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全美数学竞赛USA-AMC_12-AHSME-2005-44

发布时间:2014-01-15 16:03:41  

USA

AMC12/AHSME2005

A1Twois10%ofxand20%ofy.Whatisx?y?

(A)1(B)2(C)5(D)10(E)202Theequations2x+7=3andbx?10=?2havethesamesolutionforx.Whatisthevalueofb?

(A)?8(B)?4(C)?2(D)4(E)83Arectanglewithadiagonaloflengthxistwiceaslongasitiswide.Whatistheareaoftherectangle?

12(A)x22(B)x2(C)1x(D)x232(E)x

4Astorenormallysellswindowsat$100each.Thisweekthestoreiso?eringonefreewindowforeachpurchaseoffour.DaveneedssevenwindowsandDougneedseightwindows.Howmanydollarswilltheysaveiftheypurchasethewindowstogetherratherthanseparately?

(A)100(B)200(C)300(D)400(E)5005Theaverage(mean)of20numbersis30,andtheaverageof30othernumbersis20.Whatistheaverageofall50numbers?

(A)23(B)24(C)25(D)26(E)276JoshandMikelive13milesapart.Yesterday,JoshstartedtoridehisbicycletowardMike’shouse.AlittlelaterMikestartedtoridehisbicycletowardJosh’shouse.Whentheymet,JoshhadriddenfortwicethelengthoftimeasMikeandatfour-?fthsofMike’srate.HowmanymileshadMikeriddenwhentheymet?

(A)4(B)5(C)6(D)7(E)87SquareEFGHisinsidethesquareABCDsothateachsideofEF√GHcanbeextendedtopassthroughavertexofABCD.SquareABCDhassidelengthandBE=1.WhatistheareaoftheinnersquareEFGH?This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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AMC12/AHSME2005

A

E

H

F

DBC

(A)25(B)32(C)36(D)40(E)428LetA,M,andCbedigitswith

(100A+10M+C)(A+M+C)=2005.

WhatisA?

(A)1(B)2(C)3(D)4(E)59Therearetwovaluesofaforwhichtheequation4x2+ax+8x+9=0hasonlyonesolutionforx.Whatisthesumofthesevaluesofa?

(A)?16(B)?8(C)0(D)8(E)2010Awoodencubenunitsonasideispaintedredonallsixfacesandthencutinton3unit

cubes.Exactlyone-fourthofthetotalnumberoffacesoftheunitcubesarered.Whatisn?

(A)3(B)4(C)5(D)6(E)711Howmanythree-digitnumberssatisfythepropertythatthemiddledigitistheaverageof

the?rstandthelastdigits?

(A)41(B)42(C)43(D)44(E)4512AlinepassesthroughA(1,1)andB(100,1000).Howmanyotherpointswithintegercoordi-

natesareonthelineandstrictlybetweenAandB?

(A)0(B)2(C)3(D)8(E)913Theregular5-pointstarABCDEisdrawnandineachvertex,thereisanumber.Each

A,B,C,D,andEarechosensuchthatall5ofthemcamefromset3,5,6,7,9.Eachletterisadi?erentnumber(soonepossiblewaysisA=3,B=5,C=6,D=7,E=9).LetABbethesumThis?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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AMC12/AHSME2005

ofthenumbersinAandB.IfAB,BC,CD,DE,andEAformanarithmeticsequence(notnecessarilythisorder),?ndthevalueofCD.

(A)9(B)10(C)11(D)12(E)1314Onastandarddieoneofthedotsisremovedatrandomwitheachdotequallylikelytobe

chosen.Thedieisthenrolled.Whatistheprobabilitythatthetopfacehasanoddnumberofdots?

(A)5(B)10(C)1(D)11(E)615LetbeadiameterofacircleandCbeapointonwith2·AC=BC.LetDandE

bepointsonthecirclesuchthat⊥andisaseconddiameter.Whatistheratiooftheareaof??DCEtotheareaof??ABD?

A(A)1(B)1(C)1(D)1(E)216Threecirclesofradiussaredrawninthe?rstquadrantofthexy-plane.The?rstcircleis

tangenttobothaxes,thesecondistangenttothe?rstcircleandthex-axis,andthethirdistangenttothe?rstcircleandthey-axis.Acircleofradiusr>sistangenttobothaxesandtothesecondandthirdcircles.Whatisr/s?This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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AMC12/AHSME2005

r

s

(A)5(B)6(C)8(D)9(E)1017Aunitcubeiscuttwicetoformthreetriangularprisms,twoofwhicharecongruent,asshown

inFigure1.ThecubeisthencutinthesamemanneralongthedashedlinesshowninFigure

2.Thiscreatesninepieces.WhatisthevolumeofthepiecethatcontainsvertexW

?

