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2013年AMC10

发布时间:2013-11-25 11:33:58  

USA

AMC102013

A1Ataxiridecosts$1.50plus$0.25permiletraveled.Howmuchdoesa5-miletaxiridecost?

(A)$2.25(B)$2.50(C)$2.75(D)$3.00(E)$3.252Aliceismakingabatchofcookiesandneeds21cupsofsugar.Unforunately,hermeasuring

1cupholdsonlycupofsugar.Howmanytimesmustshe?llthatcuptogetthecorrect

amountofsugar?

(A)8(B)10(C)12(D)16(E)203SquareABCDhassidelength10.PointEison,andtheareaof??ABEis40.WhatisBE?

(A)4(B)5(C)6(D)7(E)8

BA4Asoftballteamplayedtengames,scoring1,2,3,4,5,6,7,8,9,and10runs.Theylostbyoneruninexactly?vegames.Ineachoftheothergames,theyscoredtwiceasmanyrunsastheiropponent.Howmanytotalrunsdidtheiropponentsscore?

(A)35(B)40(C)45(D)50(E)555Tom,Dorothy,andSammywentonavacationandagreedtosplitthecostsevenly.DuringtheirtripTompaid$105,Dorothypaid$125,andSammypaid$175.Inordertosharethecostsequally,TomgaveSammytdollars,andDorothygaveSammyddollars.Whatist?d?

(A)15(B)20(C)25(D)30(E)35This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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USA

AMC1020136Joeyandhis?vebrothersareages3,5,7,9,11,and13.Oneafternoontwoofhisbrotherswhoseagessumto16wenttothemovies,twobrothersyoungerthan10wenttoplaybaseball,andJoeyandthe5-year-oldstayedhome.HowoldisJoey?

(A)3(B)7(C)9(D)11(E)137AstudentmustchooseaprogramoffourcoursesfromamenuofcoursesconsistingofEnglish,Algebra,Geometry,History,Art,andLatin.ThisprogrammustcontainEnglishandatleastonemathematicscourse.Inhowmanywayscanthisprogrambechosen?

(A)6(B)8(C)9(D)12(E)1622014+22012

?2?2(A)?1(B)1(C)58Whatisthevalueof(D)2013(E)240249Inarecentbasketballgame,Shenilleattemptedonlythree-pointshotsandtwo-pointshots.Shewassuccessfulon20%ofherthree-pointshotsand30%ofhertwo-pointshots.Shenilleattempted30shots.Howmanypointsdidshescore?

(A)12(B)18(C)24(D)30(E)3610A?owerbouquetcontainspinkroses,redroses,pinkcarnations,andredcarnations.One

thirdofthepink?owersareroses,threefourthsofthered?owersarecarnations,andsixtenthsofthe?owersarepink.Whatpercentofthe?owersarecarnations?

(A)15(B)30(C)40(D)60(E)7011Astudentcouncilmustselectatwo-personwelcomingcommitteeandathree-personplanning

committeefromamongitsmembers.Thereareexactly10waystoselectatwo-personteamforthewelcomingcommittee.Itispossibleforstudentstoserveonbothcommittees.Inhowmanydi?erentwayscanathree-personplanningcommitteebeselected?

(A)10(B)12(C)15(D)18(E)2512In??ABC,AB=AC=28andBC=20.PointsD,E,andFareonsidesand

,respectively,suchthatandareparalleltoandrespectively.WhatistheperimeterofparallelogramADEF?This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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USA

AMC102013

A

D

F

B(A)48(B)52(C)56(D)60(E)7213Howmanythree-digitnumbersarenotdivisibleby5,havedigitsthatsumtolessthan20,

andhavethe?rstdigitequaltothethirddigit?

(A)52(B)60(C)66(D)68(E)7014Asolidcubeofsidelength1isremovedfromeachcornerofasolidcubeofsidelength3.

Howmanyedgesdoestheremainingsolidhave?

(A)36(B)60(C)72(D)84(E)10815Twosidesofatrianglehavelengths10and15.Thelengthofthealtitudetothethirdsideis

theaverageofthelengthsofthealtitudestothetwogivensides.Howlongisthethirdside?

(A)6(B)8(C)9(D)12(E)1816Atrianglewithvertices(6,5),(8,?3),and(9,1)isre?ectedaboutthelinex=8tocreatea

secondtriangle.Whatistheareaoftheunionofthetwotriangles?

283132(A)9(B)(C)10(D)(E)33317Daphneisvisitedperiodicallybyherthreebestfriends:Alice,Beatrix,andClaire.Alicevisits

everythirdday,Beatrixvisitseveryfourthday,andClairevisitsevery?fthday.AllthreefriendsvisitedDaphneyesterday.Howmanydaysofthenext365-dayperiodwillexactlytwofriendsvisither?

