In recent years, student olympiads and mathematics competitions have gained significant popularity as a means to stimulate scientific creativity among students. The problems presented in these olympiads are unconventional and demand not only a solid understanding of the curriculum but also an inventive and creative approach; they often illustrate profound mathematical concepts in a simplified manner. However, despite the abundance and variety of material available, there has yet to be a comprehensive and accessible collection of problems offered in these competitions. This proposed collection aims to fill that gap to some extent. It features a selection of problems from various mathematical student olympiads held across the country (Round I), problems from the Moscow City Student Olympiads (Round II), problems from the All-Union Olympiads "Student and Scientific-Technical Progress" in the mathematics section, as well as some problems from International Student Olympiads and tasks from competitions and oral exams at the Mechanics and Mathematics Faculty of Moscow University.