Various meaningful generalizations could be obtained after gaining insight into some mathematical assertions from different points of view. Generalizations of two problems from International Mathematical Olympiad papers are shown when the details are combined with the opportunities of the program software „The Geometer’s Sketchpad “ (GSP).

By using the Paskalev-Tchobanov distance formula, we give a new proof of the famous Feuerbach theorem.

What are presented are the problems and their solutions of the National competition in financial literacy held in 2020 for V - XII grade students. The problems are proposed and discussed by the National Commission at the Ministry of Education and Science in Bulgaria. The results are analyzed and possibilities for future development are shown.

A geometric relation is derived between the roots of polynomials with roots in four specially located points on a line and the roots of the corresponding derivatives. More precisely, three of the points under consideration are arbitrary while the fourth one depends on them in a special way. Some polynomials of a real variable and with real coefficients are considered as an application.

The aim of the present note is to discuss Problem 2 and Problem 5 on the IMO'2019 paper. The 60th edition of the International Mathematical Olympiad (IMO) took place in the city of Bath, United Kingdom, 11 - 22 July 2019, with the participation of 621 students from 112 countries. The event is the most prestigious scientific Olympiad for high school students. Problem 2 and Problem 5 are of mean difficulty on the paper. Problem 2 was solved fully (7 points) by 98 participants, 92 students were marked with 6 points, 3 with 5 points, 6 with 4 points, 6 with 3 points, 30 with 2 points, 135 with 1 point and 251 with 0 points. The mean result of all the 621 participants in the Olympiad is 2,399. Analogously, Problem 5 was solved fully (7 points) by 250 participants, 3 students were marked with 6 points, 7 with 5 points, 5 with 4 points, 12 with 3 points, 168 with 2 points, 20 with 1 point and 156 with 0 points. The mean result of all the 621 participants is 3,567.

The paper presents the problems for XI and XII grades from the 19th Mathematical Tournament "Perperikon", held in the city of Kardzhali on November 30, 2019. Methodological solutions of the problems are proposed. An analysis of the results achieved by the participants is done and the level of their preparation for such type of mathematical competitions is assessed.

A geometrical relation is derived for the roots of polynomials of complex variable with multiple roots in the vertices of a convex quadrilateral and the roots of the corresponding derivatives. As an application some polynomials of real variable and with real coefficients are considered.

The aim of the present note is to discuss Problem 3 on the IMO'2019 paper. The 60th edition of the International Mathematical Olympiad (IMO) took place in the city of Bath, United Kingdom, 11-22 July 2019, with the participation of 621 students from 112 countries. The event is the most prestigious scientific Olympiad for high school students. Problem 3 is one of the difficult ones on the paper. It was solved fully (7 points) by 28 participants, 4 students were marked with 6 points, 9 with 5 points, 5 with 4 points, 6 with 3 points, 3 with 2 points, 46 with 1 point and 520 with 0 points. The mean result of all the 621 participants in the Olympiad is 0, 572, which shows the high level of the problem difficulty.

Same basic properties of spiral similarity are considered. Their applications are discussed in generalizing well known plane geometry problems and a problem from the 60th International Mathematical Olympiad paper.

A geometric relation is derived between the roots of polynomials of complex variable with multiple roots in the vertices of a parallelogram and the roots of their derivatives. As an application some polynomials of real variable with real coefficients are considered.

Improving financial literacy for all segments of the population is a global problem which solving is aim of many international and national projects. It is not possible to find a sufficient solution without drawing attention to mathematical methods for adequate understanding and assessment of financial situation and taking optimal decisions. This is the goal of the Olympiad, established by two Bulgarian universities: the Higher School of Insurance and Finance (VUZF) in Sofia and the Economics University in Varna. Russia participated in the 2018-2019 edition of the Olympiad for the third time.This paper presents the Olympiad task solutions and gives information about the solution difficulties faced by the students. The authors hope that this information will be helpful for Olympiad questions developers and the participants' coaches.

It is considered a generalization of the equilateral triangles depending on their circumscribed ellipses. It is obtained a generalization of the Fermat points in the plane of a given triangle as a corollary of the corresponding construction.

A geometric relation is considered between a polynomial of fourth degree with symmetrically located roots on a line and the roots of its derivative. A special ellipse is used for the purpose, which is generated by the roots of the polynomial.

The International Mathematical Olympiad is one of the respectable events and one of the most long-lived international educational and scientific competitions. It is the largest, oldest and most prestigious scientific Olympiad for high school students. The 59th edition of the event took place in Cluj-Napoca, Romania, 3 -14 July 2018. The present paper is dedicated to the sixth problem on the Olympiad paper. A detailed analysis of the problem is proposed in a methodological way, which will be useful for students and teachers in the preparatory process for future participations in mathematical competitions.

The aim of the present note is to propose a generalization of Problem 1 on the IMO'2018 paper. The International Mathematical Olympiad (IMO) is the most prestigious scientific Olympiad for high school students. Its 59th edition took place in Cluj-Napoca, Romania, 3-14 July 2018. The problem 1 on the paper was solved fully (7 points) by 381 participants, 7 students were marked with 6 points, 7 with 5 points, 10 with 4 points, 15 with 3 points, 24 with 2 points, 54 with 1 point and 96 with 0 points. The mean result of all the 594 participants in the Olympiad from 107 countries is 4, 934, which shows that the problem is easy and has not bordered most of the contestants. Nevertheless it turns out to be interesting and originates rich in content ideas.

A geometric relation is derived concerning the mots of a special type polynomials of n-th degree and the mots of their derivatives.

Some of the basic hypotheses about the prime numbers are considered, which determine their important properties for Mathematics.