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Page 1 of 9 iCTF 2011 Team We_0wn_Y0u Vienna University of Technology, seclab “No plan survives contact with the enemy” -- Helmuth von Moltke the Elder (disciple of Carl von Clausewitz) Introduction The annual UCSB International Capture The Flag Contest is the best opportunity for our Students to prove themselves. We announce the contest for students of our Course “Advanced Internet Security”. Recently, this course experienced a massive gain in popularity, because it was included in the Masters colloquium for software engineers.

service as ‘up
competition
participants from a broader range of students with widespread abilities
image into the virtual machine
serious problems of ‘toilet
game state pusher
parts of the code
service
system
students

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RafflesInstitution

Singapore Mathematics Olympiad Training

Department of Mathematics

2010

Contents

1 ProblemSolving  Generalities, Basic Techniques11 1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 1.2 GettingStarted .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12 1.3 Methodsof Argument .. . . . . . . . . . . . . . . . . . . . . . . . . . . .15 1.3.1 Deductionand Symbolic Logic. . . . . . . . . . . . . . . . . . .15 1.3.2 Argumentby Contradiction. . . . . . . . . . . . . . . . . . . . .15 1.3.3 MathematicalInduction .. . . . . . . . . . . . . . . . . . . . . .16 1.4 Otherimportant strategies .. . . . . . . . . . . . . . . . . . . . . . . . . .18 1.5 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19 2 FundamentalTactics for Solving Problems20 2.1 Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 2.1.1 GeometricSymmetry .. . . . . . . . . . . . . . . . . . . . . . . .21 2.1.2 AlgebraicSymmetry .. . . . . . . . . . . . . . . . . . . . . . . .23 2.1.3 Symmetryin Polynomials. . . . . . . . . . . . . . . . . . . . . .24 2.2 TheExtreme Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . .25 2.3 ThePigeonhole Principle. . . . . . . . . . . . . . . . . . . . . . . . . . .25 2.4 Invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27 2.5 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28 3 Functionsand Basic Algebra30 3.1 Functionsand their inverses .. . . . . . . . . . . . . . . . . . . . . . . . .30 3.2 BasicCharacteristics of Functions. . . . . . . . . . . . . . . . . . . . . .31 3.3 CommonClasses of Functions. . . . . . . . . . . . . . . . . . . . . . . .33 3.3.1 TheModulus Function. . . . . . . . . . . . . . . . . . . . . . . .33 3.3.2 PolynomialFunctions .. . . . . . . . . . . . . . . . . . . . . . .34 3.3.3 TrigonometricFunctions . . . . . . . . . . . . . . . . . . . . . . .35 3.3.4 InverseTrigonometric Functions. . . . . . . . . . . . . . . . . . .36 3.4 Integerand Fractional Parts of a Real Number. . . . . . . . . . . . . . . .38 3.5 MiscellaneousProblems .. . . . . . . . . . . . . . . . . . . . . . . . . .39 4 Sequences41 4.1 1storder linear recurrence relation. . . . . . . . . . . . . . . . . . . . . .42 4.2 2ndorder linear recurrence relation with constant coeﬃ42. . . . . . .cients .