(A)48(B)54(C)60(D)66(E)72This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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USA

AMC10201318LetpointsA=(0,0),B=(1,2),C=(3,3),andD=(4,0).QuadrilateralABCD??iscut??printoequalareapiecesbyalinepassingthroughA.Thislineintersectsatpoint,,

wherethesefractionsareinlowestterms.Whatisp+q+r+s?

(A)54(B)58(C)62(D)70(E)7519Inbase10,thenumber2013endsinthedigit3.Inbase9,ontheotherhand,thesame

numberiswrittenas(2676)9andendsinthedigit6.Forhowmanypositiveintegersbdoesthebase-brepresentationof2013endinthedigit3?

(A)6(B)9(C)13(D)16(E)1820Aunitsquareisrotated45?aboutitscenter.Whatistheareaoftheregionsweptoutby

theinteriorofthesquare?√√√√ππ1ππ(A)1?+(B)+(C)2?+(D)+(E)1++π21Agroupof12piratesagreetodivideatreasurechestofgoldcoinsamongthemselvesas

kfollows.Thekthpiratetotakeasharetakesofthecoinsthatremaininthechest.The

numberofcoinsinitiallyinthechestisthesmallestnumberforwhichthisarrangementwillalloweachpiratetoreceiveapositivewholenumberofcoins.Howmanycoinsdoethe12thpiratereceive?

(A)720(B)1296(C)1728(D)1925(E)385022Sixspheresofradius1arepositionedsothattheircentersareattheverticesofaregular

hexagonofsidelength2.Thesixspheresareinternallytangenttoalargerspherewhosecenteristhecenterofthehexagon.Aneighthsphereisexternallytangenttothesixsmallerspheresandinternallytangenttothelargersphere.Whatistheradiusofthiseighthsphere?√√35(A)(B)(C)(D)(E)223In??ABC,AB=86,andAC=97.AcirclewithcenterAandradiusABintersectsat

pointsBandX.Moreoverandhaveintegerlengths.WhatisBC?

(A)11(B)28(C)33(D)61(E)7224CentralHighSchooliscompetingagainstNorthernHighSchoolinabackgammonmatch.

Eachschoolhasthreeplayers,andthecontestrulesrequirethateachplayerplaytwogamesagainsteachoftheother’sschool’splayers.Thematchtakesplaceinsixrounds,withthreegamesplayedsimultaneouslyineachround.Inhowmanydi?erentwayscanthematchbescheduled?

(A)540(B)600(C)720(D)810(E)900This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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AMC10201325Alldiagonalsaredrawninaregularoctagon.Athowmanydistinctpointsintheinteriorof

theoctagon(notontheboundary)dotwoormorediagonalsintersect?

(A)49(B)65(C)70(D)96(E)128This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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AMC102013

B1Whatis(A)?12+4+61+3+5?5(B)?(C)7(D)49(E)432MrGreenmeasureshisrectangulargardenbywalkingtwoofthesidesand?ndsthatitis15stepsby20steps.EachorMrGreen’sstepsistwofeetlong.MrGreenexpecthalfapoundofpotatoespersquarefootfromhisgarden.HowmanypoundsofpotatoesdoesMrGreenexpectfromhisgarden?

(A)600(B)800(C)1000(D)1200(E)14003OnaparticularJanuaryday,thehightemperatureinLincoln,Nebraska,was16degreeshigherthanthelowtemperature,andtheaverageofthehighandlowtemperatureswas3?.Indegrees,whatwasthelowtemperatureinLincolnthatday?

(A)?13(B)?8(C)?5(D)3(E)114Whencountingfrom3to201,53isthe51stnumbercounted.Whencountingbackwardsfrom201to3,53isthenthnumbercounted.Whatisn?

(A)146

(A)?20(B)147(B)?15(C)148(D)149(D)0(E)150(E)25Positiveintegersaandbareeachlessthan6.Whatisthesmallestpossiblevaluefor2·a?a·b?(C)?10

6Theaverageageof33?fth-gradersis11.Theaverageageof55oftheirparentsis33.Whatistheaverageageofalloftheseparentsand?fth-graders?

(A)22(B)23.25(C)24.75(D)26.25(E)287Sixpointsareequallyspacedaroundacircleofradius1.Threeofthesepointsaretheverticesofatrianglethatisneitherequilateralnorisosceles.Whatistheareaofthistriangle?√√√(A)(B)(C)1(D)(E)28Ray’scaraverages40milespergallonofgasoline,andTom’scaraverages10milespergallonofgasoline.RayandTomeachdrivethesamenumberofmiles.Whatisthecars’combinedrateofmilespergallonofgasoline?

(A)10(B)16(C)25(D)30(E)409Threepositiveintegersareeachgreaterthan1,haveaproductof27000,andarepairwiserelativelyprime.Whatistheirsum?

