Using the wonder of inequalities between averages for mathematics problems solving

The study presents an introductory idea of using mathematical averages as a tool for enriching mathematical problem solving. Throughout students' activities, a research was conducted on their ability to solve mathematical problems, and how to cope with a variety of mathematical tasks, in a variety of ways, using the skills, tools and experiences they have acquired. The article illustrates different mathematical averages, the relation between them, and the ability to use them as an important alternative tool for solving mathematical problems. Various examples for the use of the properties in the relations between the different averages were given for the following purposes: proving inequalities, solving algebraic and trigonometric equations, investigating functions and determining the properties of geometric shapes in the plane and in the space. Also presented was the methodological aspects of the use of averages and a minor research conducted among students, on the contribution to the knowledge of this tool for mathematical problem-solving.   Key words: Problem-solving, different proofs, combinations of fields in mathematics, various mathematical averages.