Papers

Here are a few papers from my younger days, newly polished and translated into English.

Sums of sets

This article is based on a problem from number theory. We present several generalizations and corollaries. The paper studies residues modulo a prime, sets of such residues, and a special type of summing sets. The examples are taken from past mathematical competitions.

Similar triangles inscribed in one another

This is an article I wrote in high school with the guidance of my teacher Nikolay Nikolov.

The use of complex numbers in geometry is often regarded as less
elegant than purely geometric methods. Yet, despite its lack of aesthetics, a bit of algebra can lead to surprisingly interesting geometric results. In this article, we examine families of triangles inscribed in one another.

Use of polynomials for arithmetical and combinatorial problems

Sometimes number theory and combinatorics problems can be easily translated into algebra problems by introducing suitable polynomials. The present note considers such applications in connection with some problems from various mathematical competitions and olympiads.

The article was presented at the conference of the Union of Bulgarian Mathematicians in 2002.

Certain applications of the roots of unity

The objective of this article is to illustrate the use of the roots of unity for solving problems from various areas of mathematics, including combinatorics, algebra, and number theory. The selected examples are either original or from mathematical competitions.

On the distribution of fractional parts

A short study of the fractional parts of an arithmetic progression.

Problem sets

This section contains original math problems. By “original,” I mean that I developed each problem independently, not necessarily that I was the first person ever to conceive of it. It is entirely possible that some of these problems had previously appeared in a book or competition without my knowledge. What is celebrated here, then, is not originality in that strict sense, but rather the beauty and distinctive character of the problems themselves.

Problem selection

This is a collection of original math problems and their solutions. The problems have been grouped into the standard IMO categories of algebra, combinatorics, number theory, and geometry like in an IMO Short List. Some of them have appeared in various competitions and magazines. An attempt has been made to order the problems by difficulty, but this is subjective.

Various problems

These are problems that are not suitable for competitions for one reason or another, yet they are interesting in their own right.

Training

Lecture notes

These lecture notes are a compilation of problems, ideas, useful facts, and theorems that cover the basics for preparation for math competitions. They are not meant to be exhaustive. They are not meant to provide a comprehensive mathematical foundation in mathematics. In fact, the text relies on some familiarity with the main topics discussed. It is meant as a collection of essentials needed for the IMO and other high school mathematics competitions.

This document is not complete. I hope it can be used for competitive mathematics training classes and that it will continue to evolve as interested students contribute additional topics, example problems, and, ideally, solutions — possibly in a separate document.

Selected Math Works

This is a compilation of some of the materials presented above. It has been published as a book and is awarded to students who place second in national competitions in Bulgaria.