Turnu Severin, 2013

ISBN 978-606-8332-36-9

  • Daniel Sitaru, Claudia Nănuţi: “Algebra Basis”, Ecko – Print Publishing, Drobeta Turnu Severin, 2013

ISBN 978-606-8332-45-1

  • Daniel Sitaru, Claudia Nănuţi: “Mathematical Lessons, Ecko – Print Publishing, Drobeta Turnu Severin, 2013

ISBN 978-606-8332-47-5

  • Daniel Sitaru, Claudia Nănuţi: “Basics of mathematical analysis”, Ecko – Print Publishing, Drobeta Turnu Severin, 2014
  1. Daniel Sitaru, Claudia Nănuți: “Mathematics Olympics”, Ecko – Print Publishing, Drobeta Turnu Severin, 2014
  2. Daniel Sitaru, Claudia Nănuți, Giugiuc Leonard, Diana Trăilescu -“Inequalities”, Ecko – Print Publishing, Drobeta Turnu Severin, 2015

ISBN 978-606-8332-59-8

  1. Radu Gologan, Daniel Sitaru, Leonard Giugiuc : “300 Romanian Mathematical Challenges”- Publishing House Paralela 45,

Pitesti-2016-ISBN 978-973-47-2270-9

  1. Daniel Sitaru :“Math Phenomenon”, Publishing House Paralela 45,

Pitesti-2016-ISBN 978-973-47-2271-6

  1. Daniel Sitaru : “Algebraic Phenomenon”, Publishing House Paralela 45, Pitesti-2017-ISBN 978-973-47-2523-6

 

  1. Daniel Sitaru :”Murray Klamkin’s Duality Principle for Triangle Inequalities”-The Pentagon Journal-Volume 75 NO 2-Spring 2016
  2. Daniel Sitaru, Claudia Nănuți: “Generating Inequalities using Schweitzer’s Theorem”-CRUX MATHEMATICORUM-Volume 42, NO1-January 2016
  3. Daniel Sitaru, Claudia Nănuți: “A “probabilistic” method for proving inequalities”-CRUX MATHEMATICORUM-Volume 43, NO7-September 2017

37.”Romanian Mathematical Magazine”-Interactive Journal-www.ssmrmh.ro