The reference book is devoted to tasks that for schoolchildren are considered tasks of increased difficulty, requiring non-standard methods of solutions. Methods for solving equations and inequalities based on geometric considerations, properties of functions (monotony, limitation, parity), application of the derivative are given. The book aims to acquaint schoolchildren with various methods of solving seemingly difficult problems based on the material of the secondary school program, to illustrate the wide possibilities of using well-learned school knowledge and to instill in the reader the skills to use non-standard methods of reasoning when solving problems. For schoolchildren, applicants, leaders of mathematical circles, teachers and all those who like to solve problems.