Research On Functional Equation In High School Mathematical Competition

Functional equation has a long history and is rich in content. As a branch of mathematics, it has been widely applied in practice. Since the 20th century, functional equation often appears in the International Mathematical Olympiad, which has become an important part in mathematics competitions. Due to its various and creative skills in solving problems, functional equation has won the designers'favor in different kinds of competitions and received more and more attention of the mathematics science field at home and abroad.Based on G. Polya's How to Solve It: A New Aspect of Mathematic Method as theoretical basis, this essay employs the method of analyzing documents. The Introduction of this essay focuses on the present situation of international mathematics competition, as well as competition levels. With the definition of functional equation, it points out that functional equation has received more and more attention in mathematics competitions. Chapter Two is aiming to compile functional equation and analyze the statistics in order to find out the characteristics and differences of functional equation in various types of mathematics competitions. What's more, it demonstrates that in the process of tackling problems of functional equation, the competitors should make a concrete analysis of concrete problems, though there are abundant solutions. Functional equation in the competitions may be difficult or easy while in most circumstance it is characterized by the combination of other knowledge points. Chapter Three proposes the possible solutions and the basic strategies for functional equation and illustrates the typical methods by examples such as substitution method, assignment method, Cauchy method, method of undetermined coefficients and mathematical induction. Through the analysis of successful and failed cases, Chapter Four analyzes the basic principles and the general designing methods of functional equation, including the transformation of the existing functional equation, the adaptation and the preparation of such problems. Furthermore, it analyzes Wu Weichao's designing problems of functional equation. The last chapter puts forwards to the application of functional equation in mathematics competitions.Based on this research, this essay proposes the following suggestions about the solution and design problems of functional equation in mathematics competitions for the current high school. In the process of training the students, the teachers should pay more attention to the structure and nature of the questions and improve the students'capacity of solving functional equation through the illustration of the problem-solving thinking process. On the other hand, it is useful for promoting the quality for the students to establish a sound designing system for functional equation from the perspective of solving-problems, which can not only improve the quality of functional equation but also discover the correct relationship between problem-solving and problem-designing.