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全美数学竞赛USA-AMC_12-AHSME-2006-44

发布时间:2014-01-15 17:08:24  

USA

AMC12/AHSME2006

A1SandwichesatJoe’sFastFoodcost$3eachandsodascost$2each.Howmanydollarswillitcosttopurchase5sandwichesand8sodas?

(A)31(B)32(C)33(D)34(E)352De?nex?y=x3?y.Whatish?(h?h)?

(A)?h(B)0(C)h(D)2h(E)h33TheratioofMary’sagetoAlice’sageis3:5.Aliceis30yearsold.HowoldisMary?

(A)15(B)18(C)20(D)24(E)504AdigitalwatchdisplayshoursandminuteswithAMandPM.Whatisthelargestpossiblesumofthedigitsinthedisplay?

(A)17(B)19(C)21(D)22(E)235DougandDavesharedapizzawith8equally-sizedslices.Dougwantedaplainpizza,butDavewantedanchoviesonhalfthepizza.Thecostofaplainpizzawas$8,andtherewasanadditionalcostof$2forputtinganchoviesononehalf.Daveatealltheslicesofanchovypizzaandoneplainslice.Dougatetheremainder.Eachpaidforwhathehadeaten.HowmanymoredollarsdidDavepaythanDoug?

(A)1(B)2(C)3(D)4(E)56The8×18rectangleABCDiscutintotwocongruenthexagons,asshown,insuchawaythatthetwohexagonscanberepositionedwithoutoverlaptoformasquare.Whatisy?

18AB

8

y

D188C

(A)6(B)7(C)8(D)9(E)10This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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AMC12/AHSME20067Maryis20%olderthanSally,andSallyis40%youngerthanDanielle.Thesumoftheiragesis23.2years.HowoldwillMarybeonhernextbirthday?

(A)7(B)8(C)9(D)10(E)118Howmanysetsoftwoormoreconsecutivepositiveintegershaveasumof15?

(A)1(B)2(C)3(D)4(E)59Oscarbuys13pencilsand3erasersfor$1.00.Apencilcostsmorethananeraser,andbothitemscostawholenumberofcents.Whatisthetotalcost,incents,ofonepencilandoneeraser?

(D)18(E)20??10Forhowmanyrealvaluesofxis120?aninteger?

(A)3(B)6(C)9(D)10(E)11

11Whichofthefollowingdescribesthegraphoftheequation(x+y)2=x2+y2?

(A)theemptyset(B)onepoint(C)twolines(D)acircle(E)theentireplane(A)10(B)12(C)1512Anumberoflinkedrings,each1cmthick,arehangingonapeg.Thetopringhasanoutside

diameterof20cm.Theoutsidediameterofeachoftheouterringsis1cmlessthanthatoftheringaboveit.Thebottomringhasanoutsidediameterof3cm.Whatisthedistance,incm,fromthetopofthetopringtothebottomofthebottom

ring?

20

..3

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AMC12/AHSME2006

(A)171(B)173(C)182(D)188(E)21013Theverticesofa3?4?5righttrianglearethecentersofthreemutuallyexternallytangent

circles,asshown.Whatisthesumoftheareasofthethreecircles?

A

B4C

(A)12π(B)(C)13π(D)(E)14π14Twofarmersagreethatpigsareworth$300andthatgoatsareworth$210.Whenonefarmer

owestheothermoney,hepaysthedebtinpigsorgoats,with“change”receivedintheformofgoatsorpigsasnecessary.(Forexample,a$390debtcouldbepaidwithtwopigs,withonegoatreceivedinchange.)Whatistheamountofthesmallestpositivedebtthatcanberesolvedinthisway?

(A)$5

π(B)$10π(C)$30π(D)$90(E)$21015Supposecosx=0andcos(x+z)=1/2.Whatisthesmallestpossiblepositivevalueofz?(A)(B)(C)(D)(E)

16CircleswithcentersAandBhaveradii3and8,respectively.Acommoninternaltangent

intersectsthecirclesatCandD,respectively.LinesABandCDintersectatE,andAE=5.WhatisCD?

DA

CB

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AMC12/AHSME2006

(A)13(B)44(C)√(D)√(E)5517SquareABCDhassidelengths,acirclecenteredatEhasradiusr,andrandsareboth

rational.ThecirclepassesthroughD,andDliesonPointFliesononthe??thecircle,samesideofasA.SegmentAFistangenttothecircle,andAF=9+5Whatisr/s?

F

E

AB

C(A)(B)(C)(D)(E)18Thefunctionfhasthepropertythatforeachrealnumberxinitsdomain,1/xisalsoinits

domainand????1=x.f(x)+fx

Whatisthelargestsetofrealnumbersthatcanbeinthedomainoff?(A){x|x=0}(B){x|x<0}(C){x|x>0}

(D){x|x=?1andx=0andx=1}(E){?1,1}19Circleswithcenters(2,4)and(14,9)haveradii4and9,respectively.Theequationofa

commonexternaltangenttothecirclescanbewrittenintheformy=mx+bwithm>0.Whatisb?