(A)1(B)1(C)1(D)1(E)118Callanumber”prime-looking”ifitiscompositebutnotdivisibleby2,3,or5.Thethree

smallestprime-lookingnumbersare49,77,and91.Thereare168primenumberslessthan1000.Howmanyprime-lookingnumbersaretherelessthan1000?

(A)100(B)102(C)104(D)106(E)10819Afaultycarodometerproceedsfromdigit3todigit5,alwaysskippingthedigit4,regardless

ofposition.Iftheodometernowreads002005,howmanymileshasthecaractuallytraveled?

(A)1404(B)1462(C)1604(D)1605(E)1804This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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AMC12/AHSME200520Foreachxin[0,1],de?ne

??2x,f(x)=2?2x,if0≤x≤1;

1if<x≤1.

Letf[2](x)=f(f(x)),andf[n+1](x)=f[n](f(x))foreachintegern≥2.Forhowmanyvaluesofxin[0,1]isf[2005](x)=1?

(A)0(B)2005(C)4010(D)20052(E)2200521Howmanyorderedtriplesofintegers(a,b,c),witha≥2,b≥1,andc≥0,satisfyboth

logab=c2005anda+b+c=2005?

(A)0(B)1(C)2(D)3(E)422ArectangularboxPisinscribedinasphereofradiusr.ThesurfaceareaofPis384,and

thesumofthelengthsofits12edgesis112.Whatisr?

(A)8(B)10(C)12(D)14(E)1623Twodistinctnumbersaandbarechosenrandomlyfromtheset{2,22,23,...,225}.Whatis

theprobabilitythatlogabisaninteger?

(A)2(B)31(C)13(D)7(E)124LetP(x)=(x?1)(x?2)(x?3).ForhowmanypolynomialsQ(x)doesthereexista

polynomialR(x)ofdegree3suchthatP(Q(x))=P(x)·R(x)?

(A)19(B)22(C)24(D)27(E)3225LetSbethesetofallpointswithcoordinates(x,y,z),wherex,y,andzareeachchosen

fromtheset{0,1,2}.HowmanyequilateraltriangleshavealltheirverticesinS?

(A)72(B)76(C)80(D)84(E)88This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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AMC12/AHSME2005

B1Ascouttroopbuys1000candybarsatapriceof?vefor$2.Theysellallthecandybarsatapriceoftwofor$1.Whatwastheirprot,indollars?

(A)100(B)200(C)300(D)400(E)5002Apositivenumberxhasthepropertythatx%ofxis4.Whatisx?

(A)2(B)4(C)10(D)20(E)403Briannaisusingpartofthemoneysheearnedonherweekendjobtobuyseveralequally-pricedCDs.Sheusedone?fthofhermoneytobuyonethirdoftheCDs.WhatfractionofhermoneywillshehaveleftaftershebuysalltheCDs?

(A)1(B)1(C)2(D)2(E)44Atthebeginningoftheschoolyear,LisasgoalwastoearnanAonatleast80%ofher50quizzesfortheyear.SheearnedanAon22ofthe?rst30quizzes.Ifsheistoachievehergoal,onatmosthowmanyoftheremainingquizzescansheearnagradelowerthananA?

(A)1(B)2(C)3(D)4(E)55An8-footby10-foot?ooristiledwithsquaretilesofsize1footby1foot.Eachtilehasapatternconsistingoffourwhitequartercirclesofradius1/2footcenteredateachcornerofthetile.Theremainingportionofthetileisshaded.Howmanysquarefeetofthe?oorare

shaded?

(A)80?20π(B)60?10π(C)80?10π(D)60+10π(E)80+10π6In??ABC,wehaveAC=BC=7andAB=2.SupposethatDisapointonlineABsuchthatBliesbetweenAandDandCD=8.WhatisBD?√√(A)3(B)2(C)4(D)5(E)47Whatistheareaenclosedbythegraphof|3x|+|4y|=12?

(A)6(B)12(C)16(D)24(E)25This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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AMC12/AHSME20058Forhowmanyvaluesofaisittruethattheliney=x+apassesthroughthevertexoftheparabolay=x2+a2?

