Table Of Contentn
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Advanced Problems in Mathematics
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Preparing for University s
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Stephen Siklos w
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This new and expanded edi� on is intended to help candidates prepare for entrance
examina� ons in mathema� cs and scien� fi c subjects, including STEP (Sixth Term Examina� on
Paper). STEP is an examina� on used by Cambridge Colleges for condi� onal off ers in
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mathema� cs. They are also used by some other UK universi� es and many mathema� cs t
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departments recommend that their applicants prac� ce on the past papers even if they do p
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not take the examina� on.
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Advanced Problems in Mathemati cs bridges the gap between school and university S
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mathema� cs, and prepares students for an undergraduate mathema� cs course. The l
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ques� ons analysed in this book are all based on past STEP ques� ons and each ques� on s
is followed by a comment and a full solu� on. The comments direct the reader’s a� en� on
to key points and put the ques� on in its true mathema� cal context. The solu� ons point
students to the methodology required to address advanced mathema� cal problems cri� cally Advanced Problems in
and independently.
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This book is a must read for any student wishing to apply to scien� fi c subjects at university d
level and for anyone interested in advanced mathema� cs. v Mathematics
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Stephen Siklos was a fellow of Jesus College, Cambridge. He worked in d
Cambridge University’s Faculty of Mathemati cs, where his main area of research P
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was Einstein’s theory of general relati vity and where he acted as Admissions o
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Offi cer. He had a parti cular interest in the transiti on from school to university l Preparing for University
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mathemati cs and for over thirty years presided over the STEP examinati ons, m
used as a qualifi cati on for undergraduate admissions by some of the top UK university s
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As with all Open Book publica� ons, this en� re book is available to read for free on the t
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publisher’s website. Printed and digital edi� ons, together with supplementary digital material, e
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can also be found at www.openbookpublishers.com
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Cover image: Photograph by Logan Troxell on Unsplash at h� ps://unsplash.com/photos/DsQmBlbywJ8 s
Cover design: Anna Ga�
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ebook and OA edi� ons TEPHEN IKLOS
also available
OPEN
ACCESS
OBP
ADVANCED PROBLEMS IN
MATHEMATICS
Advanced Problems in Mathematics:
Preparing for University
Stephen Siklos
http://www.openbookpublishers.com
c 2019StephenSiklos.
(cid:31)
ThisworkislicensedunderaCreativeCommonsAttribution4.0Internationallicense(CCBY4.0). This
license allows you to share, copy, distribute and transmit the work; to adapt the work and to make
commercialuseoftheworkprovidingattributionismadetotheauthor(butnotinanywaythatsuggests
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StephenSiklos,AdvancedProblemsinMathematics: PreparingforUniversity. SecondEdition. Cambridge,
UK:OpenBookPublishers,2019,https://doi.org/10.11647/OBP.0181
FurtherdetailsaboutCCBYlicensesareavailableathttp://creativecommons.org/licenses/by/4.0/
Anydigitalmaterialandresourcesassociatedwiththisvolumecanbefoundat
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STEPquestionsreproducedbykindpermissionofCambridgeAssessmentGroupArchives.Pleasesend
[email protected].
ThisisthesecondeditionofthefirstvolumeintheOBPSeriesinMathematics:
ISSN2397-1126(Print)
ISSN2397-1134(Online)
ISBNPaperback: 9781783747764
ISBNDigital(PDF):9781783747771
DOI:10.11647/OBP.0181
Cover image: Photograph c Logan Troxell. Creative Commons Attribution International 4.0, CC BY.
(cid:31)
https://unsplash.com/photos/DsQmBlbywJ8.
Coverdesign: AnnaGatti.
