Mathematics competitions: an integral part of the educational process

For a century and a half, the scene of mathematics competitions underwent a remarkable transformation from isolated and geographically scattered events to a full-scale and a full-featured vibrant global ecosystem comprising an impressive variety of competitions, school students, university students, teachers, mentors, scientists, schools, universities, research institutions, journals, websites, civil society organizations, educational authorities, parents, etc. The evolution, the current state, the functioning of this ecosystem as well as its role for the identification and development of talent, its impact on the educational process, and the institutions involved with it, is briefly reflected on. Some relatively new online competitions are presented that cultivate the use of dynamic geometry software systems for a deeper understanding of mathematical facts and phenomena, and for finding approximate numerical solutions to problems that are not part of a typical school curriculum, but often arise from real-life practice.National Scientific Program "Information and Communication Technologies for Unified Digital Market in Science, Education and Security” realized with the support of the Bulgarian Ministry of Education and Science

Uniqueness on a Residual Part of Best Approximations in Banach Spaces

[Kenderov Petar S.; Кендеров Петър С.]The paper presents some general results concerning continuity on a residual part of every semi-continuous multivalued mapping. As an application, some results are proved asserting (under some conditions) that every metric projection is single-valued on a residual part of the space. The results were announced in [6]