Winning SolutionsSpringer Science & Business Media, 6 dic 2012 - 260 páginas Problem-solving competitions for mathematically talented sec ondary school students have burgeoned in recent years. The number of countries taking part in the International Mathematical Olympiad (IMO) has increased dramatically. In the United States, potential IMO team members are identified through the USA Mathematical Olympiad (USAMO), and most other participating countries use a similar selection procedure. Thus the number of such competitions has grown, and this growth has been accompanied by increased public interest in the accomplishments of mathematically talented young people. There is a significant gap between what most high school math ematics programs teach and what is expected of an IMO participant. This book is part of an effort to bridge that gap. It is written for students who have shown talent in mathematics but lack the back ground and experience necessary to solve olympiad-level problems. We try to provide some of that background and experience by point out useful theorems and techniques and by providing a suitable ing collection of examples and exercises. This book covers only a fraction of the topics normally rep resented in competitions such as the USAMO and IMO. Another volume would be necessary to cover geometry, and there are other v VI Preface special topics that need to be studied as part of preparation for olympiad-level competitions. At the end of the book we provide a list of resources for further study. |
Índice
1 | |
3 | 14 |
4 | 19 |
5 | 35 |
6 | 46 |
4 | 113 |
Combinatorics | 141 |
5 | 178 |
6 | 188 |
7 | 195 |
8 | 202 |
Hints and Answers for Selected Exercises | 215 |
General References | 237 |
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0-1 strings a₁ algebraic AM-GM inequality arithmetic progression b₁ binomial Cauchy's inequality coefficients complex numbers congruence consecutive convex denote the number Diophantine equation divides divisible divisor element Equality holds equation x² Euclid's Lemma Euler Example Exercises for Section fact Find all solutions follows formula Gauss given graph integers mathematical induction modulo multiple natural number nonnegative integer Note nth roots obtain Olympiad pairs perfect square permutation points polynomial equation positive integers positive numbers positive real numbers prime factorization problem proof Prove quadratic nonresidue quadratic reciprocity quadratic residue r₁ rational number Rational Root Rational Root Theorem result roots of unity rows sequence Show solution set Solve subsets Suppose symmetric functions Theorem triangle USAMO vertex vertices yields zeros