AMC10美国数学竞赛讲义.docx
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AMC10美国数学竞赛讲义.docx
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AMC中的数论问题
1:
Remembertheprimebetween1to100:
235711131719232931374143475359616771
7379838991
2:
Perfectnumber:
LetPistheprimenumber.ifisalsotheprimenumber.thenistheperfectnumber.Forexample:
6,28,496.
3:
Letisthreedigitalinteger.if
ThenthenumberiscalledDaffodilsnumber.Thereareonlyfournumbers:
153370371407
Letisfourdigitalinteger.if
ThenthenumberiscalledRosesnumber.Thereareonlythreenumbers:
163482089474
4:
TheFundamentalTheoremofArithmetic
Everynaturalnumberncanbewrittenasaproductofprimesuniquelyuptoorder.
n=i=1kpiri
5:
Supposethataandbareintegerswithb=0.Thenthereexistsuniqueintegersqandrsuchthat0≤r<|b|anda=bq+r.
6:
(1)GreatestCommonDivisor:
Letgcd(a,b)=max{d∈Z:
d|aandd|b}.
Foranyintegersaandb,wehave
gcd(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:
Congruencemodulon
If,thenwecallacongruencebmodulomandwerewrite.
(1)Assumea,b,c,d,m,k∈Z(k>0,m≠0).
Ifa≡bmodm,c≡dmodmthenwehave
,
(2)Theequationax≡b(modm)hasasolutionifandonlyifgcd(a,m)dividesb.
8:
Howtofindtheunitdigitofsomespecialintegers
(1)Howmanyzeroattheendof
Forexample,when,LetNbethenumberzeroattheendofthen
(2)Findtheunitdigit.Forexample,when
9:
Palindrome,suchas83438,isanumberthatremainsthesamewhenitsdigitsarereversed.
Therearesomenumbernotonlypalindromebut112=121,222=484,114=14641
(1)Somespecialpalindromethatisalsopalindrome.Forexample:
(2)Howtocreateapalindrome?
Almostintegerplusthenumberofitsreverseddigitsandrepeatitagainandagain.Thenwegetapalindrome.Forexample:
ButwhetheranyintegerhasthisPropertyhasyettoprove
(3)Thepalindromeequationmeansthatequationfromlefttorightandrighttoleftitallsetup.
Forexample:
Letandaretwodigitalandthreedigitalintegers.Ifthedigitssatisfythe
then.
10:
Featuresofanintegerdivisiblebysomeprimenumber
Ifniseven,then2|n
一个整数的所有位数上的数字之和是3(或者9)的倍数,则被3(或者9)整除
一个整数的尾数是零,则被5整除
一个整数的后三位与截取后三位的数值的差被7、11、13整除,
则被7、11、13整除
一个整数的最后两位数被4整除,则被4整除
一个整数的最后三位数被8整除,则被8整除
一个整数的奇数位之和与偶数位之和的差被11整除,则被11整除
11.ThenumberTheoreticfunctions
If
(1)
(2)
(3)
Forexample:
Exercise
1.Thesumsofthreewholenumberstakeninpairsare12,17,and19.Whatisthemiddlenumber?
(A)4 (B)5 (C)6 (D)7 (E)8
3.Forthepositiveintegern,let
(A)6 (B)12 (C)24 (D)32 (E)36
8.Whatisthesumofallintegersolutionsto?
(A)10 (B)12 (C)15 (D)19 (E)5
10Howmanyorderedpairsofpositiveintegers(M,N)satisfytheequation
(A)6 (B)7 (C)8 (D)9 (E)10
1.Letandberelativelyprimeintegerswithand.Whatis?
(A)1 (B)2 (C)3 (D)4 (E)5
15.Thefiguresandshownarethefirstinasequenceoffigures.For,isconstructedfrombysurroundingitwithasquareandplacingonemorediamondoneachsideofthenewsquarethanhadoneachsideofitsoutsidesquare.Forexample,figurehas13diamonds.Howmanydiamondsarethereinfigure?
18.Positiveintegersa,b,andcarerandomlyandindependentlyselectedwithreplacementfromtheset{1,2,3,…,2010}.Whatistheprobabilitythatisdivisibleby3?
(A) (B) (C) (D) (E)
24.Letandbepositiveintegerswithsuchthatand.Whatis?
(A)249 (B)250 (C)251 (D)252 (E)253
5.Inmultiplyingtwopositiveintegersaandb,Ronreversedthedigitsofthetwo-digitnumbera.Hiserroneousproductwas161.Whatisthecorrectvalueoftheproductofaandb?
(A)116 (B)161 (C)204 (D)214 (E)224
23.Whatisthehundredsdigitof?
(A)1 (B)4 (C)5 (D)6 (E)9
9.Apalindrome,suchas83438,isanumberthatremainsthesamewhenitsdigitsarereversed.Thenumbersxandx+32arethree-digitandfour-digitpalindromes,respectively.Whatisthesumofthedigitsofx?
(A)20 (B)21 (C)22 (D)23 (E)24
21.Thepolynomialhasthreepositiveintegerzeros.Whatisthesmallestpossiblevalueofa?
(A)78 (B)88 (C)98 (D)108 (E)118
24.Thenumberobtainedfromthelasttwononzerodigitsof90!
Isequalton.Whatisn?
(A)12 (B)32 (C)48 (D)52 (E)68
25.Jimstartswithapositiveintegernandcreatesasequenceofnumbers.Eachsuccessivenumberisobtainedbysubtractingthelargestpossibleintegersquarelessth
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