前   言

       全國(guó)大學(xué)生數(shù)學(xué)競(jìng)賽是一項(xiàng)面向全國(guó)本科生的高水平學(xué)科競(jìng)賽，自2009年10月開(kāi)始至今已經(jīng)成功舉辦了13屆。2019年，我們編寫(xiě)了全國(guó)大學(xué)生數(shù)學(xué)競(jìng)賽輔導(dǎo)書(shū)《全國(guó)大學(xué)生數(shù)學(xué)競(jìng)賽輔導(dǎo)指南（第3版）》。此書(shū)出版后，得到了熱愛(ài)數(shù)學(xué)的廣大學(xué)子的喜愛(ài)，短短3年時(shí)間內(nèi)就重印了11次。

       全國(guó)大學(xué)生數(shù)學(xué)競(jìng)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(jì)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(zhǎng)的舞臺(tái)和與全國(guó)學(xué)子進(jìn)行數(shù)學(xué)交流的平臺(tái)，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(jī)。為了便于擬參加此項(xiàng)賽事的學(xué)生有效備考、迎戰(zhàn)競(jìng)賽，一批參賽輔導(dǎo)書(shū)也相繼問(wèn)世，其中，面向非數(shù)學(xué)專業(yè)學(xué)子的輔導(dǎo)書(shū)相對(duì)較多，為數(shù)學(xué)專業(yè)學(xué)子準(zhǔn)備的參賽輔導(dǎo)書(shū)在市面上并不多見(jiàn)。但隨著數(shù)學(xué)專業(yè)組的參賽學(xué)生越來(lái)越多，該組的競(jìng)賽規(guī)模也越來(lái)越大，數(shù)學(xué)專業(yè)組參賽學(xué)子對(duì)備賽參考資料的需求也日益高漲。為此我們專門著手編寫(xiě)了這本《全國(guó)大學(xué)生數(shù)學(xué)競(jìng)賽輔導(dǎo)（數(shù)學(xué)類）》。

       本書(shū)針對(duì)數(shù)學(xué)專業(yè)的全國(guó)大學(xué)生數(shù)學(xué)競(jìng)賽所編寫(xiě)，它既可以作為參加數(shù)學(xué)競(jìng)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(shí)還可以作為教師的教學(xué)參考用書(shū)。

       本書(shū)共分為4個(gè)部分：第一部分是各屆初賽試題及解答，以便使讀者對(duì)初賽的賽題有一個(gè)全面系統(tǒng)的認(rèn)識(shí)；第二、三部分為考點(diǎn)直擊，編寫(xiě)組結(jié)合考試大綱對(duì)數(shù)學(xué)分析和高等代數(shù)的要求，在每個(gè)專題都進(jìn)行考點(diǎn)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(shí)點(diǎn)和典型例題，先對(duì)考點(diǎn)進(jìn)行分析綜述，然后列舉相關(guān)考點(diǎn)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(kāi)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(shí)獲取競(jìng)賽的最新試題及相關(guān)信息，書(shū)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(shū)是教育部基礎(chǔ)學(xué)科拔尖計(jì)劃2.0國(guó)家級(jí)重點(diǎn)課題（課題號(hào)：2021040，項(xiàng)目名稱：拔尖計(jì)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(xiàng)課題的教師和學(xué)生為本書(shū)的編寫(xiě)作出了許多貢獻(xiàn)，不一一列舉，謹(jǐn)在此一并感謝。

鑒于作者水平有限，書(shū)中難免有不當(dāng)之處，敬請(qǐng)各位專家、讀者批評(píng)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(guó)大學(xué)生數(shù)學(xué)競(jìng)賽初賽（數(shù)學(xué)類，2009年）················ 2

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

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

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

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

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

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

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

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

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

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

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

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

第二部分考點(diǎn)直擊(數(shù)學(xué)分析)

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

      1.1基本知識(shí)點(diǎn)·································································· 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(shí)點(diǎn)································································ 114

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

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

             三、 洛必達(dá)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(shí)點(diǎn)································································ 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(wú)窮級(jí)數(shù)····································································· 167

      4.1基本知識(shí)點(diǎn)································································ 167

              一、 數(shù)項(xiàng)級(jí)數(shù)························································· 167

              二、 函數(shù)項(xiàng)級(jí)數(shù)······················································ 170

              三、 冪級(jí)數(shù)····························································· 174

              四、 傅里葉級(jí)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(shí)點(diǎn)································································ 193

              一、 常見(jiàn)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(shí)點(diǎn)································································ 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(shí)點(diǎn)································································ 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(diǎn)直擊（高等代數(shù)）

第1章多項(xiàng)式········································································· 248

       1.1基本知識(shí)點(diǎn)································································ 248

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

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

       2.1基本知識(shí)點(diǎn)································································ 262

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

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

       3.1基本知識(shí)點(diǎn)································································ 274

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

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

       4.1基本知識(shí)點(diǎn)································································ 286

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

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

       5.1基本知識(shí)點(diǎn)································································ 298

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

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

       6.1基本知識(shí)點(diǎn)································································ 311

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

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

       7.1基本知識(shí)點(diǎn)································································ 321

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

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

       8.1基本知識(shí)點(diǎn)································································ 337