(A)100(B)137(C)156(D)160(E)165This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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AMC10201310Abasketballteam’splayersweresuccessfulon50

(A)10(B)15(C)20(D)25(E)3011Realnumbersxandysatisfytheequationx2+y2=10x?6y?34.Whatisx+y?

(A)1(B)2(C)3(D)6(E)812LetSbethesetofsidesanddiagonalsofaregularpentagon.ApairofelementsofS

areselectedatrandomwithoutreplacement.Whatistheprobabilitythatthetwochosensegmentshavethesamelength?

(A)2(B)4(C)1(D)5(E)413JoandBlairtaketurnscountingfrom1toonemorethanthelastnumbersaidbytheother

person.Jostartsbysaying”1”,soBlairfollowsbysaying”1,2”.Jothensays”1,2,3”,andsoon.Whatisthe53rdnumbersaid?

(A)2(B)3(C)5(D)6(E)814De?nea?b=a2b?ab2.Whichofthefollowingdescribesthesetofpoints(x,y)forwhich

x?y=y?x?

(A)a?nitesetofpoints

(B)oneline

(C)twoparallellines

(Dtwointersectinglines

(E)threelines15Awireiscutintotwopieces,oneoflengthaandtheotheroflengthb.Thepieceoflength

aisbenttoformanequilateraltriangle,andthepieceoflengthbisbenttoformaregularhexagon.Thetriangleandthehexagonhaveequalarea.Whatisa?

√√√(A)1(B)(C)(D2(E)316In??ABC,mediansandintersectatP,PE=1.5,PD=2,andDE=2.5.What

istheareaofAEDC?

CDAEB

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AMC102013

(A)13(B)13.5(C)14(D)14.5(E)1517Alexhas75redtokensand75bluetokens.ThereisaboothwhereAlexcangivetwored

tokensandreceiveinreturnasilvertokenandabluetoken,andanotherboothwhereAlexcangivethreebluetokensandreceiveinreturnasilvertokenandaredtoken.Alexcontinuestoexchangetokensuntilnomoreexchangesarepossible.HowmanysilvertokenswillAlexhaveattheend?

(A)62(B)82(C)83(D102(E)10318Thenumber2013hasthepropertythatitsunitsdigitisthesumofitsotherdigits,thatis

2+0+1=3.Howmanyintegerslessthan2013butgreaterthan1000sharethisproperty?

(A)33(B)34(C)45(D46(E)5819Therealnumbersc,b,aformanarithmeticsequencewitha≥b≥c≥0.Thequadratic

ax2+bx+chasexactlyoneroot.Whatisthisroot?√√√√(A)?7?4(B)?2?(C)?1(D?2+(E)?7+420Thenumber2013isexpressedintheform

2013=a1!a2!···am!,b1!b2!···bn!

wherea1≥a2≥···≥amandb1≥b2≥···≥bnarepositiveintegersanda1+b1isassmallaspossible.Whatis|a1?b1|?

(A)1(B)2(C)3(D4(E)521Twonon-decreasingsequencesofnonnegativeintegershavedi?erent?rstterms.Eachse-

quencehasthepropertythateachtermbeginningwiththethirdisthesumoftheprevioustwoterms,andtheseventhtermofeachsequenceisN.WhatisthesmallestpossiblevalueofN?

(A)55(B)89(C)104(D144(E)27322TheregularoctagonABCDEFGHhasitscenteratJ.Eachoftheverticesandthecenter

aretobeassociatedwithoneofthedigits1through9,witheachdigitusedonce,insuchawaythatthesumsofthenumbersonthelinesAJE,BJF,CJG,andDJHareequal.Inhowmanywayscanthisbedone?

(A)384(B)576(C)1152(D)1680(E)3546This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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AMC102013

AB

H

JGCD

FE

23IntriangleABC,AB=13,BC=14,andCA=15.DistinctpointsD,E,andFlieonsegments,andrespectively,suchthat⊥⊥,and⊥.Thelengthofsegmentcanbewrittenasm,wheremandnarerelativelyprimepositiveintegers.Whatism+n?

(A)18(B)21(C)24(D27(E)3024Apositiveintegernisniceifthereisapositiveintegermwithexactlyfourpositivedivisors

(including1andm)suchthatthesumofthefourdivisorsisequalton.Howmanynumersintheset{2010,2011,2012,...,2019}arenice?

(A)1(B)2(C)3(D4(E)525Bernardochoosesathree-digitpositiveintegerNandwritesbothitsbase-5andbase-6

representationsonablackboard.LaterLeRoyseesthetwonumbersBernardohaswritten.Treatingthetwonumbersasbase-10integers,headdsthemtoobtainanintegerS.Forexample,ifN=749,Bernardowritesthenumbers10,444and3,245,andLeRoyobtainsthesumS=13,689.ForhowmanychoicesofNarethetworightmostdigitsofS,inorder,thesameasthoseof2N?

(A)5(B)10(C)15(D20(E)25This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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