The book is intended for people who want to deepen and expand their knowledge of mathematics before the entrance exam to a higher educational institution. It can be especially useful for students of preparatory departments of universities. Secondary school teachers will find rich material in it to study some of the key topics of the school curriculum. The book sets out some important theoretical questions, supported by a large number of disassembled competitive problems. The authors pay special attention to the logic of decisions, discuss in detail the typical errors of applicants. The book is equipped with exercises taken from the experience of admission exams. During the preparation of the second edition, the book underwent a significant revision, which aimed to more fully and consistently highlight fundamental issues, take into account the experience of the final exams of the latter, and correct the typos and inaccuracies noticed.