Advanced Problems in Mathematics: Preparing for University

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Stephen Siklos, Cambridge University

Pub Date: 2016

ISBN 13: 978-1-7837414-4-1

Publisher: Open Book Publishers

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Table of Contents

About this book
Worked Problems
Worked problem 1
Worked problem 2
P1 An integer equation
P2 Partitions of 10 and 20
P3 Mathematical deduction
P4 Divisibility
P5 The modulus function
P6 The regular Reuleaux heptagon
P7 Chain of equations
P8 Trig. equations
P9 Integration by substitution
P10 True or false
P11 Egyptian fractions
P12 Maximising with constraints
P13 Binomial expansion
P14 Sketching subsets of the plane
P15 More sketching subsets of the plane
P16 Non-linear simultaneous equations
P17 Inequalities
P18 Inequalities from cubics
P19 Logarithms
P20 Cosmological models
P21 Melting snowballs
P22 Gregory’s series
P23 Intersection of ellipses
P24 Sketching x m ( 1 − x ) n
P25 Inequalities by area estimates
P26 Simultaneous integral equations
P27 Relation between coefficients of quartic for real roots
P28 Fermat numbers
P29 Telescoping series
P30 Integer solutions of cubics
P31 The harmonic series
P32 Integration by substitution
P33 More curve sketching
P34 Trig sum
P35 Roots of a cubic equation
P36 Root counting
P37 Irrationality of e
P38 Discontinuous integrands
P39 A difficult integral
P40 Estimating the value of an integral
P41 Integrating the modulus function
P42 Geometry
P43 The t substitution
P44 A differential-difference equation
P45 Lagrange’s identity
P46 Bernoulli polynomials
P47 Vector geometry
P48 Solving a quartic
P49 Areas and volumes
P50 More curve sketching
P51 Spherical loaf
P52 Snowploughing
P53 Tortoise and hare
P54 How did the chicken cross the road?
P55 Hank’s gold mine
P56 A chocolate orange
P57 Lorry on bend
P58 Fielding
P59 Equilibrium of rod of non-uniform density
P60 Newton’s cradle
P61 Kinematics of rotating target
P62 Particle on wedge
P63 Sphere on step
P64 Elastic band on cylinder
P65 A knock-out tournament
P66 Harry the calculating horse
P67 PIN guessing
P68 Breaking plates
P69 Lottery
P70 Bodies in the fridge
P71 Choosing keys
P72 Commuting by train
P73 Collecting voles
P74 Breaking a stick
P75 Random quadratics


About the Book

This book is intended to help students prepare for entrance examinations in mathematics and scientific subjects, including STEP (Sixth Term Examination Papers). STEP examinations are used by Cambridge colleges as the basis for conditional offers in mathematics and sometimes in other mathematics-related subjects. They are also used by Warwick University, and many other mathematics departments recommend that their applicants practice on past papers to become accustomed to university-style mathematics.

Advanced Problems in Mathematics is recommended as preparation for any undergraduate mathematics course, even for students who do not plan to take the Sixth Term Examination Paper. The questions analysed in this book are all based on recent STEP questions selected to address the syllabus for Papers I and II, which is the A-level core (i.e. C1 to C4) with a few additions. Each question is followed by a comment and a full solution. The comments direct the reader’s attention to key points and put the question in its true mathematical context. The solutions point students to the methodology required to address advanced mathematical problems critically and independently.

This book is a must read for any student wishing to apply to scientific subjects at university level and for anybody interested in advanced mathematics.

About the Contributors


Stephen Siklos lectures in the Faculty of Mathematics at Cambridge University. His main area of research is Einstein's theory of General Relativity. For many years, Dr. Siklos was the Admissions Officer for the Faculty of Mathematics, dealing with issues at the interface of school and university mathematics. He has presided over the STEP examinations, used as a qualification for undergraduate admissions by some of the top British university mathematics departments, for thirty years. Dr. Siklos lives in Cambridge with his partner, who is a geologist, and their two children. He very much likes playing tennis, listening to music and reading old examination papers.