Pathfinder for Olympiad Mathematics by Vikas Tiwari and V Seshan for RMO INMO IMO Math Olympiad Foundation

Cover......Page 1
Copyright......Page 5
Brief Contents......Page 6
Contents......Page 8
Preface......Page 12
About the Authors......Page 13
Polynomial FuncTions......Page 14
Division in Polynomials......Page 15
Fundamental Theorem of Algebra......Page 16
Identity Theorem......Page 18
Rational Root Theorem......Page 20
Corollary (Integer Root Theorem)......Page 21
Vieta’s Relations......Page 22
Symmetric Functions......Page 29
Common Roots of Polynomial Equations......Page 35
Irreducibility of Polynomials......Page 37
Gauss Lemma......Page 39
Eisenstein’s Irreducibility Criterion Theorem......Page 40
Extended Eisenstein’s Irreducibility Criterion Theorem......Page 41
Solved Problems......Page 42
Check Your Understanding......Page 59
Challenge Your Understanding......Page 62
Applying a Function to Both Sides of an Inequality......Page 66
Weirstras’s InequalIty......Page 68
Triangular Inequalities......Page 69
Sum of Squares (SOS)......Page 71
Quadratic Inequality......Page 75
Derived Inequalities from AM ≥ GM ≥ HM......Page 76
Weighted Means......Page 87
Power Mean Inequality......Page 89
Rearrangement Inequality......Page 91
Chebyshev’s Inequality......Page 92
Cauchy–Schwarz Inequality......Page 94
Hölders Inequality......Page 98
The Erodos–Mordell Inequality......Page 100
Jensen’s InequalIty......Page 101
Solved Problems......Page 102
Check Your Understanding......Page 112
Challenge Your Understanding......Page 114
Proposition......Page 118
Problems of the Divisibility Type......Page 119
Problems Based on Summation of Series......Page 121
Problems Involving Inequations......Page 126
Use of Transitive Property......Page 127
Working Rule......Page 130
Solved Problems......Page 134
Check Your Understanding......Page 146
Challenge Your Understanding......Page 147
Classification......Page 150
First Order Linear Recurrence Relation......Page 152
First Order Linear, Non-homogeneouswith Constant Coefficients......Page 154
First Order Non-linear of the Form......Page 156
First Order Non-linear of the Form......Page 157
Linear Homogeneous Recurrence Relation with Constant Coefficient of Order ‘2’......Page 161
General Form of Linear Homogeneous Recurrence Relation with Constant Coefficients......Page 163
General Method For Non-Homogeneous Linear Equation......Page 164
A Special Case......Page 166
Solved Problems......Page 168
Check Your Understanding......Page 177
Challenge Your Understanding......Page 178
Some Properties of Function......Page 180
Intermediate Value Theorem......Page 181
Substitution of Variable/Function......Page 182
Isolation of Variables......Page 183
Evaluation of Function at Some Point of Domain......Page 184
Application of Properties of the Function......Page 186
Application of Mathematical Induction......Page 187
Method of Undetermined Coefficients......Page 188
Using Recurrence Relation......Page 189
Cauchy’s Functional Equation......Page 191
Equations Reducible to Cauchy’s Equations......Page 193
Using Fixed Points......Page 196
Solved Problems......Page 198
Check Your Understanding......Page 205
Challenge Your Understanding......Page 206
Properties of Divisibility......Page 208
Greatest Common Divisor (GCD)......Page 211
Properties of GCD......Page 212
Least Common Multiple......Page 213
Primes......Page 215
Euclidean Theorem......Page 216
Sophie Germain Identity......Page 218
Number of Positive Divisors of a Composite Number......Page 220
Perfect Numbers......Page 225
Properties of Congruence......Page 228
Complete Residue System (Modulo n)......Page 234
Carmichael Function......Page 235
Carmichael’s Theorem......Page 236
Chinese Remainder Theorem (CRT)......Page 237
Binomial Theorem......Page 238
Digit Sum Characteristic Theorem......Page 239
Scales of Notation......Page 242
Greatest Integer Function......Page 246
Properties of Greatest Integer Function......Page 247
Diophantine Equations......Page 252
Solved Problems......Page 266
Check Your Understanding......Page 276
Challenge Your Understanding......Page 279
Properties of Factorial......Page 284
Addition Principle......Page 285
Multiplication Principle......Page 286
Theorem......Page 296
Properties of nr; 0 ≤ r ≤ n; r, n ∈0......Page 297
Always Excluding p Particular Objects in the Selection......Page 299
Exactly or Atleast or Atmost Constraint in the Selection......Page 300
Selection of One or More Objects......Page 302
Selection of r Objects from n Objectswhen All n Objects are not Distinct......Page 306
