資料綜合amc 12b problems答案

2008AMC12BProblemAbasketballplayermadebasketsduringagame.Eachbasketwaswortheitherorpoints.Howmanydifferentnumberscouldrepresentthetotalpointsscoredbytheplayer?Ifthebasketballplayermakes three-pointshotsandtwo-pointshots,hescorespoints.Clearlyeveryvalue yieldsanumberoftotalpoints.Sincehecanmakeanynumberofthree-pointshotsbetweenandinclusive,thenumberofdifferentpointtotalsis.ProblemAblockofcalendardatesisshown.Theorderofthenumbersinthesecondrowistobereversed.Thentheorderofthenumbersinthefourthrowistobereversed.Finally,thenumbersoneachdiagonalaretobeadded.Whatwillbethepositivedifferencebetweenthetwodiagonalsums?Afterreversingthenumbersonthesecondandfourthrows,theblockwilllooklikeThedifferencebetweenthetwodiagonalsums.ProblemAsemiprobaseballleaguehasteamswithplayerseach.Leaguerulesstatethataplayermustbepaidatleastdollars,andthatthetotalofallplayers'salariesforeachteamcannotexceeddollars.Whatisthemaximumpossibllesalary,indollars,forasingleplayer?makestheminimumpossibleandthatthecombinedsalariestotalthemaximumofThemaximumanyplayercouldmakeisdollars(answerchoiceProblemOncircle,pointsandareonthesamesideofdiameter,,and.Whatistheratiooftheareaofthesmallersectortotheareaofthecircle?.Sinceacircle ,thedesiredratioisProblemAclasscollectsdollarstobuyflowersforaclassmatewhoisinthehospital.Rosescostdollarseach,andcarnationscostdollarseach.Nootherflowersaretobeused.Howmanydifferentbouquetscouldbepurchasedforexactlydollars?Theclasscouldsendjustcarnations(25ofthem).Theycouldalsosend22carnationsand2roses,19carnationsand4roses,andsoon,downto1carnationand16roses.Thereare9totalpossibilities(from0to16roses,incrementingby2ateachstep),whichisanswerchoiceC.ProblemPostmanPetehasapedometertocounthissteps.Thepedometerrecordsuptosteps,thenflipsovertoonthenextstep.Peteplanstodeterminehismileageforayear.OnJanuaryPetesetsthepedometerto.Duringtheyear,thepedometerflipsfromtoforty-fourtimes.OnDecemberthepedometerreads.Petetakesstepspermile.WhichofthefollowingisclosesttothenumberofmilesPetewalkedduringtheyear?Everytimethepedometerflips,Petehaswalkedsteps.Therefore,Petehaswalkedatotalofsteps,whichismiles,whichisclosesttoanswerchoiceProblemForreal , .WhatisProblemPointsandlieon.The istimesthelengthof,andlengthofistimesthelengthof .Thelengthofiswhatfractionofthelengthof? Problem areonacircleofradiusand.Point isthemidpointoftheminorarc .Whatisthelengthofthelinesegment Trig betheanglethatsubtendsthearcAB.BythelawofThehalf-angleformulasays,whichisanswerchoiceOtherDefineDasthemidpointofAB,andRthecenterofthecircle.R,C,andDarecollinear,andsinceDisthemidpointofAB,,andso. ,andProblemBricklayerBrendawouldtakehourstobuildachimneyalone,andbricklayerBrandonwouldtakehourstobuilditalone.Whentheyworktogethertheytalkalot,andtheircombinedoutputisdecreasedbybricksperhour.Workingtogether,theybuildthechimneyinhours.Howmanybricksareinthechimney? bethenumberofbricksintheWithouttalking,BrendaandBrandonlayandbricksperhourrespectively,sotogethertheylayperhourtogether.Sincetheyfinishthechimney hours,..ProblemAcone-shapedmountainhasitsbaseontheoceanfloorandhasaheightof8000feet.Thetopofthevolumeofthemountainisabovewater.Whatisthedepthoftheoceanatthebaseofthemountaininfeet?Inacone,radiusandheighteachvaryinverselywithincreasingheight(i.e.theradiusoftheconeformedbycuttingoffthemountainatfeetishalfthatoftheoriginalmountain).Therefore,volumevariesastheinversecubeofincreasingPlugginginourgivencondition,,answerchoiceProblemForeachpositiveinteger ,themeanofthefirst termsofasequenceis isthethtermofthesequence?LettingbethenthpartialsumoftheTheonlypossiblesequencewiththisresultisthesequenceofoddProblem ofequilateralisintheinteriorofunitsquare .Let theregionconsistingofallpointsinside andoutsidewhosedistancefrom isbetweenand.Whatistheareaof ProblemAcirclehasaradiusofandacircumferenceof.WhatisLetbethecircumferenceofthecircle,andletbetheradiusoftheUsinglogproperties,. ,Problem15(noOneachsideofaunitsquare,anequilateraltriangleofsidelength1isconstructed.Oneachnewsideofeachequilateraltriangle,anotherequilateraltriangleofsidelength1isconstructed.Theinteriorsofthesquareandthe12triangleshavenopointsincommon.Let betheregionformedbytheunionofthesquareandallthetriangles,and bethesmallestconvexpolygonthatcontains .Whatistheareaoftheregionthatisinside butoutside )最小，只有A是正確的，BCEDE的話面積都比1/4的時(shí)候大。ProblemArectangularfloormeasuresbyfeet,whereandarepositiveintegerstothesidesofthefloor.Theunpaintedpartofthefloorformsaborderofwidthfootaroundthepaintedrectangleandoccupieshalfoftheareaoftheentirefloor.Howmanypossibilitiesaretherefortheorderedpair?BySimon'sFavoriteFactoringSinceandaretheonlypositivefactoringsof.oryieldingsolutions.Noticethatbecause,thereversedpairsareProblemLetthecoordinatesof beandthecoordinatesof be.Sincethe isparallelto mustbe.Thenslopeof .Theslopeof Supposing, isperpendicularto and,itfollows,tothe asegmentofthelinex=m.Butthatwouldmeanthatthecoordinatesof,contradictingthegiventhataredistinct..Byasimilarlogic,neitherisThismeansthatand isperpendicularto .Sotheslopeof thenegativereciprocaloftheslopeof ,yielding.Becauseisthelengthofthealtitudeoftriangle ,and thelengthof ,theareaof.Since .,whosedigitssum Problem.Apyramidhasasquare and Theareaof .,andtheareasof areand ,respectively.Whatisthevolumeofthepyramid? betheheightofthepyramidand bethedistancefrom .Thesidelengthofthebaseis14.Thesidelengthsofandareand,respectively.WehaveasystemsofequationsthroughthePythagoreanTheorem:Settingthemequaltoeachotherandsimplifying.Therefore,,andthevolumeofthepyramidisProblemAfunctionisdefined forallcomplex arecomplexnumbersand.Supposethatandarebothreal.WhatisthesmallestpossiblevalueofWeneedonlyconcernourselveswiththeimaginaryportionsofand(bothofwhichmustbe0).Theseare:Sinceappearsinbothequations,weletitequal0tosimplifytheequations.Thisyieldstwosingle-variableequations.Equation1tellsusthattheimaginarypartof mustbe ,andequation2tellsusthattherealpartofmustbe. 'sabsolutevalue,welet,answerchoiceB.ProblemMichaelwalksattherateoffeetpersecondonalongstraightpath.Trashpailsarelocatedeveryfeetalongthepath.AgarbagetrucktravelsatfeetpersecondinthesamedirectionasMichaelandstopsforsecondsateachpail.AsMichaelpassesapail,henoticesthetruckaheadofhimjustleavingthenextpail.HowmanytimeswillMichaelandthetruckmeet?PickacoordinatesystemwhereMichael'sstartingpailisandtheonewheretruckstarts.bethecoordinatesofMichaelandtheafterMeetingsbetheir(signed)distanceafter.WehaveThetruckalwaysmovesforseconds,thenstandsstillfor.DuringthefirstsecondsofthecyclethetruckmovesbymetersandMichaelby,henceduringthefirst secondsofthecycleincreasesby .Duringtheremaining decreasesby.Fromthisobservationitisobviousthatafterfourfullcycles,i.e.at,wewillhaveforthefirsttime.Duringthefifthcycle,willfirstgrowfrom ,thenfallfrom HenceMichaelovertakesthetruckwhileitisstandingatthepail.Duringthesixthcycle,willfirstgrowfromto,thenfallfromto.Hencethetruckstartsmoving,overtakesMichaelontheirwaytothenextpail,andthenMichaelovertakesthetruckwhileitisstandingatthepail.Duringtheseventhcycle,willfirstgrowfromto,thenfallfromObviously,fromthispointonwillalwaysbenegative,meaningthatMichaelisalreadytoofarahead.Hencewefoundallmeetings.ThemovementofMichaelandthetruckisplottedbelow:Michaelinblue,thetruckinred.ProblemTwocirclesofradius1aretobeconstructedasfollows.Thecenterofcircle chosenuniformlyandatrandomfromthelinesegmentjoiningand.Thecenterofcircle ischosenuniformlyandatrandom,andindependentlyofthefirstchoice,fromthelinesegmentjoiningto.Whatistheprobabilitythat Twocirclesintersectifthedistancebetweentheircentersislessthanthesumoftheirradii.Inthisproblem, intersectiffInotherwords,thetwochosenX-coordinatesmustdifferbynomorethan.Tofindthisprobability,wedividetheproblemintocases: rangeforagiven (ontheleft)(ontheright)allover2(therangeofpossiblevalues).Thetotalprobabilityforthisrangeisthesumofalltheseprobabilitiesof (overtherangeof )dividedbythetotalrangeof.Thus,thetotalpossibilityforthisinterval.isontheinterval.Inthiscase,anyvalueof willdo,sotheprobabilityfortheintervalissimply.isontheinterval.Thisisidentical,bysymmetry,tocaseThetotalprobabilityistherefore SyntheticCirclescenteredatandwilloverlapifandareclosertoeachotherthanifthecirclesweretangent.Thecirclesaretangentwhenthedistancebetweentheircentersisequaltothesumoftheirradii.Thus,thedistancefromtowillbe areseparatedbyvertically,theymustbeseparatedbyhorizontally.Thus,if,thecirclesintersect.Now,plotthetworandomvariablesandonthecoordinateplane.Eachvariablerangesfrom .Thecirclesintersectifthevariablesarewithinofeachother.Thus,theareainwhichthecirclesdon'tintersectisequaltothetotalareaoftwosmalltrianglesonoppositecorners,eachofarea.WeconcludetheprobabilitythecirclesintersectProblemAparkinglothas16spacesinarow.Twelvecarsarrive,eachofwhichrequiresoneparkingspace,andtheirdriverschosespacesatrandomfromamongtheavailablespaces.AuntieEmthenarrivesinherSUV,whichrequires2adjacentspaces.Whatistheprobabilitythatsheisabletopark?AuntieEmwon'tbeabletoparkonlywhennoneofthefouravailablespotstouch.Wecanformabijectionbetweenallsuchcasesandthenumberofwaystopickfourspotsoutof13:sincenoneofthespotstouch,removeaspotfrombetweeneachofthecars.Fromtheotherdirection,givenfourspotsoutof13,simplyaddaspotbetweeneach.SotheprobabilityshecanparkProblemThesumofthebase-logarithmsofthedivisorsofis.What willbeoftheform.Usingthe ,itsufficestocountthetotalnumberof2'sand5'srunningthroughallpossible .Foreveryfactor,therewillbeanother,soitsufficestocountthetotalnumberof2'soccurringinallfactors(becauseofthissymmetry,thenumberof5'swillbeequal).Andsince,thefinalsumwillbethetotalnumberofoccurringinallfactorsofTherearechoicesfortheexponentof5ineachfactor,andforeachofthosechoices,therearefactors(eachcorrespondingtoadifferentexponentof2),yieldingtotal2's.Thetotalnumberof2'sistherefore.Plugginginouranswerchoicesintoformulayields11(answerchoice)asthecorrectForeverydivisorof, ,wehave.Therearedivisors that .Afterontheparity ,wefindthattheanswerisgiven.ProblemLet.Distinctpoints lieonthe -axis,anddistinctpointslieonthegraphof .Foreverypositiveintegerisanequilateralt