Methods of Solving Nonstandard Problems PDF
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Ellina Grigorieva Methods of Solving Nonstandard Problems Methods of Solving Nonstandard Problems Ellina Grigorieva Methods of Solving Nonstandard Problems EllinaGrigorieva MathematicsandComputerScience TexasWoman’sUniversity Denton,TX,USA ISBN978-3-319-19886-6 ISBN978-3-319-19887-3 (eBook) DOI10.1007/978-3-319-19887-3 LibraryofCongressControlNumber:2015945585 Mathematics Subject Classification (2010): 00A07, 26A09, 26A15, 26A48, 26C10, 26D05, 26D07, 97H20,97H30,97G60,08A50,11B25,11A05,11B65,11C05,11D25,11D72,11J81,65Hxx,65H04, 65H05,65H10 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof 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 dissimilarmethodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthis 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 hereinorforanyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) To my beautiful daughter, Sasha And to my wonderful parents, Natali and Valery Grigoriev Your encouragement made this book possible And to my university mentor and scientific advisor academician, Stepanov Nikolay Fedorovich Without your help and brilliant mind my career as a scientist would not be successful! Preface Many universities offer problem-solving courses for students majoring in mathematics and mathematics teaching. However, I have noticed over the years that many graduates are afraid of solving complex problems and try to escape solvingchallengingmathproblemsiftheycan.Whyshouldwebeafraid?Ourfight or flight response either leads us to take up the struggle with the problems or to avoidmathematicswithapassion. Itisnotasecretthatwetendtolikethingsthatwearenaturallygoodat.Welove something if we have visible, continuous success. Such success comes only with hardwork.WhenIwasyoung,Imyselfhaddifficultiesingeometryclass,whichI describedinmybookMethodsofSolvingComplexGeometryProblems(Springer, 2013). I later went on to win various Math Olympiads and graduate from Lomonosov Moscow State University (MGU) summa cum laude, defend a Ph.D. inMathematicalandPhysicalSciences,andpublishover60papersinthefieldsof differentialequations,gametheory, economics,andoptimal control theory. Some initialsuccessinmathematicscanleadtogreatersuccessesovertime. AtMoscowState,attheendofeachsemesterthestudentshadtopassfouroral examsgivenbyrenownedprofessors.Theprofessorscouldaskanytrickyquestion onthetopicoftheexaminationticket.WhenIwaspreparingforsuchexamsandin order to get an “A,” I did not try to memorize all the definitions and proofs, but rather I tried to develop a “global” understanding of the subject. I thought of possible questions that an examiner could ask me and tried to predict the type of problemthatIcouldbeaskedtosolve.IdevelopedmyownwayoflearningandI wanttoshareitwithyou. For over 30 years, whenever I spotted an especially interesting or tricky prob- lem,Iaddedittomynotebooksalongwithmyoriginalsolutions.I’veaccumulated thousandsoftheseproblems.Iusethemeverydayinmyteachingandincludemany of them in this book. Before accepting an academic position at a university, I workedasateacheratUrsulineAcademyofDallasandusedmyproblem-solving techniquesinmystudents’collegepreparation.Iwaspleasedtoreceiveapprecia- tionlettersfromMITandHarvardwheresomeofmystudentswereadmitted. vii viii Preface Ifyouarestrugglingwithmath,thisbookisforyou.Mostmathbooksstartfrom theoreticalfacts,giveoneortwoexamples,andthenasetofproblems.Inthisbook, almost every statement is followed by problems. You are not just memorizing a theorem—youapplytheknowledgeimmediately.Uponseeingasimilarproblemin thehomeworksection,youwillbeabletorecognizeandsolveit. Although each section of the book can be studied independently, the book is constructed to reinforce patterns developed at stages throughout the book. This helps you see how math topics are connected. The book can be helpful for self- education,forpeoplewhowanttodowellinmathclasses,orforthosepreparingfor competitions.Thebookisalsomeantformathteachersandcollegeprofessorswho wouldliketouseitasanextraresourceintheirclassroom. What Is This Book About? This book will teach you about functions and how properties of functions can be usedtosolvenonstandardequations,systemsofequations,andinequalities.When wesay“nonstandard,”onecanthinkofavarietyofproblemsthatappearunusual, intractable, or complex. However, we can