Here are a few articles from my younger days, newly polished and translated into English.
Contents
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.
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 mathematical areas, like combinatorics, algebra, number theory. The selected examples are either original or from national and international mathematical competitions.
On the distribution of fractional parts
A short study of the fractional parts of an arithmetic progression.