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

This book has been prepared in line with the requirements of national and international Olympiad examinations. The 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 by eminent Mathematicians. The exercises are designed and graded from simple to difficult level to enable the Students' to build, check and challenge their understanding.Solved ProblemsBuild-up Your UnderstandingCheck Your UnderstandingChallenge Your Understanding 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 Brand New International Paper-back Edition Same as per description, **Economy edition, May have been printed in Asia with cover stating Not for sale in US. 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