Pathfinder for Olympiad Mathematics

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