# Advanced Problems in Mathematics: Preparing for University

Stephen Siklos, Cambridge University

Pub Date: 2016

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

Publisher: Open Book Publishers

## Conditions of Use

Attribution

CC BY

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## Table of Contents

About this book

STEP

Worked Problems

Worked problem 1

Worked problem 2

Problems

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

Syllabus

## 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

### Author(s)

**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.