Occurrence of Order in Selection......Page 308
Points of Intersection between Geometrical Figures......Page 309
Formation of Subsets......Page 314
The Bijection Principle......Page 316
Combinations with Repetitions Allowed......Page 317
Theorem 1......Page 322
Theorem 2......Page 324
Theorem 3......Page 327
Permutations of n Objects Taken r at a Time whenAll n Objects are not Distinct......Page 330
Theorem 4......Page 331
Always Excluding p Particular Objects in the Arrangement......Page 333
‘p’ Particular Objects Always Separated in the Arrangement......Page 334
Rank of a Word in the Dictionary......Page 336
Theorem......Page 340
Difference between Clockwise and Anti-clockwise......Page 341
Unequal Division and Distribution of Non-identical Objects......Page 347
Equal Division and Distribution of Non-identical objects......Page 348
Equal as well as Unequal Division andDistribution of Non-identical Objects......Page 349
Number of Non-negative Integral Solutionsof a Linear Equation......Page 352
Number of Integral Solutions of a Linear Equationin x1, x2, …, xr when xi, s are Constrained......Page 354
Binomial Theorem......Page 355
Application of Generating Function......Page 356
Application of Recurrence Relations......Page 361
Principle of Inclusion and Exclusion (PIE)......Page 364
A Special Case of PIE......Page 365
Derangement......Page 376
Distinguishable Balls and Distinguishable Cells......Page 381
Identical Balls and Distinguishable Cells......Page 382
Distinguishable Balls and Identical Cells......Page 384
Identical Balls and Identical Cells......Page 385
Dirichlet’s (Or Pigeon Hole) Principle (PHP)......Page 387
Solved Problems......Page 393
Check Your Understanding......Page 429
Challenge Your Understanding......Page 433
Corresponding Angles Postulate or CA Postulate......Page 438
Angle Sum Theorem......Page 439
Right Angle Hypotenuse Side (RHS) Congruence Postulate......Page 444
Theorem 3......Page 453
Theorem 4......Page 454
Ratio and Proportion Theorem (or Area Lemma)......Page 459
Converse of Mid-point Theorem......Page 463
Basic Proportionality Theorem (Thales’ Theorem)......Page 466
Converse of Basic Proportionality Theorem......Page 468
Internal Angle Bisector Theorem......Page 472
External Bisector Theorem......Page 473
Converse of External Angle Bisector Theorem......Page 474
AAA Similarity (Angle Angle Angle Similarity)......Page 475
Area Ratio Theorem for Similar Triangles......Page 476
Converse of Baudhayana(or Pythagoras) Theorem......Page 481
Apollonius Theorem......Page 482
Stewart’s Theorem......Page 483
Lemma......Page 485
Quadrilaterals......Page 492
Parallelogram......Page 493
Trapezium......Page 498
Kite......Page 499
Carnot’s Theorem......Page 503
Ceva’s Theorem......Page 504
Converse of Ceva’s Theorem......Page 505
Menelaus Theorem......Page 518
Converse of Menelaus Theorem......Page 519
Pappus Theorem......Page 525
Alternate Segment Theorem......Page 527
Intersecting Chords Theorem......Page 530
Theorem (Converse of Intersecting Chords Theorem)......Page 531
Radical Axis......Page 532
Radical Centre......Page 533
Common Tangents to Two Circles......Page 542
Length of Transverse Common......Page 543
Corollary......Page 547
Theorem......Page 548
Simson–Wallace Line......Page 550
Ptolemy’s Theorem......Page 551
Generalization of Ptolemy’s Theorem(for All Convex Quadrilateral)......Page 552
Pitot Theorem......Page 559
Converse of Pitot Theorem......Page 560
Sine Rule......Page 564
Cosine Formula......Page 569
Projection Formula......Page 570
Napier’s Analogy (Tangent’s Rule)......Page 573
Mollweide’s Formula......Page 574
Half Angle Formulae’s......Page 575
Heron’s Formula......Page 576
m-n Theorem......Page 577
Circumcircle and Circumcentre......Page 579
Bramhagupta's Theorem......Page 581
Incircle and Incentre......Page 582
Orthocentre......Page 584
Euler Line......Page 589
Nine Point Circle......Page 590
Escribed Circles of a Triangle......Page 592
Ex-central Triangle......Page 594
Theorem 1......Page 602
Theorem 2......Page 603
Regular Polygon......Page 604
Construction of Triangles......Page 606
Summary of the Various Possibilities......Page 609
Solved Problems......Page 612
Check Your Understanding......Page 646
Challenge Your Understanding......Page 650
Answer Keys......Page 660
Glossary of Notation......Page 675
Trigonometry......Page 676
Geometry......Page 678
Inequalities......Page 685
Algebra......Page 687
Number Theory......Page 689
Combinatorics......Page 692
Glossary of Recommended Books......Page 696
Logarithms Table......Page 697
Photo Credits......Page 699