Pathfinder for Olympiad Mathematics

✍ Scribed by Vikash Tiwari, V. Seshan

• Publisher: Pearson Education
• Year: 2017
• Tongue: English
• Leaves: 700
• Category: Library

✦ Table of Contents

Cover
Copyright
Brief Contents
Contents
Preface
Acknowledgements
About the Authors
Chapter 1 Polynomials
Polynomial FuncTions
Division in Polynomials
Remainder Theorem and Factor Theorem
Remainder Theorem
Factor Theorem
Fundamental Theorem of Algebra
Identity Theorem
Polynomial Equations
Rational Root Theorem
Corollary (Integer Root Theorem)
Vieta’s Relations
Symmetric Functions
Common Roots of Polynomial Equations
Irreducibility of Polynomials
Gauss Lemma
Eisenstein’s Irreducibility Criterion Theorem
Extended Eisenstein’s Irreducibility Criterion Theorem
Solved Problems
Check Your Understanding
Challenge Your Understanding
Chapter 2 Inequalities
Basic rules
Transitivity
Addition and Subtraction
Multiplication and Division
Addition and Multiplication of Two Inequalities
Applying a Function to Both Sides of an Inequality
Weirstras’s InequalIty
Modulus Inequalities
Triangular Inequalities
Sum of Squares (SOS)
Quadratic Inequality
Arithmetic Mean ≥ Geometric Mean ≥ Harmonic Mean
Derived Inequalities from AM ≥ GM ≥ HM
Weighted Means
Power Mean Inequality
Rearrangement Inequality
Chebyshev’s Inequality
Cauchy–Schwarz Inequality
Hölders Inequality
Some Geometrical InequalItIes
Ptolemy’s Inequality
The Parallelogram Inequality
Torricelli’s (or Fermat’s) Point
The Erodos–Mordell Inequality
Leibniz’s Theorem
Jensen’s InequalIty
Solved Problems
Check Your Understanding
Challenge Your Understanding
Chapter 3 Mathematical Induction
Introduction
Proposition
First (or Weak) Principle of Mathematical Induction
Working Rule
Problems of the Divisibility Type
Problems Based on Summation of Series
Problems Involving Inequations
Use of Transitive Property
Second (or Strong) Principle of Mathematical Induction
Working Rule
Solved Problems
Check Your Understanding
Challenge Your Understanding
Chapter 4 Recurrence Relation
Introduction
Classification
First Order Linear Recurrence Relation
First Order Linear Homogeneous
First Order Linear, Non-homogeneouswith Constant Coefficients
First Order Non-linear
First Order Non-linear of the Form
First Order Non-linear of the Form
Linear Homogeneous Recurrence Relation with Constant Coefficient of Order ‘2’
General Form of Linear Homogeneous Recurrence Relation with Constant Coefficients
General Method For Non-Homogeneous Linear Equation
A Special Case
Solved Problems
Check Your Understanding
Challenge Your Understanding
Chapter 5 Functional Equations
Function
Some Properties of Function
Continuity of a Function
Intermediate Value Theorem
Functional Equation
Substitution of Variable/Function
Isolation of Variables
Evaluation of Function at Some Point of Domain
Application of Properties of the Function
Application of Mathematical Induction
Method of Undetermined Coefficients
Using Recurrence Relation
Cauchy’s Functional Equation
Equations Reducible to Cauchy’s Equations
Using Fixed Points
Solved Problems
Check Your Understanding
Challenge Your Understanding
Chapter 6 Number Theory
Divisibility of Integers
Properties of Divisibility
Euclids Division Lemma
Greatest Common Divisor (GCD)
Properties of GCD
Least Common Multiple
Primes
Euclidean Theorem
Sophie Germain Identity
Fundamental Theorem of Arithmetic
Number of Positive Divisors of a Composite Number
Perfect Numbers
Modular Arithematic
Properties of Congruence
Complete Residue System (Modulo n)
Reduced Residue System (Modulo n)
Properties
Some Important Function/theorem
Euler’s Totient Function
Carmichael Function
Fermat’s Little Theorem (FLT)
Euler’s Theorem
Carmichael’s Theorem
Wilson’s Theorem
Chinese Remainder Theorem (CRT)
Binomial Coefficient
Binomial Theorem
Digit Sum Characteristic Theorem
Scales of Notation
Greatest Integer Function
Properties of Greatest Integer Function
Diophantine Equations
Solved Problems
Check Your Understanding
Challenge Your Understanding
Chapter 7 Combinatorics
Definition of Factorial
Properties of Factorial
Basic Counting Principles
Addition Principle
Multiplication Principle
Combinations
Definition of Combination
Theorem
Properties of nr; 0 ≤ r ≤ n; r, n ∈0
Some Applications of Combinations
Always Including p Particular Objects in the Selection
Always Excluding p Particular Objects in the Selection
Exactly or Atleast or Atmost Constraint in the Selection
Selection of One or More Objects
Selection of r Objects from n Objectswhen All n Objects are not Distinct
Occurrence of Order in Selection
Points of Intersection between Geometrical Figures
Formation of Subsets
