Advanced Problems in Mathematics: Preparing for University

189 Pages·2016·1.932 MB·English

by  Stephen Siklos

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Advanced Problems in
Mathematics
Preparing for University
ADVANCED PROBLEMS IN
MATHEMATICS

Advanced Problems in Mathematics:
Preparing for University
Stephen Siklos
http://www.openbookpublishers.com
⃝c 2016StephenSiklos.
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license allows you to share, copy, distribute and transmit the work; to adapt the work and to make
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StephenSiklos,AdvancedProblemsinMathematics: PreparingforUniversity. Cambridge,UK:OpenBook
Publishers,2015. http://dx.doi.org/10.11647/OBP.0075
FurtherdetailsaboutCCBYlicensesareavailableathttp://creativecommons.org/licenses/by/4.0/
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http://www.openbookpublishers.com/isbn/9781783741427
STEPquestionsreproducedbykindpermissionofCambridgeAssessmentGroupArchives.
ThisisthefirstvolumeoftheOBPSeriesinMathematics:
ISSN2397-1126(Print)
ISSN2397-1134(Online)
ISBNPaperback: 9781783741427
ISBNDigital(PDF):9781783741441
ISBNDigitalebook(epub): 9781783741458
ISBNDigitalebook(mobi): 9781783741465
DOI:10.11647/OBP.0075
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Contents
Aboutthisbook ix
STEP 1
WorkedProblems 11
Workedproblem1 11
Workedproblem2 15
Problems 19
P1 Anintegerequation 19
P2 Partitionsof10and20 21
P3 Mathematicaldeduction 23
P4 Divisibility 25
P5 Themodulusfunction 27
P6 TheregularReuleauxheptagon 29
P7 Chainofequations 31
P8 Trig. equations 33
P9 Integrationbysubstitution 35
P10 Trueorfalse 37
P11 Egyptianfractions 39
P12 Maximisingwithconstraints 41
P13 Binomialexpansion 43
P14 Sketchingsubsetsoftheplane 45
P15 Moresketchingsubsetsoftheplane 47
P16 Non-linearsimultaneousequations 49
P17 Inequalities 51
P18 Inequalitiesfromcubics 53
P19 Logarithms 55
P20 Cosmologicalmodels 57
P21 Meltingsnowballs 59
P22 Gregory’sseries 61
P23 Intersectionofellipses 63
P24 Sketchingxm(1(cid:0)x)n 65
vi AdvancedProblemsinMathematics
P25 Inequalitiesbyareaestimates 67
P26 Simultaneousintegralequations 69
P27 Relationbetweencoefficientsofquarticforrealroots 71
P28 Fermatnumbers 73
P29 Telescopingseries 75
P30 Integersolutionsofcubics 77
P31 Theharmonicseries 79
P32 Integrationbysubstitution 81
P33 Morecurvesketching 83
P34 Trigsum 85
P35 Rootsofacubicequation 88
P36 Rootcounting 90
P37 Irrationalityofe 92
P38 Discontinuousintegrands 94
P39 Adifficultintegral 96
P40 Estimatingthevalueofanintegral 98
P41 Integratingthemodulusfunction 100
P42 Geometry 102
P43 Thetsubstitution 104
P44 Adifferential-differenceequation 106
P45 Lagrange’sidentity 108
P46 Bernoullipolynomials 110
P47 Vectorgeometry 112
P48 Solvingaquartic 114
P49 Areasandvolumes 116
P50 Morecurvesketching 118
P51 Sphericalloaf 120
P52 Snowploughing 122
P53 Tortoiseandhare 124
P54 Howdidthechickencrosstheroad? 126
P55 Hank’sgoldmine 128
P56 Achocolateorange 130
P57 Lorryonbend 132
P58 Fielding 134
P59 Equilibriumofrodofnon-uniformdensity 136
P60 Newton’scradle 138
P61 Kinematicsofrotatingtarget 140
P62 Particleonwedge 142
P63 Sphereonstep 144
StephenSiklos vii
P64 Elasticbandoncylinder 146
P65 Aknock-outtournament 148
P66 Harrythecalculatinghorse 150
P67 PINguessing 152
P68 Breakingplates 154
P69 Lottery 156
P70 Bodiesinthefridge 158
P71 Choosingkeys 160
P72 Commutingbytrain 162
P73 Collectingvoles 164
P74 Breakingastick 166
P75 Randomquadratics 168
Syllabus 170

