Problem Books in Mathematics Series editor: Peter Winkler Department of Mathematics Dartmouth College Hanover, NH 03755 USA More information about this series at http://www.springer.com/series/714
Volodymyr Brayman Alexander Kukush Undergraduate Mathematics Competitions (1995 2016) Taras Shevchenko National University of Kyiv Second Edition 123
Volodymyr Brayman Department of Mathematical Analysis Taras Shevchenko National University of Kyiv Kyiv Ukraine Alexander Kukush Department of Mathematical Analysis Taras Shevchenko National University of Kyiv Kyiv Ukraine ISSN 0941-3502 ISSN 2197-8506 (electronic) Problem Books in Mathematics ISBN 978-3-319-58672-4 ISBN 978-3-319-58673-1 (ebook) DOI 10.1007/978-3-319-58673-1 Library of Congress Control Number: 2017939622 1st edition: Publishing House Kyiv University 2015 2nd edition: Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
To our Teachers Anatoliy Dorogovtsev and Myhailo Yadrenko
Foreword The book contains the problems from the last 22 years of the Undergraduate Mathematics Competition at the Mechanics and Mathematics Faculty of Taras Shevchenko National University of Kyiv. The competition has had a long tradition going back to the 1970s. It eventually became a popular competition open to students from other colleges and universities. In the last couple of decades the winners of the competition have participated in the International Mathematical Competition for university students. The Undergraduate Mathematics Competition has provided a good training and selection venue from of the Taras Shevchenko University for composing a successful team for the IMC. The author of this Foreword also participated in the Competition when he was a student. It was a useful and interesting experience, which was very much appreciated. The problems in this collection are all original, and were mostly written by mathematicians from Kyiv University, but some were also written by mathematicians of other institutions in different countries. They cover a wide variety of areas of mathematics: calculus, algebra, combinatorics, functional analysis, etc. I would especially note that there are many interesting problems in probability theory. Problems are non-standard and solving them requires ingenuity and a deep understanding of the material. The book also contains the original solutions to the problems, many of which are very elegant and interesting to read. This is the second edition of the collection (the first was published in Ukrainian). I am sure that this book will be useful to students and professors as a source of interesting problems for competitions, for training, or even as a collection of harder problems for university courses. The authors of the book, Volodymyr Brayman and Alexander Kukush, are longtime organizers of the Competition. They are professors at the Department of Mathematical Analysis of the Mechanics and Mathematics Faculty of Taras Shevchenko National University of Kyiv, and are active in popularizing mathematics in Ukraine through mathematical olympiads, journals, and books. vii
viii Foreword They both were winners of the Undergraduate Mathematics Competition. A. Kukush, in particular, was a winner of the Competition in its early years (in 1977 and 1978). April 2017 Volodymyr Nekrashevych Professor of Mathematics at Texas A&M University, College Station, TX, USA
Preface The Mathematics Olympiad for students of the Mechanics and Mathematics Faculty has been organized at Taras Shevchenko National University of Kyiv since 1974. After a while the competition opened up to qualified students from any higher school of Kyiv and beginning in 2004, it became a nice tradition to invite the strongest mathematics students of leading Kyiv high schools to participate. Since then representatives of Ukrainian Physics and Mathematics Lyceum, Liceum No. 171 Leader, Liceum Naukova Zmina, Liceum No. 208, and Rusanivky Liceum have repeatedly become prize winners of the Olympiad. Most of the Olympiad winners are students of the Mechanics and Mathematics Faculty, but students from the following departments or institutions have also performed successfully: Institute of Physics and Technology and Institute of Applied System Analysis of National Technical University of Ukraine Igor Sikorsky Kyiv Polytechnic Institute, Faculty of Cybernetics and Faculty of Physics of Taras Shevchenko National University of Kyiv, National Pedagogical Dragomanov University, and National University of Kyiv-Mohyla Academy. Results of the Olympiad are taken into account when forming teams of All-Ukrainian students Mathematics Olympiad, International Mathematics Competition for University Students (IMC) and other student competitions. Materials and results of many mathematics competitions in which Ukrainian students take part can be found on the students page of this website of Mechanics and Mathematics Faculty http://www.mechmat.univ.kiev.ua. As a rule, first- and second-year undergraduates and third- and fourth-year undergraduate students compete separately. Along the history of the Olympiad, the number of problems distributed has changed several times. Most recently, the jury of Olympiad composed two sets of problems one for first- and second-year undergraduates and the second set for senior undergraduate students. Each set contained 7 10 problems. For first-and second-year undergraduates, problems were included for fields such as calculus, algebra, number theory, geometry, and discrete mathematics. Problem sets for third and fourth year undergraduates included additional topics in measure theory, functional analysis, probability theory, complex analysis, differential equations, etc. Solutions to all the problems do not rely on ix
