haihongyuan.com
海量文库 文档专家
全站搜索:
您现在的位置:首页 > 初中教育 > 学科竞赛学科竞赛

全美数学竞赛USA-AMC_12-AHSME-2011-44

发布时间:2014-01-16 09:50:19  

USA

AMC12/AHSME2011

A1Acellphoneplancosts$20dollarseachmonth,plus5centspertextmessagesent,plus10centsforeachminuteusedover30hours.InJanuaryMichellesent100textmessagesandtalkedfor30.5hours.Howmuchdidshehavetopay?

(A)$24.00(B)$24.50(C)$25.50(D)$28.00(E)$30.002Thereare5coinsplaced?atonatableaccordingtothe?gure.Whatistheorderofthecoinsfromtoptobottom?

(A)(C,A,E,D,B)

(D)(C,E,A,D,B)(B)(C,A,D,E,B)(E)(C,E,D,A,B)(C)(C,D,E,A,B)

A

BC

D3Asmallbottleofshampoocanhold35millilitersofshampoo,whereasalargebottlecanhold500millilitersofshampoo.Jasminewantstobuytheminimumnumberofsmallbottlesnecessarytocompletely?llalargebottle.Howmanybottlesmustshebuy?

(A)11(B)12(C)13(D)14(E)154Atanelementaryschool,thestudentsinthirdgrade,fourthgrade,and?fthgraderunanaverageof12,15,and10minutesperday,respectively.Therearetwiceasmanythirdgradersasfourthgraders,andtwiceasmanyfourthgradersas?fthgraders.Whatistheaveragenumberofminutesrunperdaybythesestudents?

(A)12(B)37(C)88(D)13(E)145Lastsummer30%ofthebirdslivingonTownLakeweregeese,25%wereswans,10%wereherons,and35%wereducks.Whatpercentofthebirdsthatwerenotswansweregeese?

(A)20(B)30(C)40(D)50(E)60This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

http://www.artofproblemsolving.com/Page1

USA

AMC12/AHSME20116Theplayersonabasketballteammadesomethree-pointshots,sometwo-pointshots,andsomeone-pointfreethrows.Theyscoredasmanypointswithtwo-pointshotsaswiththree-pointshots.Theirnumberofsuccessfulfreethrowswasonemorethantheirnumberofsuccessfultwo-pointshots.Theteam’stotalscorewas61points.Howmanyfreethrowsdidtheymake?

(A)13(B)14(C)15(D)16(E)177Amajorityofthe30studentsinMs.Demeanor’sclassboughtpencilesattheschoolbookstore.Eachofthesestudentsboughtthesamenumberofpencils,andthisnumberwasgreaterthan

1.Thecostofapencilincentswasgreaterthanthenumberofpencilseachstudentbought,andthetotalcostofallthepencilswas$17.71.Whatwasthecostofapencilincents?

(A)7(B)11(C)17(D)23(E)778Intheeight-termsequenceA,B,C,D,E,F,G,H,thevalueofCis5andthesumofanythreeconsecutivetermsis30.WhatisA+H?

(A)17(B)18(C)25(D)26(E)439Atatwinsandtripletsconvention,therewere9setsoftwinsand6setsoftriplets,allfromdi?erentfamilies.Eachtwinshookhandswithallthetwinsexcepthis/hersiblingandwithhalfthetriplets.Eachtripletshookhandswithallthetripletsexcepthis/hersiblingsandhalfthetwins.Howmanyhandshakestookplace?

(A)324(B)441(C)630(D)648(E)88210Apairofstandard6-sidedfairdiceisrolledonce.Thesumofthenumbersrolleddetermines

thediameterofacircle.Whatistheprobabilitythatthenumericalvalueoftheareaofthecircleislessthanthenumericalvalueofthecircle’scircumference?

(A)1(B)1(C)1(D)1(E)511CirclesA,B,andCeachhaveradius1.CirclesAandBshareonepointoftangency.Circle

Chasapointoftangencywiththemidpointof.WhatistheareainsidecircleCbutoutsidecircleAandcircleB?

