Pathfinder for Olympiad Mathematics

700 Pages·2017·25.291 MB·English

by  Vikash Tiwari, V. Seshan

Pathfinder
for
OLYMPIAD
MATHEMATICS

Pathfinder
Vikash Tiwari | V. Seshan
for
This book has been prepared in line with the requirements of national and international Olympiad examinations. The
OLYMPIAD
questions are carefully chosen to suit the needs of Olympiad aspirants and to provide highest level of clarity for
mathematical concepts. This book also provides deep insights about the origin of important formulae and equations P
by eminent Mathematicians. The exercises are designed and graded from simple to difficult level to enable the a
students’ to build, check and challenge their understanding. t
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EXTENSIVE PEDAGOGY fi
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 Solved Problems  Check Your Understanding
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 Build-up Your Understanding  Challenge Your Understanding MATHEMATICS
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Vikash Tiwari has been teaching students for Mathematical Olympiads (Pre RMO, RMO, INMO, and IMOTC) f
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and other examinations like KVPY and JEE for the last 20 years. He is a renowned resource person in the field
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of Mathematics in India. He conducts Olympiad training camps for several organizations such as Kendriya

Vidyalaya Sangathan, Delhi Public School (DPS), etc. He has devoted himself to the service of mankind via the O
medium of mathematics and has come up with this “first of its kind” book, which is an inventory of all
essential concepts required to ace the Mathematics Olympiad at various levels. The students have found his L
methods of problem-solving and teaching to be both insightful and intriguing. He has been instrumental in
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the success of several medal-winning students who have made our country proud in various International
Mathematics Olympiads. M
V. Seshan is one of the key resource persons nominated by CBSE to provide Olympiad training across India. P
He is a popular teacher and retd. Principal and Director of Bhartiya Vidya Bhavan, Baroda Centre. He is well
I
known for his unique ability in teaching Mathematics with utmost conceptual clarity. With teaching
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experience spanning over 40 years, he has been instrumental in setting-up the Olympiad Centre in Tata
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Additional resources available at Tiwari V. Seshan
http://www.pearsoned.co.in/Olympiad/Mathematics
Seshan
Size:203x254mm Spine: 22mm ISBN: 9789332568723 Title Sub Title Edition Authors / Editors Name With CD Red Band Territory line URL Price mQuest
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Pathfinder for Olympiad
MATHEMATICS
Vikash Tiwari
V. Seshan
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Copyright © 2017 Pearson India Education Services Pvt. Ltd
Published by Pearson India Education Services Pvt. Ltd, CIN: U72200TN2005PTC057128,
formerly known as TutorVista Global Pvt. Ltd, licensee of Pearson Education in South Asia.
No part of this eBook may be used or reproduced in any manner whatsoever without
the publisher’s prior written consent.
This eBook may or may not include all assets that were part of the print version.
The publisher reserves the right to remove any material in this eBook at any time.
ISBN: 9789332568723
eISBN: 9789352862757
Head Office: 15th Floor, Tower-B, World Trade Tower, Plot No. 1, Block-C, Sector-16,
Noida 201 301,Uttar Pradesh, India.
Registered Office: 4th Floor, Software Block, Elnet Software City, TS-140, Block 2 & 9,
Rajiv Gandhi Salai, Taramani, Chennai 600 113, Tamil Nadu, India.
Fax: [phone], Phone: [phone]
Website: in.pearson.com, Email: [e-mail]
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Brief Contents
Preface                                                 xi
Acknowledgements                                          xii
About the Authors                                           xii
1. Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1
2. Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1
3. Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1
4. Recurrence Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1
5. Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1
6. Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1
7. Combinatorics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1
8. Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1
Answer Keys                                             AK1
Appendix                                              AP1
Logarithms Table                                          LT1
Photo Credits                                            PC1
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Contents
Preface xi
Acknowledgements xii
About the Authors xii
Chapter 1 PoLyNoMIaLS 1.1
Polynomial Functions 1.1
Division in Polynomials 1.2
Remainder Theorem and Factor Theorem 1.3
Fundamental Theorem of Algebra 1.3
Polynomial Equations 1.7
Vieta’s Relations 1.9
Symmetric Functions 1.16
Common Roots of Polynomial Equations 1.22
Irreducibility of Polynomials 1.24
Chapter 2 INEquaLITIES 2.1
Basic Rules 2.1
Weirstras’s Inequality 2.3
Modulus Inequalities 2.4
Sum of Squares (SOS) 2.6
Arithmetic Mean ≥ Geometric Mean ≥ Harmonic Mean 2.11
Weighted Means 2.22
Power Mean Inequality 2.24
Rearrangement Inequality 2.26
Chebyshev’s Inequality 2.27
Cauchy–Schwarz Inequality 2.29
Hölders Inequality 2.33
Some Geometrical Inequalities 2.35
Jensen’s Inequality 2.36
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viii Contents
Chapter 3 MaThEMaTICaL INduCTIoN 3.1
Introduction 3.1
First (or Weak) Principle of Mathematical Induction 3.2
Second (or Strong) Principle of Mathematical Induction 3.13
Chapter 4 RECuRRENCE RELaTIoN 4.1
Introduction 4.1
Classification 4.1
First Order Linear Recurrence Relation 4.3
First Order Non-linear 4.7
Linear Homogeneous Recurrence Relation
with Constant Coefficient of Order ‘2’ 4.12
General Form of Linear Homogeneous Recurrence Relation
with Constant Coefficients 4.14
General Method for Non-Homogeneous
Linear Equation 4.15
Chapter 5 FuNCTIoNaL EquaTIoNS 5.1
Function 5.1
Functional Equation 5.3
Chapter 6 NuMbER ThEoRy 6.1
Divisibility of Integers 6.1
Euclids Division Lemma 6.4
Greatest Common Divisor (GCD) 6.4
Primes 6.8
Fundamental Theorem of Arithmetic 6.13
Number of Positive Divisors of a Composite Number 6.13
Modular Arithematic 6.21
Complete Residue System (Modulo n) 6.27
Some Important Function/Theorem 6.28
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Contents ix
Scales of Notation 6.35
Greatest Integer Function 6.39
Diophantine Equations 6.45
Chapter 7 CoMbINaToRICS 7.1
Definition of Factorial 7.1
Basic Counting Principles 7.2
Combinations 7.13
The Bijection Principle 7.33
Combinations with Repetitions Allowed 7.34
Definition of Permutation (Arrangements) 7.39
Introduction to Circular Permutation 7.57
Division and Distribution of Non-identical Items in Fixed Size 7.64
Number of Integral Solutions 7.69
Binomial, Multinomial and Generating Function 7.72
Application of Recurrence Relations 7.78
Principle of Inclusion and Exclusion (PIE) 7.81
Derangement 7.93
Classical Occupancy Problems 7.98
Dirichlet’s (Or Pigeon Hole) Principle (PHP) 7.104
Chapter 8 GEoMETRy 8.1
Angle 8.1
Congruent Triangles 8.7
Triangle Inequality 8.16
Ratio and Proportion Theorem (or Area Lemma) 8.22
Mid-point Theorem 8.26
Basic Proportionality Theorem (Thales’ Theorem) 8.29
Similar Triangles 8.38
Baudhayana (Pythagoras) Theorem 8.44
Quadrilaterals 8.55
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