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4.3 MiscellaneousProblems .. . . . . . . . . . . . . . . . . . . . . . . . . .44 5 Series48 5.1 TrigonometricSeries .. . . . . . . . . . . . . . . . . . . . . . . . . . . .48 5.2 PastYear SMO Problems. . . . . . . . . . . . . . . . . . . . . . . . . . .49 6 Inequalities Techniques and the Standard Few53 6.1 Overviewof some ideas .. . . . . . . . . . . . . . . . . . . . . . . . . . .53 6.1.1 Whendoes equality occur?. . . . . . . . . . . . . . . . . . . . . .53 6.1.2 Changeof variables .. . . . . . . . . . . . . . . . . . . . . . . . .53 6.1.3 Createsymmetry .. . . . . . . . . . . . . . . . . . . . . . . . . .54 6.2 Toolbox The Standard Few. . . . . . . . . . . . . . . . . . . . . . . . .54 6.2.1 TriangleInequality .. . . . . . . . . . . . . . . . . . . . . . . . .54 6.2.2 Asquare is always positive!. . . . . . . . . . . . . . . . . . . . .55 6.2.3 ThePower Means. . . . . . . . . . . . . . . . . . . . . . . . . .55 6.2.4 CauchySchwarzand Holder inequalities. . . . . . . . . . . . . .56 6.2.5 Rearrangements,Chebyshev . . . . . . . . . . . . . . . . . . . . .57 6.3 Smoothing,convexity and Jensen’s inequality. . . . . . . . . . . . . . . .58 6.4 TangentLines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61 6.5 IsolatedFudging .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61 6.6 PastYear SMO Questions .. . . . . . . . . . . . . . . . . . . . . . . . . .62 7 Inequalities The Less Standard Few68 7.1 Schur’sInequality .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68 7.2 Majorizationand Muirhead. . . . . . . . . . . . . . . . . . . . . . . . . .69 7.3n−70. . . . . . . . . . . . . . . . . . . . . . . . . .1 Equal Value Principle 8 Inequalities The Ugly Few72 8.1 Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72 8.1.1 PartialDerivatives .. . . . . . . . . . . . . . . . . . . . . . . . .72 8.1.2 Maximaand Minima. . . . . . . . . . . . . . . . . . . . . . . . .73 8.2 LagrangeMultipliers .. . . . . . . . . . . . . . . . . . . . . . . . . . . .73 8.3 Localversus Global Extremum. . . . . . . . . . . . . . . . . . . . . . . .74 9 Polynomials76 9.1 Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76 9.2 MiscellaneousProblems .. . . . . . . . . . . . . . . . . . . . . . . . . .77 10 FunctionalEquations 80 10.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80 10.2 Someinitial advice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81 10.3 Functionalequations overN,ZandQ81. . . . . . . . . . . . . . . . . . . . 10.4 Otheradvice and methods. . . . . . . . . . . . . . . . . . . . . . . . . .82 10.5 MiscellaenousProblems .. . . . . . . . . . . . . . . . . . . . . . . . . .83

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11 Additionand Multiplication; Permutations and Combinations87 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87 11.2 Additionand Multiplication Principle. . . . . . . . . . . . . . . . . . . .87 11.3 Permutationsand Combinations. . . . . . . . . . . . . . . . . . . . . . .90 11.4 MiscellaneousProblems .. . . . . . . . . . . . . . . . . . . . . . . . . .93 12 BijectionPrinciple and Examples96 12.1 BijectionPrinciple .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .96 12.2 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99 13 Recursions100 13.1 Solvingrecursive relations. . . . . . . . . . . . . . . . . . . . . . . . . .100 13.1.1 Firstorder linear recursive relations. . . . . . . . . . . . . . . . .100 13.1.2 Secondorder linear recursive relations. . . . . . . . . . . . . . . .101 13.2 Formulatingrecursive relations. . . . . . . . . . . . . . . . . . . . . . . .102 13.3 Usinga graph to setup recursive relations. . . . . . . . . . . . . . . . . .105 13.4 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106 14 Principleof Inclusion and Exclusion108 14.1 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108 14.2 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111 15 Countingin Two Ways: Fubini’s Principle112 15.1 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112 15.2 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116 16 GeneratingFunctions 117 16.1 OrdinaryGenerating Functions. . . . . . . . . . . . . . . . . . . . . . . .117 16.2 SomeModelling Problems. . . . . . . . . . . . . . . . . . . . . . . . . .118 16.3 ExponentialGenerating Functions. . . . . . . . . . . . . . . . . . . . . .120 16.4 Exponentialgenerating functions for permutations .. . . . . . . . . . . . .121 16.5 DistributionProblems .. . . . . . . . . . . . . . . . . . . . . . . . . . . .122 17 BasicGeometry Toolkit124 17.1 Transformationsof the plane. . . . . . . . . . . . . . . . . . . . . . . . .125 17.1.1 Translation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .125 17.1.2 Reﬂectionabout a point. . . . . . . . . . . . . . . . . . . . . . .125 17.1.3 Rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126 17.1.4 Reﬂectionabout a line. . . . . . . . . . . . . . . . . . . . . . . .126 17.1.5 Homothety .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .126 17.1.6 Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .127

