42. 4.2 2nd order linear recurrence relation with constant coefficients . . . . . . . . 42.
Raffles Institution - Singapore Mathematics Olympiad Training. Page 1 ...
Raffles Institution
Singapore Mathematics Olympiad Training Department of Mathematics
2010
Contents 1
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Problem Solving - Generalities, Basic Techniques 1.1 Introduction . . . . . . . . . . . . . . . . . . 1.2 Getting Started . . . . . . . . . . . . . . . . 1.3 Methods of Argument . . . . . . . . . . . . . 1.3.1 Deduction and Symbolic Logic . . . 1.3.2 Argument by Contradiction . . . . . 1.3.3 Mathematical Induction . . . . . . . 1.4 Other important strategies . . . . . . . . . . . 1.5 Problems . . . . . . . . . . . . . . . . . . .
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11 11 12 15 15 15 16 18 19
Fundamental Tactics for Solving Problems 2.1 Symmetry . . . . . . . . . . . . . . . . 2.1.1 Geometric Symmetry . . . . . . 2.1.2 Algebraic Symmetry . . . . . . 2.1.3 Symmetry in Polynomials . . . 2.2 The Extreme Principle . . . . . . . . . 2.3 The Pigeonhole Principle . . . . . . . . 2.4 Invariants . . . . . . . . . . . . . . . . 2.5 Problems . . . . . . . . . . . . . . . .
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20 20 21 23 24 25 25 27 28
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30 30 31 33 33 34 35 36 38 39
Sequences 4.1 1st order linear recurrence relation . . . . . . . . . . . . . . . . . . . . . . 4.2 2nd order linear recurrence relation with constant coefficients . . . . . . . .
41 42 42
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Functions and Basic Algebra 3.1 Functions and their inverses . . . . . . . . . . 3.2 Basic Characteristics of Functions . . . . . . 3.3 Common Classes of Functions . . . . . . . . 3.3.1 The Modulus Function . . . . . . . . 3.3.2 Polynomial Functions . . . . . . . . 3.3.3 Trigonometric Functions . . . . . . . 3.3.4 Inverse Trigonometric Functions . . . 3.4 Integer and Fractional Parts of a Real Number 3.5 Miscellaneous Problems . . . . . . . . . . .
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4.3
Miscellaneous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Series 5.1 Trigonometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Past Year SMO Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .
48 48 49
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Inequalities - Techniques and the Standard Few 6.1 Overview of some ideas . . . . . . . . . . . . . . 6.1.1 When does equality occur? . . . . . . . . 6.1.2 Change of variables . . . . . . . . . . . . 6.1.3 Create symmetry . . . . . . . . . . . . . 6.2 Toolbox - The Standard Few . . . . . . . . . . . 6.2.1 Triangle Inequality . . . . . . . . . . . . 6.2.2 A square is always positive! . . . . . . . 6.2.3 The Power Means . . . . . . . . . . . . 6.2.4 Cauchy-Schwarz and Holder inequalities 6.2.5 Rearrangements, Chebyshev . . . . . . . 6.3 Smoothing, convexity and Jensen’s inequality . . 6.4 Tangent Lines . . . . . . . . . . . . . . . . . . . 6.5 Isolated Fudging . . . . . . . . . . . . . . . . . 6.6 Past Year SMO Questions . . . . . . . . . . . . .
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53 53 53 53 54 54 54 55 55 56 57 58 61 61 62
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Inequalities - The Less Standard Few 7.1 Schur’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Majorization and Muirhead . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 n − 1 Equal Value Principle . . . . . . . . . . . . . . . . . . . . . . . . . .
68 68 69 70
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Inequalities - The Ugly Few 8.1 Calculus . . . . . . . . . . . . 8.1.1 Partial Derivatives . . 8.1.2 Maxima and Minima . 8.2 Lagrange Multipliers . . . . . 8.3 Local versus Global Extremum
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72 72 72 73 73 74
Polynomials 9.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Miscellaneous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .
76 76 77
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10 Functional Equations 10.1 Generalities . . . . . . . . . . . . . . 10.2 Some initial advice . . . . . . . . . . 10.3 Functional equations over N, Z and Q 10.4 Other advice and methods . . . . . . 10.5 Miscellaenous Problems . . . . . . .
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80 80 81 81 82 83
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11 Addition and Multiplication; Permutations and Combinations 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Addition and Multiplication Principle . . . . . . . . . . . . 11.3 Permutations and Combinations . . . . . . . . . . . . . . . 11.4 Miscellaneous Problems . . . . . . . . . . . . . . . . . . .
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87 87 87 90 93
12 Bijection Principle and Examples 12.1 Bijection Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96 96 99
13 Recursions 13.1 Solving recursive relations . . . . . . . . . . 13.1.1 First order linear recursive relations . 13.1.2 Second order linear recursive relations 13.2 Formulating recursive relations . . . . . . . . 13.3 Using a graph to set-up recursive relations . . 13.4 Problems . . . . . . . . . . . . . . . . . . .
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100 100 100 101 102 105 106
14 Principle of Inclusion and Exclusion 14.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
108 108 111
15 Counting in Two Ways: Fubini’s Principle 15.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
112 112 116
16 Generating Functions 16.1 Ordinary Generating Functions . . . . . . . . . . . 16.2 Some Modelling Problems . . . . . . . . . . . . . 16.3 Exponential Generating Functions . . . . . . . . . 16.4 Exponential generating functions for permutations . 16.5 Distribution Problems . . . . . . . . . . . . . . . .
