Mathematical Olympiad Challenges

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Springer Science & Business Media, 4 .. 2008 - 283 ˹
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Why Olympiads? Working mathematiciansoftentell us that results in the ?eld are achievedafter long experience and a deep familiarity with mathematical objects, that progress is made slowly and collectively, and that ?ashes of inspiration are mere punctuation in periods of sustained effort. TheOlympiadenvironment,incontrast,demandsarelativelybriefperiodofintense concentration,asksforquickinsightsonspeci?coccasions,andrequiresaconcentrated but isolated effort. Yet we have foundthat participantsin mathematicsOlympiadshave oftengoneontobecome?rst-classmathematiciansorscientistsandhaveattachedgreat signi?cance to their early Olympiad experiences. For many of these people, the Olympiad problem is an introduction, a glimpse into the world of mathematics not afforded by the usual classroom situation. A good Olympiad problem will capture in miniature the process of creating mathematics. Its all there: the period of immersion in the situation, the quiet examination of possible approaches, the pursuit of various paths to solution. There is the fruitless dead end, as well as the path that ends abruptly but offers new perspectives, leading eventually to the discoveryof a better route. Perhapsmost obviously,grapplingwith a goodproblem provides practice in dealing with the frustration of working at material that refuses to yield. If the solver is lucky, there will be the moment of insight that heralds the start of a successful solution. Like a well-crafted work of ?ction, a good Olympiad problem tells a story of mathematical creativity that captures a good part of the real experience and leaves the participant wanting still more. And this book gives us more.
 

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11 A Property of Equilateral Triangles
4
12 Cyclic Quadrilaterals
6
13 Power of a Point
10
14 Dissections of Polygonal Surfaces
15
15 Regular Polygons
20
16 Geometric Constructions and Transformations
25
17 Problems with Physical Flavor
27
18 Tetrahedra Inscribed in Parallelepipeds
29
12 Cyclic Quadrilaterals
110
13 Power of a Point
118
14 Dissections of Polygonal Surfaces
125
15 Regular Polygons
134
16 Geometric Constructions and Transformations
145
17 Problems with Physical Flavor
151
18 Tetrahedra Inscribed in Parallelepipeds
156
19 Telescopic Sums and Products in Trigonometry
160

19 Telescopic Sums and Products in Trigonometry
31
110 Trigonometric Substitutions
34
Algebra and Analysis
38
21 No Square Is Negative
40
22 Look at the Endpoints
42
23 Telescopic Sums and Products in Algebra
44
24 On an Algebraic Identity
48
25 Systems of Equations
50
26 Periodicity
55
27 The Abel Summation Formula
58
28 x+1x
62
29 Matrices
64
210 The Mean Value Theorem
66
Number Theory and Combinatorics
69
31 Arrange in Order
70
32 Squares and Cubes
71
33 Repunits
74
34 Digits of Numbers
76
35 Residues
79
36 Diophantine Equations with the Unknowns as Exponents
83
37 Numerical Functions
86
38 Invariants
90
39 Pell Equations
94
310 Prime Numbers and Binomial Coefficients
99
Solutions
102
Geometry and Trigonometry
103
11 A Property of Equilateral Triangles
106
110 Trigonometric Substitutions
165
Algebra and Analysis
171
21 No Square is Negative
172
22 Look at the Endpoints
176
23 Telescopic Sums and Products in Algebra
183
24 On an Algebraic Identity
188
25 Systems of Equations
190
26 Periodicity
197
27 The Abel Summation Formula
202
28 x+1x
209
29 Matrices
214
210 The Mean Value Theorem
217
Number Theory and Combinatorics
223
31 Arrange in Order
224
32 Squares and Cubes
227
33 Repunits
232
34 Digits of Numbers
235
35 Residues
242
36 Diophantine Equations with the Unknowns as Exponents
246
37 Numerical Functions
252
38 Invariants
260
39 Pell Equations
264
310 Prime Numbers and Binomial Coefficients
270
Definitions and Notation
277
A1 Glossary of Terms
278
A2 Glossary of Notation
282
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