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The article presents new effective methods of proving inequalities and problems in various mathematical Olympiads on their application. The manual is intended for gifted students of general secondary schools, academic lyceums and vocational colleges, teachers of mathematics and students of pedagogical universities. One of the basic procedures for proving inequalities is to rewrite them as a sum of squares and then according to the most elementary property that the square of real number is non negative to prove a certain inequality
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