x Preface statements out of curriculum of obligatory courses studied at Mechanics and Mathematics Faculty, but the solutions demand creative usage of obtained knowledge. Most of the problems are not technical and admit a short and elegant solution. A few complicated problems, which demand general mathematical culture and remarkable inventiveness, are included in both versions of the assignment, and this helps to compare the results of all the participants. In 1997 1999 some of the problems were borrowed from Putnam Competitions [1, 3, 4]. Almost all the problems of the last 17 years are original. Their authors are lecturers, Ph.D. students, senior students, and graduating students of the Mechanics and Mathematics Faculty, as well as colleagues from Belgium, Canada, Great Britain, Hungary, and the USA. Since 2003 participants obtain an assignment, where the author s name is indicated beside the corresponding problem. The competition lasts for 3 hours. Of course, this time interval is not enough to solve all the problems, and therefore, a participant can focus first of all on the problems, which are the most interesting for him/her. Typically, almost all the problems are solved by some of participants; a winner solves more than half of problems, and all who solve at least 2 3 problems become prize winners or get the letter of commendation. The jury of olympiad checks the works and gives a preliminary evaluation. Approximately one week later, an analysis of problems is held, appeal, and winners are awarded. For many years, until 1995, the jury leader was also the head of Mathematical Analysis Department, Prof. Anatoliy Yakovych Dorogovtsev (1935 2004), a famous expert in mathematical statistics and the theory of stochastic equations. For a long time he led a circle in calculus for first- or second-year undergraduate students (until now such circles work at Faculty of Mechanics and Mathematics and at Institute of Mathematics of the National Academy of Sciences of Ukraine). Anatoliy Yakovych proposed numerous witty problems in calculus, measure theory, and functional analysis. For a few years a jury leader was also the head of the Probability Theory and Mathematical Statistics Department as well as a Corresponding Member of the NAS of Ukraine, Myhailo Yosypovych Yadrenko (1932 2004). Myhailo Yosypovych was an outstanding expert in the theory of random fields and had authored many clever problems in probability theory and discrete mathematics. In particular years, the organizers of Olympiad were a Corresponding Member of the NAS of Ukraine Volodymyr Vladyslavovych Anisimov, lecturers Oleksiy Yuriyovych Konstantinov, Volodymyr Stepanovych Mazorchuk, and Volodymyr Volodymyrovych Nekrashevych. From 1999 until now, the permanent jury leader has also been the head of Mathematical Analysis Department, Prof. Igor Oleksandrovych Shevchuk, a famous expert in approximation theory. Members of jury for the last Olympiads were Andriy Bondarenko, Volodymyr Brayman, Alexander Kukush, Yevgen Makedonskyi, Dmytro Mitin, Oleksiy Nesterenko, Vadym Radchenko, Oleksiy Rudenko, Vitaliy Senin, Sergiy Shklyar, Sergiy Slobodyanyuk, and Yaroslav Zhurba.
Preface xi There are several famous mathematicians among the former winners of the Olympiad of Mechanics and Mathematics Faculty. In particular, Prof. O.G. Reznikov (1960 2003) used powerful methods of calculus in problems of modern geometry and was a member of London Mathematical Society. In 2016 Dr. M.S. Viazovska was awarded the Salem Prize for a conceptual breakthrough in the sphere packing problem. In 2013 Dr. A.V. Bondarenko was awarded the Vasil Popov International Prize for outstanding achievements in approximation theory. State prizes of Ukraine were awarded: to Prof. A.Ya. Dorogovtsev for a monograph in stochastic analysis; D.Sc. in Physics and Mathematics V.V. Lyubashenko for a cycle of papers in algebra; D.Sc. in Physics and Mathematics O.Yu. Teplinskyi for papers in theory of dynamical systems. Candidate of Sciences in physics and mathematics A.V. Knyazyuk (1960 2013) was a famous teacher of the Kyiv Natural Science Luceum No. 145. We mention also Professors I.M. Burban, O.Yu. Daletskyi, P.I. Etingof, M.V. Kartashov, Yu. G. Kondratyev, K.A. Kopotun, A.G. Kukush, O.M. Kulik, V.S. Mazorchuk, Yu. S. Mishura, V.V. Nekrashevych, A.Yu. Pylypenko, V.M. Radchenko, V.G. Samoylenko, G.M. Shevchenko, and B.L. Tsyagan. We apologize if we have forgotten anybody. The first part of the book contains all the problems of Olympiads dated 1995 2016. We hope that you will enjoy both self-reliant problem solving and an acquaintance with the solutions presented in the second part of the book. Some problems from earlier Olympiads can be found in the articles [2, 5, 6]. The authors are sincerely grateful to Dmytro Mitin for his long-lived fruitful cooperation, and also to Danylo Radchenko and Oleksandr Tolesnikov for useful discussions. Kyiv, Ukraine April 2017 Volodymyr Brayman Alexander Kukush
Contents Part I Problems 1995... 3 1996... 7 1997... 9 1998... 11 1999... 13 2000... 15 2001... 19 2002... 21 2003... 25 2004... 27 2005... 31 2006... 35 2007... 39 2008... 43 2009... 47 2010... 51 2011... 55 2012... 59 2013... 63 xiii
xiv Contents 2014... 65 2015... 67 2016... 69 Part II Solutions 1995... 75 1996... 83 1997... 89 1998... 95 1999... 103 2000... 107 2001... 113 2002... 119 2003... 125 2004... 133 2005... 139 2006... 145 2007... 153 2008... 161 2009... 169 2010... 175 2011... 183 2012... 189 2013... 199 2014... 205 2015... 211 2016... 217 References... 225 Thematic Index... 227