The Bijection Principle
Combinations with Repetitions Allowed
Definition of Permutation (Arrangements)
Theorem 1
Theorem 2
Theorem 3
Permutations of n Objects Taken r at a Time whenAll n Objects are not Distinct
Theorem 4
Some Miscellaneous Applications of Permutations
Always Including p Particular Objects in the Arrangement
Always Excluding p Particular Objects in the Arrangement
‘p’ Particular Objects Always Together in the Arrangement
‘p’ Particular Objects Always Separated in the Arrangement
Rank of a Word in the Dictionary
Introduction to Circular Permutation
Theorem
Difference between Clockwise and Anti-clockwise
Division and Distribution of Non-identicalItems in Fixed Size
Unequal Division and Distribution of Non-identical Objects
Equal Division and Distribution of Non-identical objects
Equal as well as Unequal Division andDistribution of Non-identical Objects
Number of Integral Solutions
Number of Non-negative Integral Solutionsof a Linear Equation
Number of Non-negative Integral Solutionsof a Linear Inequation
Number of Integral Solutions of a Linear Equationin x1, x2, …, xr when xi, s are Constrained
Binomial, Multinomial and Generating Function
Binomial Theorem
Binomial Theorem for Negative Integer Index
Multinomial Coefficients
Application of Generating Function
Application of Recurrence Relations
Principle of Inclusion and Exclusion (PIE)
A Special Case of PIE
Derangement
Classical Occupancy Problems
Distinguishable Balls and Distinguishable Cells
Identical Balls and Distinguishable Cells
Distinguishable Balls and Identical Cells
Identical Balls and Identical Cells
Dirichlet’s (Or Pigeon Hole) Principle (PHP)
Solved Problems
Check Your Understanding
Challenge Your Understanding
Chapter 8 Geometry
Angle
Complementary Angles
Supplementary Angles
Vertically Opposite Angles (VOA)
Corresponding Angles Postulate or CA Postulate
Alternate Interior Angles Theoremor AIA Theorem
Angle Sum Theorem
Congruent Triangles
Side Angle Side (SAS) Congruence Postulate
Angle Side Angle (ASA) Congruence Postulate
Angle Angle Side (AAS) Congruence Postulate
Side Side Side (SSS) Congruence Postulate
Right Angle Hypotenuse Side (RHS) Congruence Postulate
Triangle Inequality
Theorem 1
Theorem 2
Theorem 3
Theorem 4
Ratio and Proportion Theorem (or Area Lemma)
Mid-point Theorem
Converse of Mid-point Theorem
Basic Proportionality Theorem (Thales’ Theorem)
Converse of Basic Proportionality Theorem
Internal Angle Bisector Theorem
Converse of Internal Angle Bisector Theorem
External Bisector Theorem
Converse of External Angle Bisector Theorem
Similar Triangles
SSS Similarity (Side Side Side Similarity)
AAA Similarity (Angle Angle Angle Similarity)
SAS Similarity (Side Angle Side Similarity)
Area Ratio Theorem for Similar Triangles
Baudhayana (Pythagoras) Theorem
Converse of Baudhayana(or Pythagoras) Theorem
Acute Angled Triangle Theorem
Obtuse Angled Triangle Theorem
Apollonius Theorem
Stewart’s Theorem
Lemma
Quadrilaterals
Parallelogram
Rectangle
Rhombus
Square
Trapezium
Kite
Concurrency and Collinearity
Definitions
Theorem
Carnot’s Theorem
Ceva’s Theorem
Trigonometric Form of Ceva’s Theorem
Converse of Ceva’s Theorem
Menelaus Theorem
Converse of Menelaus Theorem
Pappus Theorem
Circles
Alternate Segment Theorem
The Power of a Point
Intersecting Chords Theorem
Tangent Secant Theorem
Theorem (Converse of Intersecting Chords Theorem)
Radical Axis
Radical Centre
Common Tangents to Two Circles
Centres of Similitude of Two Circles
Length of the Direct Common Tangents
Length of Transverse Common
Quadrilaterals (Cyclic and Tangential)
Cyclic Quadrilateral
Theorem
Corollary
Theorem
Simson–Wallace Line
Ptolemy’s Theorem
Generalization of Ptolemy’s Theorem(for All Convex Quadrilateral)
Tangential Quadrilateral
Pitot Theorem
Converse of Pitot Theorem
Application of Trigonometry in Geometry
Some Standard Notations
Sine Rule
Cosine Formula
Projection Formula
Napier’s Analogy (Tangent’s Rule)
Mollweide’s Formula
Half Angle Formulae’s
Area of Triangle
Heron’s Formula
m-n Theorem
Circles, Centres and the Triangle
Circumcircle and Circumcentre
Bramhagupta's Theorem
Incircle and Incentre
Orthocentre
Euler Line
Nine Point Circle
Escribed Circles of a Triangle
Ex-central Triangle
Area of a Quadrilaterals
Theorem 1
Theorem 2
Regular Polygon
Construction of Triangles
Summary of the Various Possibilities
Solved Problems
Check Your Understanding
Challenge Your Understanding
Answer Keys
Appendix Notations, Symbols and Definitions
Glossary of Notation
Glossary of Symbols
Glossary of Definitions
Trigonometry
Geometry
Inequalities
Algebra
Number Theory
Combinatorics
Glossary of Recommended Books
Logarithms Table
Photo Credits