(A)3?π(B)π(C)2(D)3π(E)1+π12ApowerboatandaraftbothleftdockAonariverandheadeddownstream.Theraftdrifted

atthespeedoftherivercurrent.Thepowerboatmaintainedaconstantspeedwithrespecttotheriver.ThepowerboatreacheddockBdownriver,thenimmediatelyturnedandtraveledbackupriver.Iteventuallymettheraftontheriver9hoursafterleavingdockA.HowmanyhoursdidittakethepowerboattogofromAtoB?

(A)3(B)3.5(C)4(D)4.5(E)5This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

http://www.artofproblemsolving.com/Page2

USA

AMC12/AHSME201113TriangleABChasside-lengthsAB=12,BC=24,andAC=18.Thelinethroughthe

incenterof??ABCparalleltointersectsatMandatN.Whatistheperimeterof??AMN?

(A)27(B)30(C)33(D)36(E)4214Supposeaandbaresingle-digitpositiveintegerschosenindependentlyandatrandom.What

istheprobabilitythatthepoint(a,b)liesabovetheparabolay=ax2?bx?

(A)11(B)13(C)5(D)17(E)1915Thecircularbaseofahemisphereofradius2restsonthebaseofasquarepyramidofheight

6.Thehemisphereistangenttotheotherfourfacesofthepyramid.Whatistheedge-lengthofthebaseofthepyramid?√√(A)3(C)4(B)13(D)6(E)1316EachvertexofconvexpentagonABCDEistobeassignedacolor.Thereare6colorsto

choosefrom,andtheendsofeachdiagonalmusthavedi?erentcolors.Howmanydi?erentcoloringsarepossible?

(A)2520(B)2880(C)3120(D)3250(E)375017Circleswithradii1,2,and3aremutuallyexternallytangent.Whatistheareaofthetriangle

determinedbythepointsoftangency?

(A)3(B)4(C)1(D)6(E)418Supposethat|x+y|+|x?y|=2.Whatisthemaximumpossiblevalueofx2?6x+y2?

(A)5(B)6(C)7(D)8(E)919AtacompetitionwithNplayers,thenumberofplayersgivenelitestatusisequalto

21+??log2(N?1)???N.

Supposethat19playersaregivenelitestatus.WhatisthesumofthetwosmallestpossiblevaluesofN?

(A)38(B)90(C)154(D)406(E)102420Letf(x)=ax2+bx+c,wherea,b,andcareintegers.Supposethatf(1)=0,50<f(7)<60,

70<f(8)<80,and5000k<f(100)<5000(k+1)forsomeintegerk.Whatisk?

(B)2(C)3(D)4(E)5√√21Letf1(x)=andforintegersn≥2,letfn(x)=fn?1(IfNisthelargest

valueofnforwhichthedomainoffnisnonempty,thedomainoffNisc.WhatisN+c?

(A)?226(B)?144(C)?20(D)20(E)144(A)1

This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

http://www.artofproblemsolving.com/Page3

USA

AMC12/AHSME201122LetRbeasquareregionandn≥4aninteger.ApointXintheinteriorofRiscalledn-ray

partitionaliftherearenraysemanatingfromXthatdivideRintontrianglesofequalarea.Howmanypointsare100-raypartitionalbutnot60-raypartitional?

(A)1500(B)1560(C)2320(D)2480(E)2500+a23Letf(z)=zandg(z)=f(f(z)),whereaandbarecomplexnumbers.Supposethat

|a|=1andg(g(z))=zforallzforwhichg(g(z))isde?ned.Whatisthedi?erencebetweenthelargestandsmallestpossiblevaluesof|b|?√√(A)0(B)?1(C)?1(D)1(E)224ConsiderallquadrilateralsABCDsuchthatAB=14,BC=9,CD=7,DA=12.

