前   言

       全國大學(xué)生數(shù)學(xué)競賽是一項面向全國本科生的高水平學(xué)科競賽，自2009年10月開始至今已經(jīng)成功舉辦了13屆。2019年，我們編寫了全國大學(xué)生數(shù)學(xué)競賽輔導(dǎo)書《全國大學(xué)生數(shù)學(xué)競賽輔導(dǎo)指南（第3版）》。此書出版后，得到了熱愛數(shù)學(xué)的廣大學(xué)子的喜愛，短短3年時間內(nèi)就重印了11次。

       全國大學(xué)生數(shù)學(xué)競賽分為數(shù)學(xué)專業(yè)組（含數(shù)學(xué)與應(yīng)用數(shù)學(xué)、信息與計算科學(xué)專業(yè)的學(xué)生）和非數(shù)學(xué)專業(yè)組，不僅激發(fā)了廣大青年學(xué)子學(xué)習(xí)數(shù)學(xué)的興趣，為其提供了一展數(shù)學(xué)特長的舞臺和與全國學(xué)子進行數(shù)學(xué)交流的平臺，也為發(fā)現(xiàn)、選拔和培養(yǎng)具有創(chuàng)新精神和創(chuàng)造能力的優(yōu)秀數(shù)學(xué)人才提供了契機。為了便于擬參加此項賽事的學(xué)生有效備考、迎戰(zhàn)競賽，一批參賽輔導(dǎo)書也相繼問世，其中，面向非數(shù)學(xué)專業(yè)學(xué)子的輔導(dǎo)書相對較多，為數(shù)學(xué)專業(yè)學(xué)子準(zhǔn)備的參賽輔導(dǎo)書在市面上并不多見。但隨著數(shù)學(xué)專業(yè)組的參賽學(xué)生越來越多，該組的競賽規(guī)模也越來越大，數(shù)學(xué)專業(yè)組參賽學(xué)子對備賽參考資料的需求也日益高漲。為此我們專門著手編寫了這本《全國大學(xué)生數(shù)學(xué)競賽輔導(dǎo)（數(shù)學(xué)類）》。

       本書針對數(shù)學(xué)專業(yè)的全國大學(xué)生數(shù)學(xué)競賽所編寫，它既可以作為參加數(shù)學(xué)競賽的師生的應(yīng)試教程，也可以為各類高校的大學(xué)生學(xué)習(xí)數(shù)學(xué)分析和高等代數(shù)準(zhǔn)備考研提供有效的參考，同時還可以作為教師的教學(xué)參考用書。

       本書共分為4個部分：第一部分是各屆初賽試題及解答，以便使讀者對初賽的賽題有一個全面系統(tǒng)的認(rèn)識；第二、三部分為考點直擊，編寫組結(jié)合考試大綱對數(shù)學(xué)分析和高等代數(shù)的要求，在每個專題都進行考點直擊，為了與課堂教學(xué)有效銜接，這兩部分專門采取分章講述的形式，每章都設(shè)置了基本知識點和典型例題，先對考點進行分析綜述，然后列舉相關(guān)考點的出題方式和解題思路，這樣安排有利于考生有效地提高數(shù)學(xué)水平；第四部分則是各屆決賽試題及解答，借此開闊讀者視野。為了拓展學(xué)生的思路，在題目解答環(huán)節(jié)，許多題目安排了多種巧妙的解題方式。為便于讀者及時獲取競賽的最新試題及相關(guān)信息，書中還設(shè)置了許多二維碼，便于讀者掃碼了解相關(guān)詳情，不遺漏任何重要信息。

       本書是教育部基礎(chǔ)學(xué)科拔尖計劃2.0國家級重點課題（課題號：2021040，項目名稱：拔尖計劃2.0中數(shù)學(xué)人才的選拔與培養(yǎng)）的成果之一。參與此項課題的教師和學(xué)生為本書的編寫作出了許多貢獻，不一一列舉，謹(jǐn)在此一并感謝。

鑒于作者水平有限，書中難免有不當(dāng)之處，敬請各位專家、讀者批評指正，便于以后改版修訂。

張?zhí)斓?nbsp;   