About this book
Thisbookhastwoaims.
(cid:15) Thegeneralaimistohelpbridgethegapbetweenschoolanduniversitymathematics.
Youmightwonderwhysuchagapexists. Thereasonisthatmathematicsistaughtatschoolfor
variouspurposes: toimprovenumeracy;tohoneproblem-solvingskills;asaserviceforstudents
goingontostudysubjectsthatrequiresomemathematicalskills(economics,biology,engineering,
chemistry—thelistislong);and,finally,toprovideafoundationforthesmallnumberofstudents
who will continue to a specialist mathematics degree. It is a very rare school that can achieve
all this, and almost inevitably the course is least successful for its smallest constituency, the real
mathematicians.
(cid:15) ThemorespecificaimistohelpyoutoprepareforSTEPorotherexaminationsrequiredforuni-
versityentranceinmathematics. TofindoutmoreaboutSTEP,readthenextsection.
Itusedtobesaidthatmathematicsandcricketwerenotspectatorsports; andthisisstilltrueofmath-
ematics. Toprogressasamathematician,youhavetostrengthenyourmathematicalmuscles. Itisnot
enoughjusttoreadbooksorattendlectures. Youhavetoworkonproblemsyourself.
Onewayofachievingthefirstoftheaimssetoutaboveistoworkonthesecond,andthatishowthis
bookisstructured. Itconsistsalmostentirelyofproblemsforyoutoworkon.
The problems are all based on STEP questions. I chose the questions either because they are ‘nice’ —
in the sense that you should get a lot of pleasure from tackling them (I did), or because I felt I had
somethinginterestingtosayaboutthem.
The first two problems (the ‘worked problems’) are in a stream of consciousness format. They are in-
tended to give you an idea how a trained mathematician would think when tackling them. This ap-
proachismuchtoolong-windedtosustainfortheremainderofthebook,butitshouldhelpyoutosee
whatsortofquestionsyoushouldbeaskingyourselfasyouworkonthelaterproblems.
Each subsequent problem occupies two pages. On the first page is the STEP question, followed by a
comment.Thecommentsmaycontainhints,theymaydirectyourattentiontokeypoints,andtheymay
includemoregeneraldiscussions. Onthenextpageisasolution;youhavetoturnover,sothatyoureye
cannotaccidentallyfallonakeylineofworking. Thesolutionsgiveenoughworkingforyoutobeable
toreadthemthroughandpickupatleastthegistofthemethod;theymaynotgiveallthedetailsofthe
calculations. Foreachproblem, thegivensolutionisofcoursejustonewayofproducingtherequired
result: theremaybemanyotherequallygoodorbetterways. Finally,ifthereisspaceonthepageafter
the solution (which is sometimes not the case, especially if diagrams have to be fitted in), there is a
postmortem. Thepostmortemsmayindicatewhataspectsofthesolutionyoushouldbereviewingand
theymaytellyouabouttheideasbehindtheproblems.
Ihopethatyouwillusethecommentsandsolutionsasspringboardsratherthanfeatherbeds. Youwill
onlyreallybenefitfromthisbookifyouhaveagoodgoateachproblembeforelookingatthecomment
and certainly before looking at the solution. The problems are chosen so that there is something for
youtolearnfromeachone,andthiswillbelosttoyouforeverifyousimplyreadthesolutionwithout
thinkingabouttheproblemonyourown.