Problems from the Book

This book is designed for students and coaches preparing for high-school-level mathematical Olympiads, national or international.The book consists of 23 chapters, each covering a topic in algebra, number theory, or combinatorics. Each chapter has two parts: a "theory and examples" section that presents the main theoretical results of the topic (with proofs if necessary) and 10-15 example problems, and a "problems for training" section with more difficult problems to challenge the reader. (Solutions to the problems in these latter sections are not provided.) Some useful substitutions Always Cauchy-Schwarz... - Look at the exponent Primes and squares T2's lemma Some classical problems in extremal graphy theory Complex combinatorics Formal series revisited A brief introduction to algebraic number theory Arithmetic properties of polynomials Lagrange interpolation formula Higher algebra in combinatorics Geometry and numbers The smaller, the better Density and regular distribution The digit sum of a positive integer At the border of analysis and number theory Quadratic reciprocity Solving elementary inequalities using integrals Pigeonhole principle revisited Some useful irreducibility criteria Cycles, paths, and other ways Some special applications of polynomials. The authors provide a combination of enthusiasm and experience which will delight any reader. In this volume they present innumerable beautiful results, intriguing problems, and ingenious solutions. The problems range from elementary gems to deep truths. A trully delightful and highly instructive book, this will prepare the engaged reader not only for any mathematics competition they may enter but also for a life time of mathematical enjoyment. A must for the bookshelves of both aspiring and seasoned mathematicians.