中学数学竞赛中的柯西不等式问题探究
发布时间：2018-12-14 03:36
【摘要】：柯西不等式在初等领域是一个非常重要的不等式。新课改后柯西不等式被纳入高中数学选修内容,而这一内容也再次成为数学竞赛的热点,只要我们能灵活的运用此不等式,就能使许多复杂的问题迎刃而解。例如用柯西不等式去证明不等式、求函数的最值和解三角形的相关问题时,其优越性显而易见。本论文主要研究的是柯西不等式的离散形式在高中奥林匹克竞赛中的应用。论文共分为四章,论文首先阐述了IMO (International Mathematical Olympiad国际奥林匹克数学竞赛)和CMO (Chinese Mathematical Olympiad中国奥林匹克数学竞赛)的源远历史与发展情况,第二章叙述了柯西不等式的表现形式,关于柯西不等式有技巧性和代表性的证明方法有二十种,本文选取了其中具有代表性的7种初等的证明方法,这样更利于高中生的理解,并对柯西不等式的变形与推广进行了深入的探究说明,这样可以将柯西不等式的应用范围加以扩大。还详细的对柯西不等式与n维不等式链的关系进行了说明。第三章主要针对IMO和CMO中关于柯西不等式的赛题进行分类整理,并对解题的方法和技巧进行分析和总结概括。第四章是基于上一章的研究成果编写的几道关于柯西不等式的赛题以供读者赏阅。本论文的价值在于详细且系统的研究了柯西不等式的相关知识以及在竞赛中的应用探究,可以为高中数学教学和数学竞赛提供参考。
[Abstract]:Cauchy inequality is a very important inequality in the elementary field. After the new curriculum reform, Cauchy inequality is included in the elective course of mathematics in senior high school, and this content has become the hot spot of mathematics competition again. As long as we can use this inequality flexibly, many complicated problems can be solved easily. For example, when we use Cauchy inequality to prove inequality, find the most value of function and solve the related problem of triangle, its superiority is obvious. This thesis mainly studies the application of the discrete form of Cauchy inequality in high school Olympiad. The thesis is divided into four chapters. Firstly, the paper expounds the history and development of the IMO (International Mathematical Olympiad International Olympiad Mathematical Competition and the CMO (Chinese Mathematical Olympiad Chinese Olympiad Mathematics Competition. The second chapter describes the manifestation of Cauchy inequality. There are twenty methods of proving Cauchy inequality with skill and representativeness. In this paper, we select 7 kinds of elementary proof methods, which are more convenient for senior high school students to understand. The deformation and extension of Cauchy inequality are discussed in depth, which can expand the application of Cauchy inequality. The relation between Cauchy inequality and n-dimensional inequality chain is also explained in detail. The third chapter classifies and summarizes the methods and techniques of solving Cauchy inequality in IMO and CMO. The fourth chapter is based on the previous chapter of the results of several Cauchy inequality competition for readers to read. The value of this thesis lies in the detailed and systematic study of the relevant knowledge of Cauchy inequality and its application in competitions, which can provide a reference for mathematics teaching and mathematics competition in senior high school.
【学位授予单位】：西北大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：G633.6
[Abstract]:Cauchy inequality is a very important inequality in the elementary field. After the new curriculum reform, Cauchy inequality is included in the elective course of mathematics in senior high school, and this content has become the hot spot of mathematics competition again. As long as we can use this inequality flexibly, many complicated problems can be solved easily. For example, when we use Cauchy inequality to prove inequality, find the most value of function and solve the related problem of triangle, its superiority is obvious. This thesis mainly studies the application of the discrete form of Cauchy inequality in high school Olympiad. The thesis is divided into four chapters. Firstly, the paper expounds the history and development of the IMO (International Mathematical Olympiad International Olympiad Mathematical Competition and the CMO (Chinese Mathematical Olympiad Chinese Olympiad Mathematics Competition. The second chapter describes the manifestation of Cauchy inequality. There are twenty methods of proving Cauchy inequality with skill and representativeness. In this paper, we select 7 kinds of elementary proof methods, which are more convenient for senior high school students to understand. The deformation and extension of Cauchy inequality are discussed in depth, which can expand the application of Cauchy inequality. The relation between Cauchy inequality and n-dimensional inequality chain is also explained in detail. The third chapter classifies and summarizes the methods and techniques of solving Cauchy inequality in IMO and CMO. The fourth chapter is based on the previous chapter of the results of several Cauchy inequality competition for readers to read. The value of this thesis lies in the detailed and systematic study of the relevant knowledge of Cauchy inequality and its application in competitions, which can provide a reference for mathematics teaching and mathematics competition in senior high school.
【学位授予单位】：西北大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：G633.6
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