摘要数学竞赛已成为国际公认的教育活动，从小学、中学到大学，参赛人数之多、范围之广、试题难度之高等均不比奥运会逊色.在对国际数学奥林匹克(International Mathematical Olympiad，简称IMO)试题的统计表明，试题范围主要稳定在数论、组合数学、数列、不等式、函数方程和几何等.与通常题目相比，这些题更多的是在考察一个人的数学思维，以及数学技巧.作为数学竞赛中的一个重要组成部分，组合数学源远流长，在数学竞赛题中出现组合问题往往表达形式上简单明了，然后求解这类问题却需要一定的技巧，这也使得此类问题收到广泛的关注.46827

本文首先介绍了容斥原理、生成函数和递推关系的基本定义和定理.并通过实例分析法，通过对历届IMO，中国数学奥林匹克（Chinese Mathematical Olympiad，简称CMO）试题进行整理，选取例题进行证明求解，更好的说明以上组合方法的运用.

毕业论文关键词：组合数学； 容斥原理； 递推关系； 生成函数； 奥林匹克数学

Abstract Math competition has become an internationally recognized education activities, from elementary school, middle school to university, the number of participating more, wide scope, item difficulty higher are not as inferior as the Olympic Games. The International Mathematical Olympiad（Abbreviation IMO） statistics show that test item scope mainly stable in number theory, combinatorial mathematics, series, inequality, functional equations and so on. Compared with the usually topic, the topic is more in one mathematical thinking, and mathematical skills. As an important part in the math competition, has a long history in the combinatorial mathematics, in the math contest questions combination problems often simple expression form, and then solve the problem but needs certain skills, it also makes such questions received widespread attention. 

This article first introduces the principle of a class, the generating function and the basic definitions and theorems of recursive relations. And through the instance analysis, through the successive IMO, Chinese Mathematical Olympiad（Abbreviation CMO）, try to arrange, selection of examples to prove, better use of the above combination method.

Keyword  Combinatorial mathematics; inclusion-exclusion principle; recurrence relations; generating function; Mathematical Olympiad

目  录

1、引言 1

1.1、 中外奥赛概述 1

1.2、 奥数中的组合数学方法 1

1.3、 研究现状 1

2、奥赛中的组合数学方法介绍 2

2.1、 容斥原理法 2

2.2、 递推公式法 4

2.3、 生成函数方法 8

3、奥赛中的组合数学方法应用 12

3.1、 容斥原理法实例分析 12

3.2、 递推公式法实例分析 15

3.3、 生成函数方法实例分析 20

4、总结 22

参考文献 23

致 谢 23

1、