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18 Circleand Triangle Geometry128 18.1 BasicResults .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128 18.1.1 Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132 18.2 TriangleGeometry, Trigonometry. . . . . . . . . . . . . . . . . . . . . .133 18.2.1 BasicResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133 18.2.2 Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .138 18.3 MiscellaneousProblems .. . . . . . . . . . . . . . . . . . . . . . . . . .139 18.3.1 Trythese yourself... .. . . . . . . . . . . . . . . . . . . . . . . . .144 19 CoordinateGeometry and Barycentric Coordinates146 19.1 CoordinateGeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .147 19.2 BarycentricCoordinates .. . . . . . . . . . . . . . . . . . . . . . . . . .152 19.3 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154 20 ComplexNumbers 160 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .160 20.2 Preliminariesand Notations .. . . . . . . . . . . . . . . . . . . . . . . . .160 20.2.1 Preliminaries .. . . . . . . . . . . . . . . . . . . . . . . . . . . .160 20.2.2 Notations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .162 20.3 UsefulResults .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .162 20.3.1 BasicAngle Properties Between Lines .. . . . . . . . . . . . . . .162 20.3.2 Propertiesof the Unit Circle. . . . . . . . . . . . . . . . . . . . .163 20.3.3 SimilarTriangles, Concyclicity and Area. . . . . . . . . . . . . .163 20.3.4 SomeSpecial Points. . . . . . . . . . . . . . . . . . . . . . . . .164 20.4 WorkedProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .165 20.5 FinalWord of advice for analytic methods. . . . . . . . . . . . . . . . . .169 20.5.1 Advantagesof the Complex Number Method. . . . . . . . . . . .169 20.5.2 Disadvantagesof the Complex Number Method. . . . . . . . . . .169 20.6 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .170 20.7 MiscellaneousProblems .. . . . . . . . . . . . . . . . . . . . . . . . . .171 21 InversiveGeometry 172 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .172 21.2 Preliminariesand Notations .. . . . . . . . . . . . . . . . . . . . . . . . .172 21.3 UsefulResults .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .173 21.3.1 Polesand Polars. . . . . . . . . . . . . . . . . . . . . . . . . . .174 21.4 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .175 22 Divisibility,Prime Numbers and Arithmetic Functions176 22.1 SomeBasic Stuﬀ176. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Greatestcommon divisor (gcd) and lowest common multiple (lcm). . . . .177 22.3 EuclideanAlgorithm .. . . . . . . . . . . . . . . . . . . . . . . . . . . .178 22.3.1 EuclideanAlgorithm and Bezout’s Theorem .. . . . . . . . . . . .178 22.3.2 Gauss’sLemma and consequences. . . . . . . . . . . . . . . . . .179

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22.4 PrimeNumbers .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .180 22.4.1 PrimeNumbers and some important results. . . . . . . . . . . . .180 22.4.2p. . . . . . . . . . . . . . . . . . . . . . . . . . .adic Valuation180 22.5 Arithmeticfunctions .. . . . . . . . . . . . . . . . . . . . . . . . . . . .181 22.5.1 Thedivisor functiond(n) and the sum of divisorsσ(n181. . . . . .) . 22.5.2 Euler’sϕ. . . . . . . . . . . . . . . . . . . . . . . . . .182function . 22.6 MiscellaenousProblems .. . . . . . . . . . . . . . . . . . . . . . . . . .184 23 Congruences187 23.1 Deﬁnition,some initial properties. . . . . . . . . . . . . . . . . . . . . .187 23.2 Orderof an element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .188 23.3 ChineseRemainder Theorem. . . . . . . . . . . . . . . . . . . . . . . . .188 23.4 Congruencesmodulop189. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.5 MiscellaneousProblems .. . . . . . . . . . . . . . . . . . . . . . . . . .190 24 DiophantineEquations 191 24.1 Somereﬂexes .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .191 24.2 UsingCongruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .194 24.3 InﬁniteDescent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .195 24.4 DiscriminantMethod .. . . . . . . . . . . . . . . . . . . . . . . . . . . .196 24.5 Vieta’srelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .196 24.6 2ndorder equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .198 24.7 MiscellaneousProblems .. . . . . . . . . . . . . . . . . . . . . . . . . .199

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Foreword

The International Mathematical Olympiad (IMO) is the most important and prestigious math ematical 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 Mathemat ical 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 Raﬄof them have gone on to receive other prestigious awards like theesians. Many President’s Scholarship and Public Service Commission Overseas Merit Scholarship. Char maine 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 Insti tution. Ihave 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 solv ing 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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