全国大学生数学竞赛是一项面向全国本科生的高水平学科竞赛，自2009年10月开始至今已经成功举办了13届。2019年，我们编写了全国大学生数学竞赛辅导书《全国大学生数学竞赛辅导指南（第3版）》。此书出版后，得到了热爱数学的广大学子的喜爱，短短3年时间内就重印了11次。
全国大学生数学竞赛分为数学专业组（含数学与应用数学、信息与计算科学专业的学生）和非数学专业组，不仅激发了广大青年学子学习数学的兴趣，为其提供了一展数学特长的舞台和与全国学子进行数学交流的平台，也为发现、选拔和培养具有创新精神和创造能力的优秀数学人才提供了契机。为了便于拟参加此项赛事的学生有效备考、迎战竞赛，一批参赛辅导书也相继问世，其中，面向非数学专业学子的辅导书相对较多，为数学专业学子准备的参赛辅导书在市面上并不多见。但随着数学专业组的参赛学生越来越多，该组的竞赛规模也越来越大，数学专业组参赛学子对备赛参考资料的需求也日益高涨。为此我们专门着手编写了这本《全国大学生数学竞赛辅导（数学类）》。
本书针对数学专业的全国大学生数学竞赛所编写，它既可以作为参加数学竞赛的师生的应试教程，也可以为各类高校的大学生学习数学分析和高等代数准备考研提供有效的参考，同时还可以作为教师的教学参考用书。
本书共分为4个部分：第一部分是各届初赛试题及解答，以便使读者对初赛的赛题有一个全面系统的认识；第二、三部分为考点直击，编写组结合考试大纲对数学分析和高等代数的要求，在每个专题都进行考点直击，为了与课堂教学有效衔接，这两部分专门采取分章讲述的形式，每章都设置了基本知识点和典型例题，先对考点进行分析综述，然后列举相关考点的出题方式和解题思路，这样安排有利于考生有效地提高数学水平；第四部分则是各届决赛试题及解答，借此开阔读者视野。为了拓展学生的思路，在题目解答环节，许多题目安排了多种巧妙的解题方式。为便于读者及时获取竞赛的最新试题及相关信息，书中还设置了许多二维码，便于读者扫码了解相关详情，不遗漏任何重要信息。
本书是教育部基础学科拔尖计划2.0国家级重点课题（课题号：2021040，项目名称：拔尖计划2.0中数学人才的选拔与培养）的成果之一。参与此项课题的教师和学生为本书的编写作出了许多贡献，不一一列举，谨在此一并感谢。
鉴于作者水平有限，书中难免有不当之处，敬请各位专家、读者批评指正，便于以后改版修订。
张天德
2022年6月
第一届全国大学生数学竞赛初赛（数学类，2009年）················ 2
第二届全国大学生数学竞赛初赛（数学类，2010年）················ 7
第三届全国大学生数学竞赛初赛（数学类，2011年）·············· 11
第四届全国大学生数学竞赛初赛（数学类，2012年）·············· 17
第五届全国大学生数学竞赛初赛（数学类，2013年）·············· 21
第六届全国大学生数学竞赛初赛（数学类，2014年）·············· 25
第七届全国大学生数学竞赛初赛（数学类，2015年）·············· 29
第八届全国大学生数学竞赛初赛（数学类，2016年）·············· 33
第九届全国大学生数学竞赛初赛（数学类，2017年）·············· 37
第十届全国大学生数学竞赛初赛（数学类，2018年）·············· 42
第十一届全国大学生数学竞赛初赛（数学类，2019年）··········· 46
第十二届全国大学生数学竞赛初赛（数学类，2020年）··········· 57
第十三届全国大学生数学竞赛初赛（数学类，2021年）··········· 67
第1章极限与连续···································································· 78
1.1基本知识点·································································· 78
一、 极限·································································· 78
二、 连续·································································· 82
1.2典型例题····································································· 83
第2章一元函数微分学··························································· 114
2.1基本知识点································································ 114
一、 导数与微分······················································ 114
二、 微分中值定理·················································· 115
三、 洛必达法则······················································ 116
2.2典型例题···································································· 116
第3章一元函数积分学··························································· 130
3.1基本知识点································································ 130
一、 不定积分························································· 130
二、 定积分····························································· 132
三、 反常积分························································· 136
3.2典型例题···································································· 140
第4章无穷级数····································································· 167
4.1基本知识点································································ 167
一、 数项级数························································· 167
二、 函数项级数······················································ 170
三、 幂级数····························································· 174
四、 傅里叶级数······················································ 175
4.2典型例题···································································· 176
第5章多元函数微分学··························································· 193
5.1基本知识点································································ 193
一、 常见的几种关系··············································· 193
二、 泰勒公式、隐函数定理···································· 197
三、 二元函数的极值··············································· 199
5.2典型例题···································································· 200
第6章含参变量积分······························································ 210
6.1基本知识点································································ 210
一、 含参变量的定积分的性质································· 210
二、 含参变量的反常积分········································ 211
6.2典型例题···································································· 215
第7章多元函数积分学··························································· 224
7.1基本知识点································································ 224
一、 重积分····························································· 224
二、 曲线积分························································· 226
三、 曲面积分························································· 227
7.2典型例题···································································· 229
第1章多项式········································································· 248
1.1基本知识点································································ 248
1.2典型例题···································································· 253
第2章行列式········································································· 262
2.1基本知识点································································ 262
2.2典型例题···································································· 265
第3章线性方程组·································································· 274
3.1基本知识点································································ 274
3.2典型例题···································································· 277
第4章矩阵············································································ 286
4.1基本知识点································································ 286
4.2典型例题···································································· 289
第5章二次型········································································· 298
5.1基本知识点································································ 298
5.2典型例题···································································· 299
第6章线性空间····································································· 311
6.1基本知识点································································ 311
6.2典型例题···································································· 313
第7章线性变换····································································· 321
7.1基本知识点································································ 321
7.2典型例题···································································· 323
第8章λ-矩阵······································································· 337
8.1基本知识点································································ 337
8.2典型例题···································································· 338
第9章欧几里得空间······························································ 346
9.1基本知识点································································ 346
9.2典型例题···································································· 348
第一届全国大学生数学竞赛决赛（数学类,2010年）··············· 358
第二届全国大学生数学竞赛决赛（数学类，2011年）············ 365
第三届全国大学生数学竞赛决赛（数学类,2012年）··············· 370
第四届全国大学生数学竞赛决赛（数学类，2013年）············ 376
第五届全国大学生数学竞赛决赛（数学类,2014年）··············· 381
第六届全国大学生数学竞赛决赛（数学类,2015年）··············· 385
第七届全国大学生数学竞赛决赛（数学类，2016年）············ 389
第八届全国大学生数学竞赛决赛（数学类,2017年）··············· 393
第九届全国大学生数学竞赛决赛（数学类,2018年）··············· 398
第十届全国大学生数学竞赛决赛（数学类，2019年）············ 404
第十一届全国大学生数学竞赛决赛（数学类,2021年）············ 409
第十二届全国大学生数学竞赛决赛（数学类，2021年）········· 417
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