Advanced Problems in Mathematics: Preparing for University - cover image

Book Series


Stephen Siklos

Published On





  • English

Print Length

186 pages (186)


Paperback210 x 10 x 297 mm(8.268" x 0.4" x 11.693")


Paperback1028g (36.26oz)

OCLC Number



  • PB
  • YQM
  • PBK
  • PBM
  • PBT


  • MAT000000
  • MAT030000
  • MAT005000
  • MAT012000
  • MAT029000


  • Advanced mathematical problems
  • STEP examinations
  • undergraduate mathematics course
  • calculus
  • geometry
  • probability and statistics

    Advanced Problems in Mathematics

    Preparing for University

    This book is intended to help candidates prepare for entrance examinations in mathematics and scientific subjects, including STEP (Sixth Term Examination Paper). STEP is an examination used by Cambridge colleges as the basis for conditional offers. They are also used by Warwick University, and many other mathematics departments recommend that their applicants practice on the past papers even if they do not take the examination.
    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.

    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