Mathematical olympiad treasures [1 ed.] 0817643052, 9780817643058, 3764343052
Mathematical Olympiad Treasures
343 68 1MB
English Pages 244 Year 2004
Table of contents :
Front cover ......Page 1
Title page ......Page 3
Date-line ......Page 4
Contents ......Page 5
I Problems ......Page 7
1.1 An Algebraic Identity ......Page 9
1.2 Cauchy-Schwartz revisited ......Page 13
1.3 Easy ways through absolute values ......Page 17
1.4 Parameters ......Page 21
1.5 Take the conjugate! ......Page 23
1.6 Inequalities with convex functions ......Page 27
1.7 Induction at work ......Page 31
1.8 Roots and coefficients ......Page 34
2.1 Geometric inequalities ......Page 41
2.2 An interesting locus ......Page 45
2.3 Cyclic Quads ......Page 50
2.4 Equiangular polygons ......Page 57
2.5 More on equilateral triangles ......Page 61
2.6 The "carpets" theorem ......Page 66
2.7 Quadrilaterals with an inscribed circle ......Page 70
2.8 Dr. Trig learns complex numbers ......Page 75
3.1 Arrays of numbers ......Page 79
3.2 Functions defined on sets of points ......Page 82
3.3 Count twice! ......Page 86
3.4 Sequences of integers ......Page 88
3.5 Equations with infinitely many solutions ......Page 92
3.6 Equations with no solutions ......Page 94
3.7 Powers of 2 ......Page 96
3.8 Progressions ......Page 99
II Solutions ......Page 103
4.1 An Algebraic Identity ......Page 105
4.2 Cauchy-Schwartz revisited ......Page 111
4.3 Easy ways through absolute values ......Page 115
4.4 Parameters ......Page 120
4.5 Take the conjugate ......Page 124
4.6 Inequalities with convex functions ......Page 130
4.7 Induction at work - ......Page 133
4.8 Roots and coef cients ......Page 136
5.1 Geometric inequalities ......Page 143
5.2 An interesting locus ......Page 151
5.3 Cyclic Quads ......Page 159
5.4 Equiangular polygons ......Page 167
5.5 More on equilateral triangles ......Page 177
5.6 The "carpets" theorem ......Page 184
5.7 Quadrilaterals with an inscribed circle ......Page 190
5.8 Dr. Trig learns complex numbers ......Page 196
6.1 Arrays of numbers ......Page 201
6.2 Functions defined on sets of points ......Page 205
6.3 Count twice! ......Page 210
6.4 Sequences of integers ......Page 215
6.5 Equations with infinitely many solutions ......Page 218
6.6 Equations with no solutions ......Page 222
6.7 Powers of 2 ......Page 225
6.8 Progressions ......Page 233
Glossary ......Page 239
Index of notations ......Page 241
Index to the problems ......Page 243
Back cover ......Page 244
Front cover ......Page 1
Title page ......Page 3
Date-line ......Page 4
Contents ......Page 5
I Problems ......Page 7
1.1 An Algebraic Identity ......Page 9
1.2 Cauchy-Schwartz revisited ......Page 13
1.3 Easy ways through absolute values ......Page 17
1.4 Parameters ......Page 21
1.5 Take the conjugate! ......Page 23
1.6 Inequalities with convex functions ......Page 27
1.7 Induction at work ......Page 31
1.8 Roots and coefficients ......Page 34
2.1 Geometric inequalities ......Page 41
2.2 An interesting locus ......Page 45
2.3 Cyclic Quads ......Page 50
2.4 Equiangular polygons ......Page 57
2.5 More on equilateral triangles ......Page 61
2.6 The "carpets" theorem ......Page 66
2.7 Quadrilaterals with an inscribed circle ......Page 70
2.8 Dr. Trig learns complex numbers ......Page 75
3.1 Arrays of numbers ......Page 79
3.2 Functions defined on sets of points ......Page 82
3.3 Count twice! ......Page 86
3.4 Sequences of integers ......Page 88
3.5 Equations with infinitely many solutions ......Page 92
3.6 Equations with no solutions ......Page 94
3.7 Powers of 2 ......Page 96
3.8 Progressions ......Page 99
II Solutions ......Page 103
4.1 An Algebraic Identity ......Page 105
4.2 Cauchy-Schwartz revisited ......Page 111
4.3 Easy ways through absolute values ......Page 115
4.4 Parameters ......Page 120
4.5 Take the conjugate ......Page 124
4.6 Inequalities with convex functions ......Page 130
4.7 Induction at work - ......Page 133
4.8 Roots and coef cients ......Page 136
5.1 Geometric inequalities ......Page 143
5.2 An interesting locus ......Page 151
5.3 Cyclic Quads ......Page 159
5.4 Equiangular polygons ......Page 167
5.5 More on equilateral triangles ......Page 177
5.6 The "carpets" theorem ......Page 184
5.7 Quadrilaterals with an inscribed circle ......Page 190
5.8 Dr. Trig learns complex numbers ......Page 196
6.1 Arrays of numbers ......Page 201
6.2 Functions defined on sets of points ......Page 205
6.3 Count twice! ......Page 210
6.4 Sequences of integers ......Page 215
6.5 Equations with infinitely many solutions ......Page 218
6.6 Equations with no solutions ......Page 222
6.7 Powers of 2 ......Page 225
6.8 Progressions ......Page 233
Glossary ......Page 239
Index of notations ......Page 241
Index to the problems ......Page 243
Back cover ......Page 244
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