PREPARING FOR MATHEMATICAL COMPETITIONS: A PROBLEM SERIES ON METRIC RATIOS IN A QUADRANGLE

Abstract

Solving of competitive problems by pupils and students is a good foundation and preparation for future practical and scientific activities, as mastering the methods of solving competitive problems requires them to work hard, actively and focused, as well as develops their creativity and raises level of interest in mathematics. The article reveals the mathematical aspects of preparing students to solve competitive problems on the example of one geometric problem (the ratio between the areas of triangles formed by the intersection of diagonals of a convex quadrilateral), which is the basis of many competitive problems in geometry; the problem is solved using the facts of elementary mathematics, available to students of the eighth form of secondary school; an analysis of the range of competitive problems of various mathematical competitions, for which the considered reference problem is a key subtask in the solution. An author's competitive problem for high school students has been created, which allows integrating a purely theoretical-numerical problem into the geometric shell with the study of simplicity of elements, divisibility of a product by a prime number, mutual simplicity of elements, with the need to find solutions of Diophantine equations in natural numbers. The article combines a problem series of a large number of different competitive geometric problems around one reference problem, presents the methodological aspects of preparing students to solve competitive problems on the example of this problem; attention is paid to checking the correctness of the obtained results, which avoids erroneous solutions; the tasks which urge to find and realize ways of their fulfillment are analyzed; examples of different tasks in terms of age capabilities of researchers are selected; the problems of competitions of regional levels with geometric and theoretical-numerical filling are considered; the competitive task on the given subject is created. Further research will be aimed at creating a broader series of tasks for the considered reference problem, including problems with integration into related competitive topics. The article emphasizes the problem content and structuring according to the age capabilities of students on the research topic.