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117 117 118 120 121 122
17 Basic Geometry Toolkit 17.1 Transformations of the plane . . 17.1.1 Translation . . . . . . . 17.1.2 Reflection about a point 17.1.3 Rotation . . . . . . . . . 17.1.4 Reflection about a line . 17.1.5 Homothety . . . . . . . 17.1.6 Problems . . . . . . . .
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124 125 125 125 126 126 126 127
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18 Circle and Triangle Geometry 18.1 Basic Results . . . . . . . . . . . 18.1.1 Problems . . . . . . . . . 18.2 Triangle Geometry, Trigonometry 18.2.1 Basic Results . . . . . . . 18.2.2 Problems . . . . . . . . . 18.3 Miscellaneous Problems . . . . . 18.3.1 Try these yourself... . . . .
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128 128 132 133 133 138 139 144
19 Coordinate Geometry and Barycentric Coordinates 19.1 Coordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Barycentric Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
146 147 152 154
20 Complex Numbers 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 20.2 Preliminaries and Notations . . . . . . . . . . . . . . . 20.2.1 Preliminaries . . . . . . . . . . . . . . . . . . 20.2.2 Notations . . . . . . . . . . . . . . . . . . . . 20.3 Useful Results . . . . . . . . . . . . . . . . . . . . . . 20.3.1 Basic Angle Properties Between Lines . . . . . 20.3.2 Properties of the Unit Circle . . . . . . . . . . 20.3.3 Similar Triangles, Concyclicity and Area . . . 20.3.4 Some Special Points . . . . . . . . . . . . . . 20.4 Worked Problems . . . . . . . . . . . . . . . . . . . . 20.5 Final Word of advice for analytic methods . . . . . . . 20.5.1 Advantages of the Complex Number Method . 20.5.2 Disadvantages of the Complex Number Method 20.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . 20.7 Miscellaneous Problems . . . . . . . . . . . . . . . .
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160 160 160 160 162 162 162 163 163 164 165 169 169 169 170 171
21 Inversive Geometry 21.1 Introduction . . . . . . . . . 21.2 Preliminaries and Notations . 21.3 Useful Results . . . . . . . . 21.3.1 Poles and Polars . . 21.4 Problems . . . . . . . . . .
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172 172 172 173 174 175
22 Divisibility, Prime Numbers and Arithmetic Functions 22.1 Some Basic Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Greatest common divisor (gcd) and lowest common multiple (lcm) 22.3 Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 22.3.1 Euclidean Algorithm and Bezout’s Theorem . . . . . . . . 22.3.2 Gauss’s Lemma and consequences . . . . . . . . . . . . .
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176 176 177 178 178 179
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22.4 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 22.4.1 Prime Numbers and some important results . . . . . . 22.4.2 p-adic Valuation . . . . . . . . . . . . . . . . . . . . 22.5 Arithmetic functions . . . . . . . . . . . . . . . . . . . . . . 22.5.1 The divisor function d(n) and the sum of divisors σ(n) 22.5.2 Euler’s ϕ function . . . . . . . . . . . . . . . . . . . . 22.6 Miscellaenous Problems . . . . . . . . . . . . . . . . . . . .
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180 180 180 181 181 182 184
23 Congruences 23.1 Definition, some initial properties 23.2 Order of an element . . . . . . . . 23.3 Chinese Remainder Theorem . . . 23.4 Congruences modulo p . . . . . . 23.5 Miscellaneous Problems . . . . .
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187 187 188 188 189 190
24 Diophantine Equations 24.1 Some reflexes . . . . . . 24.2 Using Congruences . . . 24.3 Infinite Descent . . . . . 24.4 Discriminant Method . . 24.5 Vieta’s relations . . . . . 24.6 2nd order equations . . . 24.7 Miscellaneous Problems
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191 191 194 195 196 196 198 199
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Foreword The International Mathematical Olympiad (IMO) is the most important and prestigious mathematical competition for high-school students. The first IMO was held in Brasov, Romania in July, 1959. This year will be the 51st edition, to be held in Astana, Kazakhstan. Singapore has taken part in the IMO since 1988. So this year will be the 22nd year we are taking part. Each year, 6 students will be chosen to represent Singapore. Currently, the national team is selected through a National Team Selection Test in April/May from the training team, which comprises the top 20-25 2nd round results in the Singapore Mathematical Olympiad (SMO). In the 22 years of Singapore’s participation, we have had 1 single gold medal, from Senkodan Thevendran in IMO 1996. Needless to say, this haul is not the most impressive. However, the IMO is probably the “hardest” science olympiad, in the sense where creativity, insight and perhaps even talent, are required. The drill and mug mode (i.e. regurgitation and rote learning) which we Singaporeans are so good at, which probably explains why we excel at Physics, Chemistry and Biology olympiads, is less applicable here. Nevertheless in the 22 years, 72 of the 132 students that have represented Singapore have been Rafflesians. Many of them have gone on to receive other prestigious awards like the President’s Scholarship and Public Service Commission Overseas Merit Scholarship. Charmaine Sia has also been recently awarded the Alice T Schafer Prize for the most outstanding undergraduate woman in mathematics in the United States. Are you ready to be the next?
Why this set of notes? This set of notes first came about in 2008 from the SMO training sessions at Raffles Institution. I have tried to draw the ideas for this set of notes from my experiences attending SIMO training sessions during 1998-1999 as well as conducting training sessions for SIMO during 2001-2003. I have also used quite a number of other resources, such as problem solving books, mathematical journals and online forums. Each chapter has worked examples, exercises and solutions. Included are also past year SMO questions, to which some have
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