One of the first thoughts that led us to write this book is a gap we noticed in the literature: There is an abundance of sources with introductory material of low difficulty or with inaccessible problems aimed at experienced readers, while a large part of the students already possesses the first and needs tools to arrive at the second. This step, namely the transition from school mathematics to competitive mathematics, is also the most difficult.

It requires the reader to understand previously unknown areas of mathematics, as well as to write complex solutions in a structured manner, unfolding their reasoning in mathematical rather than intuitive terms. The latter is a very common problem in written middle school competitions, where the dividing line between observation and proof is thin in the eyes of a student, whose only experience is the school exercises that require simple application of given methods. Consequently, one of the main goals of our book is for the reader to understand how to write something that is a solution, not just a collection of ideas and observations that resembles a solution.

Precisely for this reason, we present the necessary theory, as well as practice exercises, which aim to teach the "how" and the "why" we should think this way in order to solve a problem.

As you will discover while reading, many of the solutions have a preamble with the reasoning of the solution, sometimes with "failed" attempts at solving. The theory is presented briefly and accompanied by applications, as well as practice exercises that are solved at the back of the book. The general exercises are intentionally written without a specific order so that the reader approaches them without preconceptions about the level of difficulty or the method of solving, just as would happen in a competition.

The categories of exercises in mathematical competitions have been established to be drawn from the areas: Algebra, Number Theory, Geometry, Combinatorics. This book covers the first two, and we will soon attempt to publish a similar effort for the other two.

We hope that our book will help students get to know the world of mathematical competitions and give them the opportunity to discover the beauty of mathematical science.

Specifically, we hope it will be useful for young students from Greece and Cyprus who will represent our countries abroad, as well as for those who assist them in their preparation.

We would like to warmly thank our teachers Christos Patilas and Babis Stergiou for their timely advice on the content of the book, as well as our esteemed friends Sotiris Loizias and Achilleas Synefacopoulos for their comments on drafts of this book. Finally, we would like to thank our publisher Giannis Korfias, who supported this effort.