Problem solving tactics : lessons from the Australian Mathematical Olympiad Committee training program

Problem Solving Tactics is a compilation of tricks and tactics useful in solving mathematical problems at the Olympiad level. More than 150 ideas are illustrated in the fields of number theory, geometry, algebra and combinatorics. With an informal style, clear diagrams and hundreds of practice problems, this book will be attractive to those aspiring to Olympiad training, mathematically able students and others interested in problem solving. The authors, all research mathematicians and past Australian IMO medallists, are members of the training team for the Australian Mathematical Olympiad Committee's School of Excellence. About this book......Page 6 Problems......Page 14 Converse......Page 18 If and only if......Page 19 Contrapositive......Page 20 Proof by contradiction......Page 21 Proof by induction......Page 22 Strong induction......Page 24 Proof by exhaustion (case bashing)......Page 25 Pigeonhole principle......Page 26 Advanced pigeonhole principle......Page 28 Extremal principle......Page 29 Telescoping......Page 31 Problems......Page 32 Fundamental theorem of arithmetic......Page 37 Dealing with digits......Page 38 Floor function......Page 39 Square roots and conjugates......Page 40 Powers of two......Page 41 Euclid's algorithm......Page 42 Integers base-n......Page 44 Construction problems......Page 45 Chinese remainder theorem......Page 47 From Fermat to Euler......Page 48 Existence of a generator......Page 50 Problems......Page 52 Monotonicity......Page 56 Bounding arguments......Page 57 Polynomial modulus......Page 58 Modular arithmetic......Page 59 Divisibility and gcds......Page 60 Reduction of variables......Page 61 Infinite descent......Page 62 Vieta jumping......Page 63 Cyclotomic recognition......Page 64 Plane geometry......Page 66 Problems......Page 67 Cyclic quadrilaterals......Page 72 One step at a time......Page 73 Triangle centres......Page 75 Constructions......Page 77 Extend to the circumcircle......Page 79 Reverse reconstruction......Page 81 Trigonometry......Page 82 Areas......Page 83 Relate to known diagrams......Page 85 Create beautiful pictures......Page 86 Important configurations in geometry......Page 88 Angle bisector and perpendicular bisector......Page 89 Pivot theorem......Page 90 Radical axis theorem......Page 91 Similar switch......Page 92 Radical axis bisects common tangent......Page 93 Perpendicularity......Page 94 Alternate segment switch......Page 95 Ratios for collinearity......Page 96 Points of contact of incircle and excircle......Page 97 Circumcircle, incentre and excentre......Page 98 Simson line......Page 99 Pascal's theorem......Page 100 Desargues' theorem......Page 101 Quadrilateral and incircle......Page 102 Nine-point circle......Page 103 Euler line......Page 104 Four lines and four circles......Page 105 Newton–Gauss line......Page 106 Alternative characterisation of symmedian......Page 107 Convex cyclic hexagon and diagonals......Page 108 Quadrilateral, triangles and incircles......Page 109 Median, inradius and chord of incircle......Page 110 Incentre and midpoints......Page 111 Incentre, excentre, midpoint and contact points......Page 112 Incentre and chord of incircle......Page 113 Harmonic quadrilateral......Page 114 Incentre and mixtilinear incircle......Page 115 Butterfly theorem......Page 116 Problems......Page 118 Collinear points......Page 121 Menelaus' theorem......Page 122 Concurrent lines......Page 123 Ceva's theorem......Page 124 Concyclic points......Page 126 Power of a point......Page 130 Radical axes......Page 131 Ellipses......Page 132 Pascal's theorem......Page 135 Transformation geometry......Page 138 Problems......Page 139 Rotations......Page 143 Dilations......Page 144 Spiral symmetries......Page 146 Affine transformations......Page 147 Problems......Page 150 Addition ideas......Page 153 Angles......Page 154 Multiplication ideas......Page 155 Similarity ideas......Page 157 Roots of unity......Page 159 Polynomials......Page 162 Problems......Page 163 Division algorithm......Page 167 Fundamental theorem of algebra......Page 168 Vieta's formulas......Page 169 Integer polynomials......Page 171 Algebraic trickery......Page 173 Irreducibility......Page 175 Polynomials modulo p (upstairs–downstairs)......Page 176 Lagrange interpolation......Page 179 Root focus......Page 180 Problems......Page 182 Cauchy's functional equation......Page 186 Guess and hope......Page 187 Substitutions......Page 188 Injective, surjective and bijective......Page 189 The associative trick......Page 191 Involutions......Page 192 Fixed points......Page 193 Somewhere versus everywhere......Page 195 Completely multiplicative functions......Page 196 Well-ordering of N+......Page 197 Problems......Page 200 Squares are non-negative......Page 203 Rearrangement inequality......Page 204 Cauchy–Schwarz inequality......Page 206 Power means inequality......Page 208 Jensen's inequality......Page 209 Substitutions......Page 210 Addition and multiplication of inequalities......Page 211 Expand and conquer......Page 213 Homogeneous inequalities......Page 214 Muirhead's inequality......Page 215 Weighted inequalities......Page 216 Problems......Page 220 Reflection principle......Page 223 Transformations......Page 226 Trigonometry......Page 227 Parametrisation......Page 228 Ptolemy's inequality......Page 229 Locus and tangency......Page 232 Isoperimetric inequalities......Page 234 Triangle formulas......Page 235 Problems......Page 238 Subtraction and division......Page 241 Binomial identities......Page 242 Bijections......Page 244 The supermarket principle......Page 245 Pigeonhole principle......Page 246 Principle of inclusion–exclusion......Page 247 Double counting......Page 248 Injections......Page 250 Recursion......Page 251 Double counting via tables......Page 253 Combinatorial reciprocal principle......Page 254 Graph theory......Page 256 Problems......Page 257 Degree......Page 260 Connected graphs, cycles and trees......Page 261 Pigeonhole principle......Page 263 Euler trails......Page 264 Paths......Page 266 Extremal principle......Page 267 Count and count again......Page 268 Planar graphs......Page 269 Polyhedra......Page 270 Graph theory and inequalities......Page 271 Problems......Page 274 Number invariants......Page 280 Parity......Page 281 Modular arithmetic invariants......Page 282 Colouring invariants......Page 283 Monovariants......Page 285 Invariants as cost......Page 286 Permutation parity......Page 287 Combinatorial games......Page 288 Position analysis......Page 289 Pairing strategies......Page 290 Strategy stealing......Page 291 Problems......Page 294 Extremal principle......Page 298 Perturbation......Page 300 Discrete intermediate value theorem......Page 301 Convex hull......Page 302 Euler's formula......Page 303 Pigeonhole principle......Page 304 Colouring......Page 305 How do complex numbers work?......Page 306 Function notation......Page 309 Directed angles......Page 310 Some useful triangle formulas......Page 311 Index......Page 312