Contents
Aboutthisbook ix
STEP 1
WorkedProblems 11
Workedproblem1 11
Workedproblem2 15
Problems 19
P1 Anintegerequation(1993PaperI) 19
P2 Partitionsof10and20(1997PaperI) 21
P3 Mathematicaldeduction(1994PaperI) 23
P4 Divisibility(1999PaperI) 25
P5 Themodulusfunction(1997PaperI) 27
P6 TheregularReuleauxheptagon(1987SpecimenPaperI) 29
P7 Chainofequations(1997PaperII) 31
P8 Trig. equations(1997PaperII) 33
P9 Integrationbysubstitution(1998PaperI) 35
P10 Trueorfalse(1998PaperI) 37
P11 Egyptianfractions(2000PaperII) 39
P12 Maximisingwithconstraints(1998PaperI) 41
P13 Binomialexpansion(1998PaperII) 43
P14 Sketchingsubsetsoftheplane(1999PaperI) 45
P15 Moresketchingsubsetsoftheplane(1995PaperI) 47
P16 Non-linearsimultaneousequations(1996PaperII) 49
P17 Inequalities(2001PaperI) 51
P18 Inequalitiesfromcubics(2001PaperI) 53
P19 Logarithms(2000PaperI) 55
P20 Cosmologicalmodels(2001PaperI) 57
P21 Meltingsnowballs(1991PaperI) 59
P22 Gregory’sseries(1991PaperII) 61
P23 Intersectionofellipses(2002PaperI) 63
P24 Sketchingxm(1(cid:0)x)n(2002PaperI) 65
vi AdvancedProblemsinMathematics
P25 Inequalitiesbyareaestimates(2002PaperI) 67
P26 Simultaneousintegralequations(2002PaperI) 69
P27 Relationbetweencoefficientsofquarticforrealroots(1997PaperIII) 71
P28 Fermatnumbers(2002PaperII) 73
P29 Telescopingseries(1998PaperII) 75
P30 Integersolutionsofcubics(1998PaperII) 77
P31 Theharmonicseries(1999PaperI) 79
P32 Integrationbysubstitution(1999PaperII) 81
P33 Morecurvesketching(1999PaperII) 83
P34 Trig. sum(1999PaperII) 85
P35 Rootsofacubicequation(1999PaperIII) 87
P36 Rootcounting(1999PaperIII) 89
P37 Irrationalityofe(1997PaperIII) 91
P38 Discontinuousintegrands(2000PaperI) 93
P39 Adifficultintegral(1996PaperII) 95
P40 Estimatingthevalueofanintegral(2000PaperI) 97
P41 Integratingthemodulusfunction(2000PaperI) 99
P42 Geometry(2015PaperII) 101
P43 Thetsubstitution(2000PaperII) 103
P44 Adifferential-differenceequation(1990PaperII) 105
P45 Lagrange’sidentity(1987PaperII) 107
P46 Bernoullipolynomials(1987PaperIII) 109
P47 Vectorgeometry(2000PaperII) 111
P48 Solvingaquartic(2000PaperIII) 113
P49 Areasandvolumes(1987PaperII) 115
P50 Morecurvesketching(2001PaperII) 117
P51 Sphericalloaf(2001PaperI) 119
P52 Snowploughing(1987SpecimenPaperIII) 121
P53 Tortoiseandhare(1999PaperI) 123
P54 Howdidthechickencrosstheroad? (1997PaperI) 125
P55 Hank’sgoldmine(1998PaperI) 127
P56 Achocolateorange(1987SpecimenPaperII) 129
P57 Lorryonbend(2002PaperI) 131
P58 Fielding(1998PaperII) 133
P59 Equilibriumofrodofnon-uniformdensity(2002PaperII) 135
P60 Newton’scradle(1999PaperII) 137
P61 Kinematicsofrotatingtarget(1999PaperII) 139
P62 Particleonwedge(1998PaperII) 141
P63 Sphereonstep(1997PaperII) 143
StephenSiklos vii
P64 Elasticbandoncylinder(2001PaperI) 145
P65 Aknock-outtournament(1987SpecimenPaperII) 147
P66 Harrythecalculatinghorse(1997PaperII) 149
P67 PINguessing(2002PaperI) 151
P68 Breakingplates(2001PaperI) 153
P69 Lottery(2001PaperII) 155
P70 Bodiesinthefridge(1987PaperII) 157
P71 Choosingkeys(2000PaperII) 159
P72 Commutingbytrain(2000PaperII) 161
P73 Collectingvoles(2000PaperII) 163
P74 Breakingastick(1999PaperII) 165
P75 Randomquadratics(1988PaperII) 167
Syllabus 169
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, future
mathematicsundergraduates.
(cid:15) ThemorespecificaimistohelpyoutoprepareforSTEPorotherexaminationsrequiredforuni-
versityentranceinmathematics. TofindoutmoreaboutSTEP,readthenextsection.
Itusedtobesaidthatmathematicsandcricketwerenotspectatorsports;thisisstilltrueofmathematics.
To progress as a mathematician, you have to strengthen your mathematical muscles. It is not enough
justtoreadbooksorattendlectures. 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 of how a trained mathematician would think when tackling them. This
approachismuchtoolong-windedtosustainfortheremainderofthebook, butitshouldhelpyouto
seewhatsortofquestionsyououghttobeaskingyourselfasyouworkonthelaterproblems.
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.