9

(14,9)4

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USA

AMC12/AHSME2006

(A)908(B)909(C)130(D)911(E)91220Abugstartsatonevertexofacubeandmovesalongtheedgesofthecubeaccordingto

thefollowingrule.Ateachvertexthebugwillchoosetotravelalongoneofthethreeedgesemanatingfromthatvertex.Eachedgehasequalprobabilityofbeingchosen,andallchoicesareindependent.Whatistheprobabilitythataftersevenmovesthebugwillhavevisitedeveryvertexexactlyonce?

(A)21Let

and(B)(C)(D)(E)S1={(x,y)|log10(1+x2+y2)≤1+log10(x+y)}

S2={(x,y)|log10(2+x2+y2)≤2+log10(x+y)}.

(B)99(C)100(D)101(E)102WhatistheratiooftheareaofS2totheareaofS1?(A)98

22Acircleofradiusrisconcentricwithandoutsidearegularhexagonofsidelength2.The

probabilitythatthreeentiresidesofhexagonarevisiblefromarandomlychosenpointonthecircleis1/2.Whatisr?√√√√√√√√(A)2√+√2(B)3+(C)2+(D)3+(E)6?23Givena?nitesequenceS=(a1,a2,...,an)ofnrealnumbers,letA(S)bethesequence

????an?1+ana1+a2a2+a3,,...,222

ofn?1realnumbers.De?neA1(S)=A(S)and,foreachintegerm,2≤m≤n?1,de?neAm(S)=A(Am?1(S)).Supposex>0,andletS=(1,x,x2,...,x100).IfA100(S)√=(1/250),√√√1thenwhatisx?(A)1?(B)?1(C)(D)2?(E)24Theexpression

(x+y+z)2006+(x?y?z)2006

issimpli?edbyexpandingitandcombiningliketerms.Howmanytermsareinthesimpli?edexpression?

(A)6018(B)671,676(C)1,007,514(D)1,008,016(E)2,015,028This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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AMC12/AHSME200625Howmanynon-emptysubsetsSof{1,2,3,...,15}havethefollowingtwoproperties?

(1)NotwoconsecutiveintegersbelongtoS.(2)IfScontainskelements,thenScontainsnonumberlessthank.

(A)277(B)311(C)376(D)377(E)405This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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AMC12/AHSME2006

B1Whatis(?1)1+(?1)2+···+(?1)2006?

(A)?2006(B)?1(C)0(D)1(E)20062Forrealnumbersxandy,de?nex?y=(x+y)(x?y).Whatis3?(4?5)?

(A)?72(B)?27(C)?24(D)24(E)723Afootballgamewasplayedbetweentwoteams,theCougarsandthePanthers.Thetwoteamsscoredatotalof34points,andtheCougarswonbyamarginof14points.HowmanypointsdidthePanthersscore?

(A)10(B)14(C)17(D)20(E)244Maryisabouttopayfor?veitemsatthegrocerystore.Thepricesoftheitemsare$7.99,$

4.99,$2.99,$1.99,and$0.99.Marywillpaywithatwenty-dollarbill.Whichofthefollowingisclosesttothepercentageofthe$20.00thatshewillreceiveinchange?

(A)5(B)10(C)15(D)20(E)255Johniswalkingeastataspeedof3milesperhour,whileBobisalsowalkingeast,butataspeedof5milesperhour.IfBobisnow1milewestofJohn,howmanyminuteswillittakeforBobtocatchuptoJohn?

(A)30(B)50(C)60(D)90(E)1206Francescauses100gramsoflemonjuice,100gramsofsugar,and400gramsofwatertomakelemonade.Thereare25caloriesin100gramsoflemonjuiceand386caloriesin100gramsofsugar.Watercontainsnocalories.Howmanycaloriesarein200gramsofherlemonade.

(A)129(B)137(C)174(D)223(E)4117Mr.andMrs.Lopezhavetwochildren.Whentheygetintotheirfamilycar,twopeoplesitinthefront,andtheothertwositintheback.EitherMr.LopezorMrs.Lopezmustsitinthedriver’sseat.Howmanyseatingarrangementsarepossible?

(A)4(B)12

3(C)16(D)24(E)4898Thelinesx=y+aandy=x+bintersectatthepoint(1,2).Whatisa+b?(A)0(B)(C)1(D)2(E)9Howmanyeventhree-digitintegershavethepropertythattheirdigits,readlefttoright,areinstrictlyincreasingorder?

(A)21(B)34(C)51(D)72(E)150This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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AMC12/AHSME200610Inatrianglewithintegersidelengths,onesideisthreetimesaslongasasecondside,and

thelengthofthethirdsideis15.Whatisthegreatestpossibleperimeterofthetriangle?

