117 Polynomial Problems
From the AwesomeMath Summer Program

For any serious math olympiad aspirant, a firm grip of polynomials is a must as its a gateway to many advanced topics in algebra, geometry, functions and calculus. The usual preparatory books for Mathematical Olympiads typically lump Algebra as one section with a melange of problems which end up lacking an exhaustive coverage specifically of polynomials thereby creating a gap in developing structured thinking and techniques to approach high order thinking polynomial problems and advanced topics in polynomials that bleed into other areas like number theory and functions. The 117 Polynomial problems from the AMSP is a fantastic resource because: 1. This book is written with a math olympiad aspirant in mind as it includes curated essential and important topics in polynomials. This book will help you stay grounded and learn what you need to continue to develop in your olympiad journey 2. It is focused on creating a ground up understanding of important olympiad topics on polynomials including developing many advanced topics that you will not find together in a single olympiad book (e.g. Number theory applications, higher degree polynomials) 2. Like all other books from Awsomemath, each chapter has profound theory to be read carefully and closely, useful lemmas which are a huge bonus and excellent examples that teach you non-standard and efficient techniques for polynomials beyond standard theorems that we find in textbooks or online. 3. Selection of Introductory and Advanced problems is unique because it helps build your thinking and approach in a graded manner and also arms you out with several new/useful tools to breakdown complex problems. This book is a must for students/enthusiasts who want to acquire a firm grip of an important area in math olympiads i.e. polynomials.

This book is the first one in a three-book series and proposes a very detailed and

complete journey through the amazing world of polynomials.

This first volume successfully presents some of the basic properties of polynomials,

with emphasis on some factorings, on the division of the polynomials, on the odd and

even polynomials, as well as on the integer and rational roots of the polynomials. Also,

the volume discusses in a very methodical way the specific properties of the

polynomials of second, third and fourth degree, with a focus on Vieta’s formulas and

number of roots in each of these particular cases. Roots of polynomials are further

discussed with an accent on Vieta’s general formulas and other inequalities between

coefficients and roots.

All topics of the book are carefully explained from a theoretical point of view and many

examples are used to reinforce some of the concepts. Further on, the authors present

many significant problems that were used in several past mathematical competitions

from around the world with complete solutions and comments. Problems are carefully

selected and range in difficulty level from medium to hard.

The uniqueness of this book consist in the multitude of the algebraic techniques that

are explained and exposed in a very coherent way with an effort on creating a

continuum from the beginning to the end.

The part that I personally treasure in this the book is the section concerning number

theory and polynomials, with all the subsequent original proposed problems.

As the authors use to do in many of their books, this volume contains also two rich

sections of proposed problems : introductory and advanced. All problems have

detailed solutions and comments to guide the reader through the process of learning

and understanding the methods.

I strongly recommend this book to all the young minds that train for writing AMC 10-12,

AIME or USAMO tests, or to all other students that want to explore properties of

polynomials beyond the limits of the regular academical school curriculum. Also, the

book could be a very strong tool for teachers that work to train students for competing

in mathematical competitions.