also say that nonstandard can indicate amethod thatisoppositetoastandardorcommonwayofthinking.Forexample, mostofthetimesweneedtofindthemaximalorminimalvalueofafunction,the standard method would be to use the derivative of a function. However, under certain conditions, maximum and minimum problems can be solved through knowledge of some properties, such as boundedness of functions, and perhaps with the application of known inequalities. Another example of a nonstandard problem would be a word problem whose solution is restricted to the integers or thatmaybereducedtoanonlinearsystemwithmorevariablesthanthenumberof the equations. A nonstandard problem is one that does not yield easily to direct solution.Thenonstandardmethodofproblemsolvingistheprocessofsynthesizing connections between seemingly disassociated areas of mathematics and selecting appropriate generalizations, so that known constraints coincide to yield the solu- tion.TheyaretheSudokupuzzlesofmathematics. Standard methods and relevant formulas make up the context for the problem sets and are presented in each chapter together with simple problems for illustra- tion.Basicknowledgeofsecondaryschoolmathematicsisassumed.Forexample, if a probleffiffimffiffi is to solve a quadratic equation x2(cid:2)7xþ2¼0, then its roots, p x ¼7(cid:3) 41, can be found using the well-known quadratic formula. What if I 1,2 2 change the problem a little bit and ask you nowto prove that for a new quadratic equation,x2þaxþ1(cid:2)b¼0withnaturalroots,thequantitya2þb2 cannotbea prime number? Would standard methods and the quadratic formula help to solve thisproblem? Preface ix The problem has two parameters a and b, and is restricted to the set of natural numbers.Hence,inordertosolvethisproblem,weneedtoknowmorethanjusta quadraticformula;weneedtohaveamethodthatwillprovideanotherconstraintof the solution. In this case, for example, Vieta’s formula and the knowledge of elementarynumbertheorymightbehelpful. Letmegiveyounowthefollowingproblem: Solvetheinequality ðx2þ2xþ2Þx (cid:4)1. In this problem, we do not have any parameters, but the problem is no less difficult than the previous one. The expression inside parentheses is a quadratic raised to the power of x. Would knowledge of solving standard quadratic or exponential inequalities help here? Do we need to do anything with the unit on the right-hand side? Did you hear anything about monotonic functions in the past thatmightbehelpful? Let’sconsider thesolutionofcubicequations.Many ofmystudentsknow that the Rational Zero Theorem or the Fundamental Theorem of Algebra might be of help.Somewhotookacourseonthehistoryofmathematicshaveheardaboutthe Cardanoformula.Theseformulasmaybeapplicable,buttheymaynotbeadequate tosolvethequestion. Many interesting Olympiad problems can be solved by using nonstandard and otherwisenonobviousapproaches.Forexample,whatwouldyoudoifIaskyouto findthevalueofaparameteraforwhichthecubicfunction fðxÞ¼x3(cid:2)3x(cid:2)ahas preciselytwox-intercepts?Whatconditiononawouldbenecessarysothatitwould haveoneorthreex-intercepts?Playingwithagraphingcalculatormightgiveyoua hint,butnocalculatorsareallowedintheMathematicsOlympiad. Knowledgeofnonstandardmethodsofproblemsolvingisimportantbecausewe developadeeperunderstandingofmathematicsfromtheseoddquestions.Mathe- matics is not a disjointed collection of topics but rather a unified whole. The connections between fields are what tie them together. The structures of these mathematicalfieldscanbelearnedinamorepowerfulwaybyunderstandingthese connectionsgainedbytheexplorationofthesenonstandardproblems. There is a method for developing solutions to nonstandard problems. After solving some especially interesting problem, look for a generalization and try to see an application of that method in the solution of other problems. Usually, a nonstandardproblemrequiresknowledgeofseveralaspectsofmathematicsandcan be solved only with the knowledge of some particular fact from a seemingly disassociatedfield. If students see an elegant solution but do not apply the approach to other problems, they will not remember it, just as nobody remembers phone numbers thesedays.However,ifateacherusesandreusesthesameapproachthroughoutthe entire curriculum, students will remember it and learn to value the beauty of the method.ThisiswhatIpracticeinmyteachingandsharewithyouinthisbook.
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