دانلود کتاب A Primer for Mathematics Competitions

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Cover\r......Page 1
Title: A Primer for Mathematics Competitions......Page 4
Copyright\r......Page 5
Contents......Page 6
Preface......Page 10
1 Geometry......Page 20
1.1 Brief reminder of basic geometry......Page 22
1.1.1 Geometry of straight lines......Page 23
1.1.2 Geometry of polygons......Page 25
1.1.3 Geometry of the fundamental polygon – the triangle......Page 27
1.1.4 Geometry of circles and circular arcs......Page 32
1.2 Advanced geometry of the triangle......Page 42
1.3 Advanced circle geometry......Page 64
1.4 Problems......Page 73
1.5 Solutions......Page 84
2 Algebraic inequalities and mathematical induction......Page 108
2.1 The method of induction......Page 109
2.2 Elementary inequalities......Page 118
2.3 Harder inequalities......Page 120
2.4 The discriminant of a quadratic expression......Page 129
2.5 The modulus function......Page 131
2.6 Problems......Page 137
2.7 Solutions......Page 139
3.1 Introduction......Page 144
3.2 Division algorithm and greatest common divisor......Page 145
3.3 Euclidean algorithm......Page 146
3.4.1 Finding a particular solution of ax + by = c......Page 147
3.4.2 Finding the general solution of ax + by = c......Page 149
3.5 Euclidean reduction, or ‘divide and conquer’......Page 150
3.6 Some simple nonlinear Diophantine equations......Page 154
3.7 Problems......Page 157
3.8 Solutions......Page 158
4 Number theory......Page 164
4.1 Divisibility, primes and factorization......Page 165
4.2 Tests for divisibility......Page 166
4.3 The congruence notation: finding remainders......Page 167
4.4 Residue classes......Page 171
4.5 Two useful theorems......Page 173
4.6 The number of zeros at the end of n!......Page 178
4.7 The Unique Factorization Theorem......Page 180
4.8 The Chinese Remainder Theorem......Page 181
4.9 Problems......Page 185
4.10 Solutions......Page 188
5 Trigonometry......Page 200
5.1 Angles and their measurement......Page 201
5.2 Trigonometric functions of acute angles......Page 205
5.3 Trigonometric functions of general angles......Page 208
5.4 Graphs of sine and cosine functions......Page 213
5.5.1 The Pythagorean set of identities......Page 216
5.5.2 Addition formulas......Page 217
5.5.3 Double angle formulas......Page 219
5.5.4 Product formulas......Page 220
5.6 Trigonometric equations......Page 221
5.7 Problems......Page 226
5.8 Solutions......Page 227
6 Sequences and series......Page 232
6.2 The summation notation......Page 233
6.3 Arithmetic Progressions......Page 235
6.4 Geometric Progressions......Page 237
6.5 Sum to infinity of a Geometric Progression......Page 239
6.6 Formulas for sums of squares and cubes......Page 241
6.7 Problems......Page 245
6.8 Solutions......Page 247
7 Binomial Theorem......Page 254
7.1 Pascal’s triangle......Page 255
7.2 A formula for the coefficients......Page 258
7.3 Some properties of Pascal’s triangle......Page 261
7.4 Problems......Page 263
7.5 Solutions......Page 265
8 Combinatorics (counting techniques)......Page 270
8.1 The fundamental principle of enumeration......Page 271
8.2 Factorial arithmetic......Page 273
8.3.2 The general partitioning formula......Page 275
8.3.3 The general permutation formula......Page 277
8.3.4 Circular permutations......Page 278
8.4 Combinations......Page 284
8.5 Derangements......Page 287
8.6 The exclusion–inclusion principle......Page 291
8.7 The pigeon-hole principle......Page 299
8.8 Problems......Page 302
8.9 Solutions......Page 304
9.1 Problems......Page 312
9.2 Solutions......Page 325
Further Training Resources......Page 356
Index......Page 358