(A)0(B)1(C)2(D)10(E)in?nitelymany9Onacertainmathexam,10%ofthestudentsgot70points,25%got80points,20%got85points,15%got90points,andtherestgot95points.Whatisthedi?erencebetweenthemeanandthemedianscoreonthisexam?

(A)0(B)1(C)2(D)4(E)510The?rsttermofasequenceis2005.Eachsucceedingtermisthesumofthecubesofthe

digitsofthepreviousterms.Whatisthe2005thtermofthesequence?

(A)29(B)55(C)85(D)133(E)25011Anenvelopecontainseightbills:2ones,2?ves,2tens,and2twenties.Twobillsaredrawn

atrandomwithoutreplacement.Whatistheprobabilitythattheirsumis$20ormore?

(A)1(B)2(C)3(D)1(E)212Thequadraticequationx2+mx+n=0hasrootsthataretwicethoseofx2+px+m=0,

andnoneofm,n,andpiszero.Whatisthevalueofn/p?

(A)1(B)2(C)4(D)8(E)1613Supposethat4x1=5,5x2=6,6x3=7,...,127x124=128.Whatisx1x2···x124?

(A)2(B)5(C)3(D)7(E)414Acirclehavingcenter(0,k),withk>6,istangenttothelinesy=x,y=?xandy=6.

Whatistheradiusofthiscircle?√√√(A)6?6(B)6(C)6(D)12(E)6+615Thesumoffourtwo-digitnumbersis221.Noneoftheeightdigitsis0andnotwoofthem

aresame.Whichofthefollowingisnotincludedamongtheeightdigits?

(A)1(B)2(C)3(D)4(E)516Eightspheresofradius1,oneperoctant,areeachtangenttothecoordinateplanes.Whatis

theradiusofthesmallestsphere,centeredattheorigin,thatcontainstheseeightspheres?√√√√(A)(B)(C)1+(D)1+(E)317Howmanydistinctfour-tuples(a,b,c,d)ofrationalnumbersaretherewith

alog102+blog103+clog105+dlog107=2005?

(A)0(B)1(C)17(D)2004(E)in?nitelymanyThis?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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AMC12/AHSME200518LetA(2,2)andB(7,7)bepointsintheplane.De?neRastheregioninthe?rstquadrant

consistingofthosepointsCsuchthat??ABCisanacutetriangle.WhatistheclosestintegertotheareaoftheregionR?

(A)25(B)39(C)51(D)60(E)8019Letxandybetwo-digitintegerssuchthatyisobtainedbyreversingthedigitsofx.The

integersxandysatisfyx2?y2=m2forsomepositiveintegerm.Whatisx+y+m?

(A)88(B)112(C)116(D)144(E)15420Leta,b,c,d,e,f,gandhbedistinctelementsintheset

{?7,?5,?3,?2,2,4,6,13}.

Whatistheminimumpossiblevalueof

(a+b+c+d)2+(e+f+g+h)2

(A)30(B)32(C)34(D)40(E)5021Apositiveintegernhas60divisorsand7nhas80divisors.Whatisthegreatestintegerk

suchthat7kdividesn?

(A)0(B)1(C)2(D)3(E)422Asequenceofcomplexnumbersz0,z1,z2,....isde?nedbytherule

zn+1=izn

nwherenisthecomplexconjugateofznandi2=?1.Supposethat|z0|=1andz2005=1.Howmanypossiblevaluesarethereforz0?

(A)1(B)2(C)4(D)2005(E)2200523LetSbethesetoforderedtriples(x,y,z)ofrealnumbersforwhich

log10(x+y)=zandlog10(x2+y2)=z+1.

Therearerealnumbersaandbsuchthatforallorderedtriples(x,y,z)inSwehavex3+y3=a·103z+b·102z.Whatisthevalueofa+b?

(A)15(B)29(C)15(D)39(E)2424Allthreeverticesofanequilateraltriangleareontheparabolay=x2,andoneofitssides

hasaslopeof2.Thex-coordinatesofthethreeverticeshaveasumofm/n,wheremandnarerelativelyprimepositiveintegers.Whatisthevalueofm+n?

(A)14(B)15(C)16(D)17(E)18This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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AMC12/AHSME200525Sixantssimultaneouslystandonthesixverticesofaregularoctahedron,witheachantata

di?erentvertex.Simultaneouslyandindependently,eachantmovesfromitsvertextooneofthefouradjacentvertices,eachwithequalprobability.Whatistheprobabilitythatnotwoantsarriveatthesamevertex?

(A)5(B)21(C)11(D)23(E)3This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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