Undergraduate Mathematics Competitions (1995–2016): Taras Shevchenko National University of Kyiv (Problem Books in Mathematics)

214 Pages·2017·2.089 MB·English

by  Volodymyr Brayman, Alexander Kukush

Problem Books in Mathematics
Volodymyr Brayman
Alexander Kukush
Undergraduate
Mathematics
Competitions
(1995–2016)
Taras Shevchenko National University
of Kyiv
Second Edition
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
(cid:129)
Undergraduate Mathematics
–
Competitions (1995 2016)
Taras Shevchenko National University
of Kyiv
Second Edition
123
VolodymyrBrayman Alexander Kukush
Department ofMathematical Analysis Department ofMathematical Analysis
Taras ShevchenkoNational University Taras ShevchenkoNational University
ofKyiv ofKyiv
Kyiv Kyiv
Ukraine Ukraine
ISSN 0941-3502 ISSN 2197-8506 (electronic)
Problem Booksin Mathematics
ISBN978-3-319-58672-4 ISBN978-3-319-58673-1 (eBook)
DOI 10.1007/978-3-319-58673-1
LibraryofCongressControlNumber:2017939622
1stedition:©PublishingHouse“KyivUniversity”2015
2ndedition:©SpringerInternationalPublishingAG2017
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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
ShevchenkoNationalUniversityofKyiv.Thecompetitionhashadalongtradition
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 mathemati-
ciansofotherinstitutionsindifferent countries.Theycoverawidevarietyofareas
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,manyofwhichareveryelegantandinterestingtoread.Thisisthesecond
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 uni-
versity 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,inparticular,wasawinneroftheCompetitioninitsearlyyears(in1977
and 1978).
April 2017 Volodymyr Nekrashevych
Professor of Mathematics at Texas
A&M University, College Station, TX, USA
Preface
TheMathematicsOlympiadforstudentsoftheMechanicsandMathematicsFaculty
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
SikorskyKyivPolytechnicInstitute”,FacultyofCyberneticsandFacultyofPhysics
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 stu-
dentstakepartcanbefoundonthestudents’pageofthiswebsiteofMechanicsand
Mathematics Faculty http://www.mechmat.univ.kiev.ua.
As a rule, first- and second-year undergraduates and third- and fourth-year
undergraduatestudentscompeteseparately.AlongthehistoryoftheOlympiad,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
contained7–10problems.Forfirst-andsecond-yearundergraduates,problemswere
includedforfieldssuchascalculus,algebra,numbertheory,geometry,anddiscrete
mathematics. Problem sets for third and fourth year undergraduates included
additional topics in measure theory, functional analysis, probability theory, com-
plexanalysis,differentialequations,etc.Solutionstoalltheproblemsdonotrelyon
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.Afewcomplicatedproblems,whichdemandgeneralmathematicalculture
andremarkableinventiveness,areincludedinbothversionsoftheassignment,and
this helps to compare the results of all the participants.
In1997–1999someoftheproblemswereborrowedfromPutnamCompetitions
[1,3,4].Almostalltheproblemsofthelast17yearsareoriginal.Theirauthorsare
lecturers,Ph.D.students,seniorstudents,andgraduatingstudentsoftheMechanics
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 notenough 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,andallwhosolveatleast2–3problemsbecomeprizewinnersorgetthe
letter of commendation. The jury of olympiad checks the works and gives a pre-
liminaryevaluation.Approximatelyoneweeklater,ananalysisofproblemsisheld,
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
famousexpertinmathematicalstatisticsandthetheoryofstochasticequations.For
a long time he led a circle in calculus for first- or second-year undergraduate
students(untilnowsuchcirclesworkatFacultyofMechanicsandMathematicsand
at Institute of Mathematics of the National Academy of Sciences of Ukraine).
Anatoliy Yakovych proposed numerous witty problems in calculus, measure the-
ory,andfunctional analysis.Forafewyears ajuryleader wasalsotheheadofthe
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 approxi-
mation 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.ViazovskawasawardedtheSalemPrizeforaconceptualbreakthroughin
the sphere packing problem. In 2013 Dr. A.V. Bondarenko was awarded the Vasil
Popov International Prize for outstanding achievements in approximation theory.
StateprizesofUkrainewereawarded:toProf.A.Ya.Dorogovtsevforamonograph
in stochastic analysis; D.Sc. in Physics and Mathematics V.V. Lyubashenko for a
cycleofpapersinalgebra;D.Sc.inPhysicsandMathematicsO.Yu.Teplinskyifor
papers in theory of dynamical systems. Candidate of Sciences in physics and
mathematicsA.V.Knyazyuk(1960–2013)wasafamousteacheroftheKyivNatural
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 Volodymyr Brayman
April 2017 Alexander Kukush