2022年6月

目錄

第一部分初賽試題及參考解答

第一屆全國大學(xué)生數(shù)學(xué)競賽初賽（數(shù)學(xué)類，2009年）················ 2

第二屆全國大學(xué)生數(shù)學(xué)競賽初賽（數(shù)學(xué)類，2010年）················ 7

第三屆全國大學(xué)生數(shù)學(xué)競賽初賽（數(shù)學(xué)類，2011年）·············· 11

第四屆全國大學(xué)生數(shù)學(xué)競賽初賽（數(shù)學(xué)類，2012年）·············· 17

第五屆全國大學(xué)生數(shù)學(xué)競賽初賽（數(shù)學(xué)類，2013年）·············· 21

第六屆全國大學(xué)生數(shù)學(xué)競賽初賽（數(shù)學(xué)類，2014年）·············· 25

第七屆全國大學(xué)生數(shù)學(xué)競賽初賽（數(shù)學(xué)類，2015年）·············· 29

第八屆全國大學(xué)生數(shù)學(xué)競賽初賽（數(shù)學(xué)類，2016年）·············· 33

第九屆全國大學(xué)生數(shù)學(xué)競賽初賽（數(shù)學(xué)類，2017年）·············· 37

第十屆全國大學(xué)生數(shù)學(xué)競賽初賽（數(shù)學(xué)類，2018年）·············· 42

第十一屆全國大學(xué)生數(shù)學(xué)競賽初賽（數(shù)學(xué)類，2019年）··········· 46

第十二屆全國大學(xué)生數(shù)學(xué)競賽初賽（數(shù)學(xué)類，2020年）··········· 57

第十三屆全國大學(xué)生數(shù)學(xué)競賽初賽（數(shù)學(xué)類，2021年）··········· 67

第二部分考點直擊(數(shù)學(xué)分析)

第1章極限與連續(xù)···································································· 78

      1.1基本知識點·································································· 78

             一、 極限·································································· 78

             二、 連續(xù)·································································· 82

      1.2典型例題····································································· 83

第2章一元函數(shù)微分學(xué)··························································· 114

      2.1基本知識點································································ 114

             一、 導(dǎo)數(shù)與微分······················································ 114

             二、 微分中值定理·················································· 115

             三、 洛必達法則······················································ 116

      2.2典型例題···································································· 116

第3章一元函數(shù)積分學(xué)··························································· 130

      3.1基本知識點································································ 130

              一、 不定積分························································· 130

              二、 定積分····························································· 132

              三、 反常積分························································· 136

      3.2典型例題···································································· 140

第4章無窮級數(shù)····································································· 167

      4.1基本知識點································································ 167

              一、 數(shù)項級數(shù)························································· 167

              二、 函數(shù)項級數(shù)······················································ 170

              三、 冪級數(shù)····························································· 174

              四、 傅里葉級數(shù)······················································ 175

      4.2典型例題···································································· 176

第5章多元函數(shù)微分學(xué)··························································· 193

      5.1基本知識點································································ 193

              一、 常見的幾種關(guān)系··············································· 193

              二、 泰勒公式、隱函數(shù)定理···································· 197

              三、 二元函數(shù)的極值··············································· 199

      5.2典型例題···································································· 200

第6章含參變量積分······························································ 210

      6.1基本知識點································································ 210

              一、 含參變量的定積分的性質(zhì)································· 210

              二、 含參變量的反常積分········································ 211

      6.2典型例題···································································· 215

第7章多元函數(shù)積分學(xué)··························································· 224

      7.1基本知識點································································ 224

              一、 重積分····························································· 224

              二、 曲線積分························································· 226

              三、 曲面積分························································· 227

       7.2典型例題···································································· 229

第三部分考點直擊（高等代數(shù)）

第1章多項式········································································· 248

       1.1基本知識點································································ 248

       1.2典型例題···································································· 253

第2章行列式········································································· 262

       2.1基本知識點································································ 262

       2.2典型例題···································································· 265

第3章線性方程組·································································· 274

       3.1基本知識點································································ 274

       3.2典型例題···································································· 277

第4章矩陣············································································ 286

       4.1基本知識點································································ 286

       4.2典型例題···································································· 289

第5章二次型········································································· 298

       5.1基本知識點································································ 298

       5.2典型例題···································································· 299

第6章線性空間····································································· 311

       6.1基本知識點································································ 311

       6.2典型例題···································································· 313

第7章線性變換····································································· 321

       7.1基本知識點································································ 321

       7.2典型例題···································································· 323

第8章λ-矩陣······································································· 337