Whatistheradiusofthelargestpossiblecirclethat?tsinsideorontheboundaryofsuchaquadrilateral?√√√√(A)(B)(C)2(D)5(E)225TriangleABChas∠BAC=60?,∠CBA≤90?,BC=1,andAC≥AB.LetH,I,andO

betheorthocenter,incenter,andcircumcenterof??ABC,respectively.AssumethattheareaofthepentagonBCOIHisthemaximumpossible.Whatis∠CBA?

(A)60?(B)72?(C)75?(D)80?(E)90?This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

http://www.artofproblemsolving.com/Page4

USA

AMC12/AHSME2011

B1Whatis2+4+61+3+5??1+3+52+4+6

(B)5(A)?1(C)7(D)147(E)432Josanna’stestscorestodateare90,80,70,60,and85.Hergoalistoraisehertestaverageatleast3pointswithhernexttest.Whatistheminimumtestscoreshewouldneedtoaccomplishthisgoal?

(A)80(B)82(C)85(D)90(E)953LeRoyandBernardowentonaweek-longtriptogetherandagreedtosharethecostsequally.Overtheweek,eachofthempaidforvariousjointexpensessuchasgasolineandcarrental.AttheendofthetripitturnedoutthatLeRoyhadpaidAdollarsandBernardohadpaidBdollars,whereA<B.HowmanydollarsmustLeRoygivetoBernardosothattheysharethecostsequally?

(A)(B)(C)(D)B?A(E)A+B4Inmultiplyingtwopositiveintegersaandb,Ronreversedthedigitsofthetwo-digitnumbera.Hiserrorneousproductwas161.Whatisthecorrectvalueoftheproductofaandb?

(A)116(B)161(C)204(D)214(E)2245LetNbethesecondsmallestpositiveintegerthatisdivisiblebyeverypositiveintegerlessthan7.WhatisthesumofthedigitsofN?

(A)3(B)4(C)5(D)6(E)96TwotangentstoacirclearedrawnfromapointA.ThepointsofcontactBandCdividethecircleintoarcswithlengthsintheratio2:3.Whatisthedegreemeasureof∠BAC?

(A)24(B)30(C)36(D)48(E)607Letxandybetwo-digitpositiveintegerswithmean60.Whatisthemaximumvalueoftheratio?

(A)3(B)33(C)39(D)9(E)998Keikowalksoncearoundatrackatexactlythesameconstantspeedeveryday.Thesidesofthetrackarestraight,andtheendsaresemicircles.Thetrackhaswidth6meters,andThis?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

http://www.artofproblemsolving.com/Page5

USA

AMC12/AHSME2011

ittakesher36secondslongertowalkaroundtheoutsideedgeofthetrackthanaroundtheinsideedge.WhatisKeiko’sspeedinmeterspersecond?

(A)π(B)2π(C)π(D)4π(E)5π9Tworealnumbersareselectedindependentlyatrandomfromtheinterval[-20,10].Whatistheprobabilitythattheproductofthosenumbersisgreaterthanzero?

(A)1(B)1(C)4(D)5(E)210RectangleABCDhasAB=6andBC=3.PointMischosenonsideABsothat∠AMD=

∠CMD.Whatisthedegreemeasureof∠AMD?

(A)15(B)30(C)45(D)60(E)7511Afroglocatedat(x,y),withbothxandyintegers,makessuccessivejumpsoflength5and

alwayslandsonpointswithintegercoordinates.Supposethatthefrogstartsat(0,0)andendsat(1,0).Whatisthesmallestpossiblenumberofjumpsthefrogmakes?

(A)2(B)3(C)4(D)5(E)612Adartboardisaregularoctagondividedintoregionsasshown.Supposethatadartthrown

attheboardisequallylikelytolandanywhereontheboard.Whatisprobabilitythatthedartlandswithinthecenter

square?

(A)1(B)1(C)√2?√(D)(E)2?√13Brianwritesdownfourintegersw>x>y>zwhosesumis44.Thepairwisepositive

di?erencesofthesenumbersare1,3,4,5,6,and9.Whatisthesumofthepossiblevaluesforw?

