【摘要】：柯西不等式在初等領域是一個非常重要的不等式。新課改后柯西不等式被納入高中數(shù)學選修內容,而這一內容也再次成為數(shù)學競賽的熱點,只要我們能靈活的運用此不等式,就能使許多復雜的問題迎刃而解。例如用柯西不等式去證明不等式、求函數(shù)的最值和解三角形的相關問題時,其優(yōu)越性顯而易見。本論文主要研究的是柯西不等式的離散形式在高中奧林匹克競賽中的應用。論文共分為四章,論文首先闡述了IMO (International Mathematical Olympiad國際奧林匹克數(shù)學競賽)和CMO (Chinese Mathematical Olympiad中國奧林匹克數(shù)學競賽)的源遠歷史與發(fā)展情況,第二章敘述了柯西不等式的表現(xiàn)形式,關于柯西不等式有技巧性和代表性的證明方法有二十種,本文選取了其中具有代表性的7種初等的證明方法,這樣更利于高中生的理解,并對柯西不等式的變形與推廣進行了深入的探究說明,這樣可以將柯西不等式的應用范圍加以擴大。還詳細的對柯西不等式與n維不等式鏈的關系進行了說明。第三章主要針對IMO和CMO中關于柯西不等式的賽題進行分類整理,并對解題的方法和技巧進行分析和總結概括。第四章是基于上一章的研究成果編寫的幾道關于柯西不等式的賽題以供讀者賞閱。本論文的價值在于詳細且系統(tǒng)的研究了柯西不等式的相關知識以及在競賽中的應用探究,可以為高中數(shù)學教學和數(shù)學競賽提供參考。
[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.
【學位授予單位】：西北大學
【學位級別】：碩士
【學位授予年份】：2016
【分類號】：G633.6

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