AMC10美国数学竞赛讲义

1、AMC中的数论问题1：Remembertheprimebetween1to100：23571113171923293137414347535961677173798389912：Perfectnumber:LetPistheprimenumber.if21pisalsotheprimenumber.then1(21)2ppistheperfectnumber.Forexample:6,28,496.3:Let,0nabcaisthreedigitalinteger.if333nabcThenthenumberniscalledDaffodilsnumber.Thereareonlyfournumbers:153370371407Let,0nabcdaisfourdigitalinteger.if4444dnabcThenthenumberniscalledRosesnumber.Thereareonlythreenumbers:1634820894744：TheFundamentalTheoremofArithmeticEverynaturalnumber。

2、canbewrittenasaproductofprimesuniquelyuptoorder.=∏piriki=15：Supposethataandbareintegerswithb=0.Thenthereexistsuniqueintegersqandrsuchthat0≤r|b|anda=bq+r.6：(1)GreatestCommonDivisor:Letgcd(a,b)=max{d∈Z:d|aandd|b}.Foranyintegersaandb,wehavegcd(a,b)=gcd(b,a)=gcd(±a,±b)=gcd(a,b−a)=gcd(a,b+a).Forexample:gcd(150,60)=gcd(60,30)=gcd(30,0)=30(2)Leastcommonmultiple:Letlcm(a,b)=min{d∈Z:a|dandb|d}.(3)Wehavethat:ab=gcd(a,b)lcm(a,b)7：CongruencemodulonIf,0abmqm,thenwecallacongruencebmodulomandwerewritemodabm。

3、.(1)Assumea,b,c,d,m,k∈Z(k0,m≠0).Ifa≡bmodm,c≡dmodmthenwehavemodacbdm,modacbdm,modkkabm(2)Theequationax≡b(modm)hasasolutionifandonlyifgcd(a,m)dividesb.8：Howtofindtheunitdigitofsomespecialintegers(1)Howmanyzeroattheendof!nForexample,when100n,LetNbethenumberzeroattheendof100!then10010010020424525125N(2),,anZFindtheunitdigitna.Forexample,when100,3na9：Palindrome,suchas83438,isanumberthatremainsthesamewhenitsdigitsarereversed.Therearesomenumbernotonlypalindromebut112=1。

4、21，222=484，114=14641(1)Somespecialpalindromenthat2nisalsopalindrome.Forexample:222221111121111123211111123432111111111112345678987654321(2)Howtocreateapalindrome?Almostintegerplusthenumberofitsreverseddigitsandrepeatitagainandagain.Thenwegetapalindrome.Forexample:87781651655617267266271353135335314884ButwhetheranyintegerhasthisPropertyhasyettoprove(3)Thepalindromeequationmeansthatequationfromlefttorightandrighttoleftitallsetup.Forexample：1242242112231132211121241388888831421211。

5、Letabandcdearetwodigitalandthreedigitalintegers.Ifthedigitssatisfythe,,9acbedced,thenabcdeedcba.10:FeaturesofanintegerdivisiblebysomeprimenumberIfniseven，then2|n一个整数n的所有位数上的数字之和是3（或者9）的倍数，则n被3（或者9）整除一个整数n的尾数是零，则n被5整除一个整数n的后三位与截取后三位的数值的差被7、11、13整除，则n被7、11、13整除一个整数n的最后两位数被4整除，则n被4整除一个整数n的最后三位数被8整除，则n被8整除一个整数n的奇数位之和与偶数位之和的差被11整除，则n被11整除11.ThenumberTheoreticfunctionsIf312123trrrrtnpppp(1)12()#0:|(1)(1)(1)tnaanrrr(2)12222111222|()(1)(1)(1)trrrtttannappppppppp(3)。

6、11221111122()#:,gcd(,)1()()()ttrrrrrrttnaNananppppppForexample:2(12)(23)(21)(11)622(12)(23)(122)(13)2822(12)(23)(22)(31)4Exercise1.Thesumsofthreewholenumberstakeninpairsare12,17,and19.Whatisthemiddlenumber?(A)4(B)5(C)6(D)7(E)83.Forthepositiveintegern,letndenotethesumofallthepositivedivisorsofnwiththeexceptionofnitself.Forexample,4=1+2=3and12=1+2+3+4+6=16.Whatis6?(A)6(B)12(C)24(D)32(E)368.Whatisthesumofallintegersolutionsto21(x-2)25?(A)10(B)12(C)15(D)19(E)510H。

7、owmanyorderedpairsofpositiveintegers(M,N)satisfytheequation6=6MN(A)6(B)7(C)8(D)9(E)101.Letaandbberelativelyprimeintegerswith0aband333-73=(-)3abab.Whatis-ab?(A)1(B)2(C)3(D)4(E)515．Thefigures123,,FFFand4Fshownarethefirstinasequenceoffigures.For3n,nFisconstructedfrom-1nFbysurroundingitwithasquareandplacingonemorediamondoneachsideofthenewsquarethan-1nFhadoneachsideofitsoutsidesquare.Forexample,figure3Fhas13diamonds.Howmanydiamondsarethereinfigure20F?18.Positiveintegersa,b,andcarerandomlyandindepend。

8、entlyselectedwithreplacementfromtheset{1,2,3,…,2010}.Whatistheprobabilitythatabcabaisdivisibleby3?(A)13(B)2981(C)3181(D)1127(E)132724.Let,abandcbepositiveintegerswithabcsuchthat222-b-c+=2011aaband222+3b+3c-3-2-2=-1997aabacbc.Whatisa?(A)249(B)250(C)251(D)252(E)2535.Inmultiplyingtwopositiveintegersaandb,Ronreversedthedigitsofthetwo-digitnumbera.Hiserroneousproductwas161.Whatisthecorrectvalueoftheproductofaandb?(A)116(B)161(C)204(D)214(E)22423.Whatisthehundredsdigitof20112011?(A)1(B)4(C)5(D)6(E)9。

9、9.Apalindrome,suchas83438,isanumberthatremainsthesamewhenitsdigitsarereversed.Thenumbersxandx+32arethree-digitandfour-digitpalindromes,respectively.Whatisthesumofthedigitsofx?(A)20(B)21(C)22(D)23(E)2421.Thepolynomial322010xaxbxhasthreepositiveintegerzeros.Whatisthesmallestpossiblevalueofa?(A)78(B)88(C)98(D)108(E)11824.Thenumberobtainedfromthelasttwononzerodigitsof90!Isequalton.Whatisn?(A)12(B)32(C)48(D)52(E)6825.Jimstartswithapositiveintegernandcreatesasequenceofnumbers.Eachsuccessivenumberis。

10、obtainedbysubtractingthelargestpossibleintegersquarelessthanorequaltothecurrentnumberuntilzeroisreached.Forexample,ifJimstartswithn=55,thenhissequencecontain5numbers:5555-72=66-22=22-12=11-12=0LetNbethesmallestnumberforwhichJim’ssequencehas8numbers.WhatistheunitsdigitofN?(A)1(B)3(C)5(D)7(E)921．Whatistheremainderwhen01220093+3+3++3isdividedby8?(A)0(B)1(C)2(D)4(E)65．Whatisthesumofthedigitsofthesquareof111,111,111?(A)18(B)27(C)45(D)63(E)8125．For0k,letk2ABC222I10064333ADaSaROAa24。