       8.1基本知識點································································ 337

       8.2典型例題···································································· 338

第9章歐幾里得空間······························································ 346

       9.1基本知識點································································ 346

       9.2典型例題···································································· 348

第四部分決賽試題及參考解答

第一屆全國大學(xué)生數(shù)學(xué)競賽決賽（數(shù)學(xué)類,2010年）··············· 358

第二屆全國大學(xué)生數(shù)學(xué)競賽決賽（數(shù)學(xué)類，2011年）············ 365

第三屆全國大學(xué)生數(shù)學(xué)競賽決賽（數(shù)學(xué)類,2012年）··············· 370

第四屆全國大學(xué)生數(shù)學(xué)競賽決賽（數(shù)學(xué)類，2013年）············ 376

第五屆全國大學(xué)生數(shù)學(xué)競賽決賽（數(shù)學(xué)類,2014年）··············· 381

第六屆全國大學(xué)生數(shù)學(xué)競賽決賽（數(shù)學(xué)類,2015年）··············· 385

第七屆全國大學(xué)生數(shù)學(xué)競賽決賽（數(shù)學(xué)類，2016年）············ 389

第八屆全國大學(xué)生數(shù)學(xué)競賽決賽（數(shù)學(xué)類,2017年）··············· 393

第九屆全國大學(xué)生數(shù)學(xué)競賽決賽（數(shù)學(xué)類,2018年）··············· 398

第十屆全國大學(xué)生數(shù)學(xué)競賽決賽（數(shù)學(xué)類，2019年）············ 404

第十一屆全國大學(xué)生數(shù)學(xué)競賽決賽（數(shù)學(xué)類,2021年）············ 409

第十二屆全國大學(xué)生數(shù)學(xué)競賽決賽（數(shù)學(xué)類，2021年）········· 417

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簽收商品時，務(wù)必當(dāng)著派件員的面仔細(xì)檢查書籍是否破損、漏發(fā)及發(fā)錯，并仔細(xì)檢查外包裝是否完好，若發(fā)現(xiàn)快遞外包裝袋破損請保留好快遞包裝袋拍照留證據(jù)，（無包裝袋拍照我沒沒法處理）并盡快聯(lián)系我們售后客服處理，書籍破損的訂單協(xié)商處理（注意書籍破損構(gòu)不成退貨條件，不可退貨，只能補發(fā)或者協(xié)商補償處理，）若執(zhí)意要退貨的我們視為你個人原因退貨，其寄回賣家的快遞費需由買家承擔(dān)，而且退款中會扣掉10元快遞費差價（差價就賣家寄給買家的快遞費）請小伙伴悉知。

【退換須知】 

已收到書的同學(xué)需要退換的，其快遞費需自理，就是寄回商家的快遞費需買家承擔(dān)，等我們收到書后檢查書籍沒有破損，書寫的痕跡，方可退款或換書，而且退款中會扣掉10元快遞費差價（差價就賣家寄給買家的快遞費）

凡退換貨要求書籍外觀完好，沒有書寫過，不影響二次銷售的前提可以退換貨，（快遞費自理）本店拒絕到付。退貨時請將商品、贈品等一同寄回，并在包裹中注明收件人姓名、電話、訂單號、退貨原因等信息方便我們核實。個人信息提供錯誤，姓名，電話，地址，錯誤的，未接聽電話或長時間未取件，導(dǎo)致被退回的，自己買錯書，多買、退換書的，等等），需補10元快遞費方可補發(fā)。

【退款說明】

商品退款會有一個第三方收取的0.006 %的手續(xù)費。

1  ，商品在未發(fā)貨前退款，請聯(lián)系在線客服并選擇合適理由申請，我們會盡快給您退款。退款會有一個第三方收取的0.006 %的手續(xù)費。
2  ，商品發(fā)貨途中申請退款，請聯(lián)系售后給你的快遞做攔截，等書回來后聯(lián)系我們退款，退款中會扣掉10元快遞費差價和第三方收取的0.006 %的手續(xù)費。

3  ，收到商品后退款的，請聯(lián)系售后客服要退貨地址，其寄回商家的快遞費自理（到付拒收）等收到商品后賣家檢查沒有破損 書寫等痕跡方退款，退款中會扣掉10元快遞費差價+0.006%的手續(xù)費，請各位小伙伴悉知！

友情提示：購買商品時從下單之日起10個工作日如果收不到快遞，盡快聯(lián)系客服，我們會第一時間解決，如果超過30個工作日，快遞單號失效后無法處理退款，請悉知。

電子+紙質(zhì)的產(chǎn)品不享受以上退貨待遇，（例如題庫的課程送書）都是不可以退款的。