(A)43(B)44(C)45(D)46(E)4711JoeandJoAnneachbought12ouncesofco?eeina16-ouncecup.Joedrank2ouncesofhis

co?eeandthenadded2ouncesofcream.JoAnnadded2ouncesofcream,stirredtheco?eewell,andthendrank2ounces.WhatistheresultingratiooftheamountofcreaminJoe’sco?eetothatinJoAnn’sco?ee?

(A)(B)(C)1(D)(E)12Theparabolay=ax2+bx+chasvertex(p,p)andy-intercept(0,?p),wherep=0.What

isb?

(A)?p(B)0(C)2(D)4(E)p13RhombusABCDissimilartorhombusBFDE.TheareaofrhombusABCDis24,and

∠BAD=60?.WhatistheareaofrhombusBFDE?

D

F

E

A

√CB√(E)6(A)6(B)4(C)8(D)9

14ElmomakesNsandwichesforafundraiser.ForeachsandwichheusesBglobsofpeanut

butterat4centsperglobandJblobsofjamat5centsperglob.Thecostofthepeanutbutterandjamtomakeallthesandwichesis$2.53.AssumethatB,J,andNareallpositiveintegerswithN>1.WhatisthecostofthejamElmousestomakethesandwiches?

(A)$1.05(B)$1.25(C)$1.45(D)$1.65(E)$1.8515CircleswithcentersOandPhaveradii2and4,respectively,andareexternallytangent.

PointsAandBareonthecirclecenteredatO,andpointsCandDareonthecirclecenteredatP,suchthatandarecommonexternaltangentstothecircles.WhatistheareaofhexagonAOBCPD?This?lewasdownloadedfromtheAoPSMathOlympiadResourcesPage

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AMC12/AHSME2006

D

A

2

O

B

C

√(A)18√(B)24√(D)24√(E)324P(C)36

16RegularhexagonABCDEFhasverticesAandCat(0,0)and(7,1),respectively.Whatis

itsarea?√√√√(A)20(B)22(C)25(D)27(E)5017Foraparticularpeculiarpairofdice,theprobabilitiesofrolling1,2,3,4,5and6oneach

dieareintheratio1:2:3:4:5:6.Whatistheprobabilityofrollingatotalof7onthetwodice?

(A)(B)(C)(D)(E)18Anobjectintheplanemovesfromonelatticepointtoanother.Ateachstep,theobjectmay

moveoneunittotheright,oneunittotheleft,oneunitup,oroneunitdown.Iftheobjectstartsattheoriginandtakesaten-steppath,howmanydi?erentpointscouldbethe?nalpoint?

(A)120(B)121(C)221(D)230(E)23119Mr.Joneshaseightchildrenofdi?erentages.Onafamilytriphisoldestchild,whois9,

spotsalicenseplatewitha4-digitnumberinwhicheachoftwodigitsappearstwotimes.”Look,daddy!”sheexclaims.”Thatnumberisevenlydivisiblebytheageofeachofuskids!””That’sright,”repliesMr.Jones,”andthelasttwodigitsjusthappentobemyage.”WhichofthefollowingisnottheageofoneofMr.Jones’schildren?

(A)4(B)5(C)6(D)7(E)820Letxbechosenatrandomfromtheinterval(0,1).Whatistheprobabilitythat

??log104x?????log10x??=0

Here??x??denotesthegreatestintegerthatislessthanorequaltox.

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AMC12/AHSME200621RectangleABCDhasarea2006.Anellipsewitharea2006πpassesthroughAandCand

hasfociatBandD.Whatistheperimeteroftherectangle?(Theareaofanellipseisπab,where2aand2barethelengthsofitsaxes.)√√√√16321003(A)(B)(C)8(D)6(E)22Supposea,b,andcarepositiveintegerswitha+b+c=2006,anda!b!c!=m·10n,wherem

andnareintegersandmisnotdivisibleby10.Whatisthesmallestpossiblevalueofn?

(A)489(B)492(C)495(D)498(E)50123Isosceles??ABChasarightangleatC.PointP??isinside??ABC,suchthatPA=11,PB=7,andPC=6.Legsandhavelengths=a+bwhereaandbarepositiveintegers.

Whatisa+b?

PCA

(A)85(B)91(C)108(D)121(E)127

π24LetSbethesetofallpoints(x,y)inthecoordinateplanesuchthat0≤x≤

WhatistheareaofthesubsetofSforwhich

sin2x?sinxsiny+sin2y≤

(A)π2

and0≤y≤π.34(B)π2

(C)π2

(D)3π2

(E)2π2

25Asequencea1,a2,...ofnon-negativeintegersisde?nedbytherulean+2=|an+1?an|forn≥1.Ifa1=999,a2<999,anda2006=1,howmanydi?erentvaluesofa2arepossible?

(A)165(B)324(C)495(D)499(E)660

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