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

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

       9.1基本知識(shí)點(diǎn)································································ 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(guó)大學(xué)生數(shù)學(xué)競(jìng)賽決賽（數(shù)學(xué)類,2010年）··············· 358

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

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

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

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

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

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

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

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

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

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

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

圖書(shū)照片 

【售后客服】

  退換貨，無(wú)法收到書(shū) 書(shū)籍退款等問(wèn)題

                       請(qǐng)微信掃碼聯(lián)系售后客服企業(yè)微信                       

【注意】訂單簽收后超過(guò)一周，無(wú)法退貨。

下單之后如果長(zhǎng)時(shí)間未收到快遞，請(qǐng)?jiān)?個(gè)月之內(nèi)聯(lián)系我們，超過(guò)1個(gè)月的訂單我們無(wú)法辦理退款或補(bǔ)發(fā)。（預(yù)售訂單不包括在內(nèi)）

【關(guān)于物流】
由賽氪發(fā)貨，包郵（首次包郵，退書(shū)者不享受包郵待遇，退款中會(huì)扣掉10元快遞費(fèi)）本店會(huì)根據(jù)派送區(qū)域?yàn)槟x取合適的快遞方式，暫不接受指定快遞。
【發(fā)貨時(shí)間】
下午5點(diǎn)之前下單當(dāng)日發(fā)貨，5點(diǎn)后下單次日發(fā)貨，如遇爆單的情況，會(huì)按下單的先后順序發(fā)貨，有可能會(huì)延至到2個(gè)工作日發(fā)貨，預(yù)售書(shū)籍發(fā)貨以實(shí)際到貨時(shí)間為準(zhǔn)。
【關(guān)于簽收】
簽收商品時(shí)，務(wù)必當(dāng)著派件員的面仔細(xì)檢查書(shū)籍是否破損、漏發(fā)及發(fā)錯(cuò)，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(qǐng)保留好快遞包裝袋拍照留證據(jù)，（無(wú)包裝袋拍照我沒(méi)沒(méi)法處理）并盡快聯(lián)系我們售后客服處理，書(shū)籍破損的訂單協(xié)商處理（注意書(shū)籍破損構(gòu)不成退貨條件，不可退貨，只能補(bǔ)發(fā)或者協(xié)商補(bǔ)償處理，）若執(zhí)意要退貨的我們視為你個(gè)人原因退貨，其寄回賣家的快遞費(fèi)需由買家承擔(dān)，而且退款中會(huì)扣掉10元快遞費(fèi)差價(jià)（差價(jià)就賣家寄給買家的快遞費(fèi)）請(qǐng)小伙伴悉知。

【退換須知】 

已收到書(shū)的同學(xué)需要退換的，其快遞費(fèi)需自理，就是寄回商家的快遞費(fèi)需買家承擔(dān)，等我們收到書(shū)后檢查書(shū)籍沒(méi)有破損，書(shū)寫(xiě)的痕跡，方可退款或換書(shū)，而且退款中會(huì)扣掉10元快遞費(fèi)差價(jià)（差價(jià)就賣家寄給買家的快遞費(fèi)）

凡退換貨要求書(shū)籍外觀完好，沒(méi)有書(shū)寫(xiě)過(guò)，不影響二次銷售的前提可以退換貨，（快遞費(fèi)自理）本店拒絕到付。退貨時(shí)請(qǐng)將商品、贈(zèng)品等一同寄回，并在包裹中注明收件人姓名、電話、訂單號(hào)、退貨原因等信息方便我們核實(shí)。個(gè)人信息提供錯(cuò)誤，姓名，電話，地址，錯(cuò)誤的，未接聽(tīng)電話或長(zhǎng)時(shí)間未取件，導(dǎo)致被退回的，自己買錯(cuò)書(shū)，多買、退換書(shū)的，等等），需補(bǔ)10元快遞費(fèi)方可補(bǔ)發(fā)。

【退款說(shuō)明】

商品退款會(huì)有一個(gè)第三方收取的0.006 %的手續(xù)費(fèi)。

1  ，商品在未發(fā)貨前退款，請(qǐng)聯(lián)系在線客服并選擇合適理由申請(qǐng)，我們會(huì)盡快給您退款。退款會(huì)有一個(gè)第三方收取的0.006 %的手續(xù)費(fèi)。
2  ，商品發(fā)貨途中申請(qǐng)退款，請(qǐng)聯(lián)系售后給你的快遞做攔截，等書(shū)回來(lái)后聯(lián)系我們退款，退款中會(huì)扣掉10元快遞費(fèi)差價(jià)和第三方收取的0.006 %的手續(xù)費(fèi)。

3  ，收到商品后退款的，請(qǐng)聯(lián)系售后客服要退貨地址，其寄回商家的快遞費(fèi)自理（到付拒收）等收到商品后賣家檢查沒(méi)有破損 書(shū)寫(xiě)等痕跡方退款，退款中會(huì)扣掉10元快遞費(fèi)差價(jià)+0.006%的手續(xù)費(fèi)，請(qǐng)各位小伙伴悉知！

友情提示：購(gòu)買商品時(shí)從下單之日起10個(gè)工作日如果收不到快遞，盡快聯(lián)系客服，我們會(huì)第一時(shí)間解決，如果超過(guò)30個(gè)工作日，快遞單號(hào)失效后無(wú)法處理退款，請(qǐng)悉知。

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