What can a new book of problems in elementary mathematics possibly contribute to the vast existing collection of journals, articles, and books? This was our main concern when we decided to write this book. The inevitability of this question does not facilitate the answer, because after five years of writing and rewriting we still had something to add. It could be a new problem, a comment we considered pertinent, or a solution that escaped our rationale until this predictive moment, when we were supposed to bring it under the scrutiny of a specialist in the field.
A mere perusal of this book should be sufficient to identify its target audience: students and coaches preparing for mathematical Olympiads, national or international. It takes more effort to realize that these are not the only potential beneficiaries of this work. While the book is rife with problems collected from various mathematical competitions and journals, one cannot neglect the classical results of mathematics, which naturally exceed the level of time-constrained competitions. And no, classical does not mean easy! These mathematical beauties are more than just proof that elementary mathematics can produce jewels. They serve as an invitation to mathematics beyond competitions, regarded by many to be the “true mathematics”. In this context, the audience is more diverse than one might think. Even so, as it will be easily discovered, many of the problems in this book are very difficult. Thus, the theoretical portions are short, while the emphasis is squarely placed on the problems. Certainly, more subtle results like quadratic reciprocity and existence of primitive roots are related to the basic results in linear algebra or mathematical analysis. Whenever their proofs are particularly useful, they are provided. We will assume of the reader a certain familiarity with classical theorems of elementary mathematics, which we will use freely. The selection of problems was made by weighing the need for rou-tine exercises that engender familiarity with the joy of the difficult problems in which we find the truly beautiful ideas. We strove to select only those problems, easy and hard, that best illustrate the ideas we wanted to exhibit. Allow us to discuss in brief the structure of the book. What will most likely surprise the reader when browsing just the table of contents is the absence of any chapters on geometry. This book was not intended to be an exhaustive treatment of elementary mathematics; if ever such a book appears, it will be a sad day for mathematics. Rather, we tried to assemble problems that enchanted us in order to give a sense of techniques and ideas that become leitmotifs not just in problem solving but in all of mathematics.
Furthermore, there are excellent books on geometry, and it was not hard to realize that it would be beyond our ability to create something new to add to this area of study. Thus, we preferred to elaborate more on three important fields of elementary mathematics: algebra, number theory, and combinatorics. Even after this narrowing of focus there are many topics that are simply left out, either in consideration of the available space or else because of the fine existing literature on the subject. This is, for example, the fate of functional equations, a field which can spawn extremely difficult, intriguing problems, but one which does not have obvious recurring themes that tie everything together.Hoping that you have not abandoned the book because of these omissions,Hoping that you have not abandoned the book because of these omissions,which might be considered major by many who do not keep in mind thestated objectives, we continue by elaborating on the contents of the chapters.To start out, we ordered the chapters in ascending order of difficulty of themathematical tools used. Thus, the exposition starts out lightly with someclassical substitution techniques in algebra, emphasizing a large number ofexamples and applications. These are followed by a topic dear to us: the Cauchy-Schwarz inequality and its variations. A sizable chapter presents ap-plications of the Lagrange interpolation formula, which is known by most only through rote, straightforward applications. The interested reader will findsome genuine pearls in this chapter, which should be enough to change his orher opinion about this useful mathematical tool. Two rather difficult chapters,in which mathematical analysis mixes with algebra, are given at the end ofthe book. One of them is quite original, showing how simple consideration of integral calculus can solve very difficult inequalities. The other discusses properties of equidistribution and dense numerical series. Too many books consider the Weyl equidistribution theorem to be “much too difficult” to include, and we cannot resist contradicting them by presenting an elementary proof. Furthermore, the reader will quickly realize that for elementary problems we have not shied away from presenting the so-called non-elementary solutions which use mathematical analysis or advanced algebra. It would be a crime to consider these two types of mathematics as two different entities, and it would be even worse to present laborious elementary solutions without admitting the possibility of generalization for problems that have conceptual and easy non-elementary solutions. In the end we devote a whole chapter to discussing criteria for polynomial irreducibility. We observe that some extremely efficient criteria (like those of Peron and Capelli) are virtually unknown, even though they are more efficient than the well-known Eisenstein criterion.