(A)16(B)31(C)48(D)62(E)93This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

http://www.artofproblemsolving.com/Page6

USA

AMC12/AHSME201114AsegmentthroughthefocusFofaparabolawithvertexVisperpendiculartoand

intersectstheparabolainpointsAandB.Whatiscos(∠AVB)?

(A)?√3(B)?√2(C)?4(D)?3(E)?115Howmanypositivetwo-digitintegersarefactorsof224?1?

(A)4(B)8(C)10(D)12(E)1416RhombusABCDhassidelength2and∠B=120?.RegionRconsistsofallpointsinsidethe

rhombusthatareclosertovertexBthananyoftheotherthreevertices.WhatistheareaofR?√(A)√(B)(C)√2√(D)1+(E)217Letf(x)==log10,h1(x)=g(f(x)),andhn(x)=h1(hn?1(x))forinte-

gersn≥2.Whatisthesumofthedigitsofh2011(1)?

(A)16,081(B)16,089(C)18,089(D)18,098(E)18,0991010x,g(x)??x??

18Apyramidhasasquarebasewithsidesoflength1andhaslateralfacesthatareequilateral

triangles.Acubeisplacedwithinthepyramidsothatonefaceisonthebaseofthepyramidanditsoppositefacehasallitsedgesonthelateralfacesofthepyramid.Whatisthevolumeofthiscube?√√√√√2(A)5?7(B)7?4(C)(D)(E)19Alatticepointinanxy-coordinatesystemisanypoint(x,y)wherebothxandyareintegers.

Thegraphofy=mx+2passesthroughnolatticepointwith0<x≤100forallmsuchthat1<m<a.Whatisthemaximumpossiblevalueofa?

(A)51(B)50(C)51(D)52(E)1320TriangleABChasAB=13,BC=14,andAC=15.ThepointsD,E,andFarethemid-

pointsofandrespectively.LetX=Ebetheintersectionofthecircumcirclesof??BDEand??CEF.WhatisXA+XB+XC?√√√12969195(A)24(B)14(C)(D)(E)21Thearithmeticmeanoftwodistinctpositiveintegersxandyisatwo-digitinteger.The

geometricmeanofxandyisobtainedbyreversingthedigitsofthearithmeticmean.Whatis|x?y|?

(A)24(B)48(C)54(D)66(E)7022LetT1beatrianglewithsides2011,2012,and2013.Forn≥1,ifTn=??ABCandD,E,

andFarethepointsoftangencyoftheincircleof??ABCtothesidesAB,BCandAC,This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

http://www.artofproblemsolving.com/Page7

USA

AMC12/AHSME2011

respectively,thenTn+1isatrianglewithsidelengthsAD,BE,andCF,ifitexists.Whatistheperimeterofthelasttriangleinthesequence(Tn)?

(A)1509(B)1509(C)1509(D)1509(E)150923Abugtravelsinthecoordinateplane,movingonlyalongthelinesthatareparalleltothe

x-axisory-axis.LetA=(?3,2)andB=(3,?2).ConsiderallpossiblepathsofthebugfromAtoBoflengthatmost20.Howmanypointswithintegercoordinateslieonatleastoneofthesepaths?

(B)185(C)195(D)227(E)255√√24LetP(z)=z8+(4+6)z4?(4+7).Whatistheminimumperimeteramongallthe

8-sidedpolygonsinthecomplexplanewhoseverticesarepreciselythezerosofP(z)?√√√√√√√(A)4+4(B)8(C)3+3(D)4+4(E)4+625Foreverymandkintegerswithkodd,denoteby[]theintegerclosesttointegerk,letP(k)betheprobabilitythat

n100?n100]=[][]+[kkk

foranintegernrandomlychosenfromtheinterval1≤n≤99!.WhatistheminimumpossiblevalueofP(k)overtheoddintegersKintheinterval1≤k≤99?

(A)1.(A)161Foreveryodd(B)50(C)44(D)34(E)7This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

http://www.artofproblemsolving.com/Page8

网站首页网站地图 站长统计
All rights reserved Powered by 海文库
copyright ©right 2010-2011。
文档资料库内容来自网络,如有侵犯请联系客服。zhit326@126.com