Undergraduate mathematics competitions (1995--2016). Taras Shevchenko National University of Kyiv (Q522936)

This book contains the problems given from 1995 to 2016 at the Undergraduate Mathematics Competition organized by the Taras Shevchenko National University of Kyiv. The problems included in this book vary in difficulty and cover most of the basic courses given at the undergraduate level. The problems proposed in Part I of this volume have detailed solutions in Part II. Most of the problems in this book are not technical and allow for a short and elegant solution. The key features of this volume are the following: it (i) contains a collection of challenging problems in undergraduate mathematics; (ii) is self-contained and assumes only a basic knowledge but opens the path to competitive research in the field; (iii) uses competition-like problems as a platform for training typical inventive skills; (iv) develops basic valuable techniques for solving problems; (v) includes interesting and valuable account of ideas and methods in algebra, calculus, discrete mathematics, measure theory, complex analysis, differential equations, probability theory, geometry, and number theory. A problem book review would be incomplete without the reviewer's favorite problem in the collection. I have chosen the following Problem 11 (p. 32, proposed in 2005 to 3--4 years students): Find all \(\lambda\in {\mathbb C}\) such that every sequence \(\{a_n,\;n\geq 1\}\subset {\mathbb C}\), which satisfies \(|\lambda a_{n+1}-\lambda^2a_n|<1\) for each \(n\geq 1\), is bounded. The reviewer recommends this book to all students curious about undergraduate problems for competitions. Teachers would find this book to be a welcome resource for organizing their activities at a high scientific level. The book under review is very strongly recommended to undergraduate students, PhD students, and instructors.