## Mathematical Olympiad ChallengesMathematical Olympiad Challenges is a rich collection of problems put together by two experienced and well-known professors and coaches of the U.S. International Mathematical Olympiad Team. Hundreds of beautiful, challenging, and instructive problems from algebra, geometry, trigonometry, combinatorics, and number theory were selected from numerous mathematical competitions and journals. An important feature of the work is the comprehensive background material provided with each grouping of problems. The problems are clustered by topic into self-contained sections with solutions provided separately. All sections start with an essay discussing basic facts and one or two representative examples. A list of carefully chosen problems follows and the reader is invited to take them on. Additionally, historical insights and asides are presented to stimulate further inquiry. The emphasis throughout is on encouraging readers to move away from routine exercises and memorized algorithms toward creative solutions to open-ended problems. Aimed at motivated high school and beginning college students and instructors, this work can be used as a text for advanced problem- solving courses, for self-study, or as a resource for teachers and students training for mathematical competitions and for teacher professional development, seminars, and workshops. |

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### Table des matières

11 A Property of Equilateral Triangles | 4 |

12 Cyclic Quadrilaterals | 6 |

13 Power of a Point | 9 |

14 Dissections of Polygonal Surfaces | 13 |

15 Regular Polygons | 16 |

16 Geometric Constructions and Transformations | 20 |

17 Problems with Physical Flavor | 22 |

18 Tetrahedra Inscribed in Parallelepipeds | 24 |

13 Power of a Point | 99 |

14 Dissections of Polygonal Surfaces | 106 |

15 Regular Polygons | 114 |

16 Geometric Constructions and Transformations | 124 |

17 Problems with Physical Flavor | 129 |

18 Tetrahedra Inscribed in Parallelepipeds | 136 |

19 Telescopic Sums and Products in Trigonometry | 140 |

110 Trigonometric Substitutions | 145 |

19 Telescopic Sums and Products in Trigonometry | 26 |

110 Trigonometric Substitutions | 29 |

Algebra and Analysis | 33 |

21 No Square is Negative | 34 |

22 Look at the Endpoints | 36 |

23 Telescopic Sums and Products in Algebra | 38 |

24 On an Algebraic Identity | 41 |

25 Systems of Equations | 43 |

26 Periodicity | 47 |

27 The Abel Summation Formula | 50 |

28 𝑥 + 1𝑥 | 52 |

29 Matrices | 54 |

210 The Mean Value Theorem | 55 |

Number Theory and Combinatorics | 59 |

31 Arrange in Order | 60 |

32 Squares and Cubes | 62 |

33 Repunits | 64 |

34 Digits of Numbers | 66 |

35 Residues | 69 |

36 Diophantine Equations with the Unknowns as Exponents | 72 |

37 Numerical Functions | 74 |

38 Invariants | 77 |

39 Pell Equations | 80 |

310 Prime Numbers and Binomial Coefficients | 84 |

Solutions | 87 |

Geometry and Trigonometry | 89 |

11 A Property of Equilateral Triangles | 90 |

12 Cyclic Quadrilaterals | 93 |

Algebra and Analysis | 151 |

21 No Square is Negative | 152 |

22 Look at the Endpoints | 156 |

23 Telescopic Sums and Products in Algebra | 159 |

24 On an Algebraic Identity | 164 |

25 Systems of Equations | 166 |

26 Periodicity | 172 |

27 The Abel Summation Formula | 176 |

28 𝑥 + l𝑥 | 183 |

29 Matrices | 188 |

210 The Mean Value Theorem | 191 |

Number Theory and Combinatorics | 197 |

31 Arrange in Order | 198 |

32 Squares and Cubes | 201 |

33 Repunits | 206 |

34 Digits of Numbers | 209 |

35 Residues | 216 |

36 Diophantine Equations with Unknowns as Exponents | 221 |

37 Numerical Functions | 226 |

38 Invariants | 233 |

39 Pell Equations | 237 |

310 Prime Numbers and Binomial Coefficients | 244 |

Definitions and Notation | 251 |

AI Glossary of Terms | 252 |

A2 Glossary of Notation | 257 |

About the Authors | 259 |