The section dedicated to number theory is the largest. Some introductory chapters related to prime numbers of the form 4k + 3 and to the order of an element are included to provide a better understanding of fundamental results which are used later in the book. A large chapter develops a tool which is as simple as it is useful: the exponent of a prime in the factorization of an integer. Some mathematical diamonds belonging to Paul Eras and others appear within. And even though quadratic reciprocity is brought up in many books, we included an entire chapter on this topic because the problems available to us were too ingenious to exclude. Next come some difficult chapters concerning arithmetic properties of polynomials, the geometry of numbers (in which we present some arithmetic applications of the famous Minkowski’s theorem), and the properties of algebraic numbers. A special chapter studies some applications of the extremely simple idea that a convergent series of integers is eventually stationary! The reader will have the chance to realize that in mathematics even simple ideas have great impact: consider, for example, the fundamental idea that in the interval (-1, 1) the only integer is 0. But how many fantastic results concerning irrational numbers follow simply from that! Another chapter dear to us concerns the sum of digits, a subject that always yields unexpected and fascinating problems, but for which we could not find a unique approach.
Finally, some words about the combinatorics section. The reader will immediately observe that our presentation of this topic takes an algebraic slant, which was, in fact, our intention. In this way we tried to present some unexpected applications of complex numbers in combinatorics, and a whole chapter is dedicated to useful formal series. Another chapter shows how useful linear algebra can be when solving problems on set combinatorics. Of course, we are traditional in presenting applications of Turan’s theorem and of graph theory in general, and the pigeonhole principle could not be omitted. We faced difficulties here, because this topic is covered extensively in other books, though rarely in a satisfying way. For this reason, we tried to present lesser-known problems, because this topic is so dear to elementary mathematics lovers. At the end, we included a chapter on special applications of polynomials in number theory and combinatorics, emphasizing the Combinatorial Nullstellensatz, a recent and extremely useful theorem by Noga Alon.
We end our description with some remarks on the structure of the chapters. In general, the main theoretical results are stated, and if they are sufficiently profound or obscure, a proof is given. Following the theoretical part, we present between ten and fifteen examples, most from mathematical contests or from journals such as Kvant, Komal, and American Mathematical Monthly. Others are new problems or classical results. Each chapter ends with a series of problems, the majority of which stem from the theoretical results.
Finally, a change that will please some and scare others: the end-of-chapter problems do not have solutions! We had several reasons for this. The first and most practical consideration was minimizing the mass of the book. But the second and more important factor was this: we consider solving problems to necessarily include the inevitably lengthy process of trial and research to which the inclusion of solutions provides perhaps too tempting of a shortcut. Keeping this in mind, the selection of the problems was made with the goal that the diligent reader could solve about a third of them, make some progress in the second third and have at least the satisfaction of looking for a solution in the remainder. We come now to the most delicate moment, the one of saying thank you.
First and foremost, we thank Marin Tetiva and Paul Stanford, whose close reading of the manuscript uncovered many errors that we would not have liked in this final version. We thank them for the great effort they put into reviewing the book. All of the remaining mistakes are the responsibility of the authors, who would be grateful for reports of errors so that in a future edition they will disappear. Many thanks to Radu Sorici for giving the book the look it has now and for the numerous suggestions for improvement. We thank Adrian Zahariuc for his help in writing the sections on the sums of digits and graph theory. Several solutions are either his own or the fruit of his experience. Special thanks are due to Valentin Vornicu for creating Mathlinks, which has generated many of the problems we have included. His website, mathlinks ro, hosts a treasure trove of problems, and we invite every passionate mathematician to avail themselves of this fact. We would also like to thank Ravi Boppana, Vesselin Dimitrov, and Richard Stong for the
excellent problems, solutions, and comments they provided. Lastly, we have surely forgotten many others who helped throughout the writing process; our thanks and apologies go out to them.
