## Mathematical Olympiad ChallengesWhy 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. It’s 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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4 | |

6 | |

10 | |

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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Abel summation Andreescu angle assume binomial coefficients bisector Cauchy–Schwarz inequality circle circumcircle circumradius collinear color complex numbers compute conclusion follows congruent construct convex cubes cyclic quadrilateral deduce Denote digits dissection divides divisible endpoints equilateral triangle exists factors Figure Find formula function f Gazeta Matematică Gelca given hence identity implies induction infinitely integer solutions intersection interval isosceles Let f Matematică Mathematical Olympiad Mathematics Gazette matrix mean value theorem midpoint modulo multiple nonnegative integers Note obtain orthocenter orthogonal pairs parallel parallelepiped Pell equation perfect square perpendicular plane polynomial positive integers prime number proposed Putnam Mathematical Competition quadratic residue radical axis rational real numbers rectangle relatively prime repunit residues modulo Romanian Mathematical rotation satisfies Second solution segment sequence shows side tangent tetrahedron theorem transform triangle ABC trigonometry USAMO vector vertex vertices W.L. Putnam Mathematical yields