2021年度AMC12真题及答案

AMC12AProblem1Whatisthevalueof ?HYPERLINK""\o"AMC12AProblems/Problem1"SolutionProblem2Forwhatvalueof  does ?HYPERLINK""\o"AMC12AProblems/Problem2"SolutionProblem3Theremaindercanbedefinedforallrealnumbers  and  with  bywhere  denotesthegreatestintegerlessthanorequalto .Whatisthevalueof ?HYPERLINK""\o"AMC12AProblems/Problem3"SolutionProblem4Themean，median，andmodeofthe  datavalues  areallequalto .Whatisthevalueof ?HYPERLINK""\o"AMC12AProblems/Problem4"SolutionProblem5Goldbach'sconjecturestatesthateveryevenintegergreaterthan2canbewrittenasthesumoftwoprimenumbers(forexample, ).Sofar，noonehasbeenabletoprovethattheconjectureistrue，andnoonehasfoundacounterexampletoshowthattheconjectureisfalse.Whatwouldacounterexampleconsistof?HYPERLINK""\o"AMC12AProblems/Problem5"SolutionProblem6Atriangulararrayof  coinshas  coininthefirstrow,  coinsinthesecondrow,  coinsinthethirdrow，andsoonupto  coinsinthe throw.Whatisthesumofthedigitsof  ?HYPERLINK""\o"AMC12AProblems/Problem6"SolutionProblem7Whichofthesedescribesthegraphof  ?HYPERLINK""\o"AMC12AProblems/Problem7"SolutionProblem8Whatistheareaoftheshadedregionofthegiven  rectangle?HYPERLINK""\o"AMC12AProblems/Problem8"SolutionProblem9Thefivesmallshadedsquaresinsidethisunitsquarearecongruentandhavedisjointinteriors.Themidpointofeachsideofthemiddlesquarecoincideswithoneoftheverticesoftheotherfoursmallsquaresasshown.Thecommonsidelengthis ，where  and  arepositiveintegers.Whatis  ?HYPERLINK""\o"AMC12AProblems/Problem9"SolutionProblem10Fivefriendssatinamovietheaterinarowcontaining  seats，numbered  to  fromlefttoright.(Thedirections"left"and"right"arefromthepointofviewofthepeopleastheysitintheseats.)DuringthemovieAdawenttothelobbytogetsomepopcorn.Whenshereturned，shefoundthatBeahadmovedtwoseatstotheright，Cecihadmovedoneseattotheleft，andDeeandEdiehadswitchedseats，leavinganendseatforAda.InwhichseathadAdabeensittingbeforeshegotup?HYPERLINK""\o"AMC12AProblems/Problem10"SolutionProblem11Eachofthe  studentsinacertainsummercampcaneithersing，dance，oract.Somestudentshavemorethanonetalent，butnostudenthasallthreetalents.Thereare  studentswhocannotsing,  studentswhocannotdance，and  studentswhocannotact.Howmanystudentshavetwoofthesetalents?HYPERLINK""\o"AMC12AProblems/Problem11"SolutionProblem12In , , ，and .Point  lieson ，and  bisects .Point  lieson ，and bisects .Thebisectorsintersectat .Whatistheratio  : ?HYPERLINK""\o"AMC12AProblems/Problem12"SolutionProblem13Let  beapositivemultipleof .Oneredballand  greenballsarearrangedinalineinrandomorder.Let  betheprobabilitythatatleast  ofthegreenballsareonthesamesideoftheredball.Observethat  andthat approaches  as  growslarge.Whatisthesumofthedigitsoftheleastvalueof  suchthat ?HYPERLINK""\o"AMC12AProblems/Problem13"SolutionProblem14Eachvertexofacubeistobelabeledwithanintegerfrom  through ，witheachintegerbeingusedonce，insuchawaythatthesumofthefournumbersontheverticesofafaceisthesameforeachface.Arrangementsthatcanbeobtainedfromeachotherthroughrotationsofthecubeareconsideredtobethesame.Howmanydifferentarrangementsarepossible?HYPERLINK""\o"AMC12AProblems/Problem14"SolutionProblem15Circleswithcenters  and ，havingradii  and ，respectively，lieonthesamesideofline  andaretangentto  at  and ，respectively，with  between  and .Thecirclewithcenter  isexternallytangenttoeachoftheothertwocircles.Whatistheareaoftriangle ?HYPERLINK""\o"AMC12AProblems/Problem15"SolutionProblem16Thegraphsof  and  areplottedonthesamesetofaxes.Howmanypointsintheplanewithpositive -coordinateslieontwoormoreofthegraphs?HYPERLINK""\o"AMC12AProblems/Problem16"SolutionProblem17Let  beasquare.Let  and  bethecenters，respectively，ofequilateraltriangleswithbases  and eachexteriortothesquare.Whatistheratiooftheareaofsquare  totheareaofsquare ?HYPERLINK""\o"AMC12AProblems/Problem17"SolutionProblem18Forsomepositiveinteger  thenumber  has  positiveintegerdivisors，including  andthenumber  Howmanypositiveintegerdivisorsdoesthenumber  have?HYPERLINK""\o"AMC12AProblems/Problem18"SolutionProblem19Jerrystartsat  ontherealnumberline.Hetossesafaircoin  times.Whenhegetsheads，hemoves  unitinthepositivedirection；whenhegetstails，hemoves  unitinthenegativedirection.Theprobabilitythathereaches  atsometimeduringthisprocessis  where  and  arerelativelyprimepositiveintegers.Whatis  (Forexample，hesucceedsifhissequenceoftossesis )HYPERLINK""\o"AMC12AProblems/Problem19"SolutionProblem20Abinaryoperation  hasthepropertiesthat  andthat  forallnonzerorealnumbers  and  (Herethedot  representstheusualmultiplicationoperation.)Thesolutiontotheequation  canbewrittenas  where  and  arerelativelyprimepositiveintegers.Whatis HYPERLINK""\o"AMC12AProblems/Problem20"SolutionProblem21Aquadrilateralisinscribedinacircleofradius  Threeofthesidesofthisquadrilateralhavelength  Whatisthelengthofitsfourthside?HYPERLINK""\o"AMC12AProblems/Problem21"SolutionProblem22Howmanyorderedtriples  ofpositiveintegerssatisfy  and ?HYPERLINK""\o"AMC12AProblems/Problem22"SolutionProblem23Threenumbersintheinterval  arechosenindependentlyandatrandom.Whatistheprobabilitythatthechosennumbersarethesidelengthsofatrianglewithpositivearea?HYPERLINK""\o"AMC12AProblems/Problem23"SolutionProblem24Thereisasmallestpositiverealnumber  suchthatthereexistsapositiverealnumber  suchthatalltherootsofthepolynomial  arereal.Infact，forthisvalueof  thevalueof  isunique.Whatisthevalueof HYPERLINK""\o"AMC12AProblems/Problem24"SolutionProblem25Let  beapositiveinteger.BernardoandSilviataketurnswritinganderasingnumbersonablackboardasfollows：Bernardostartsbywritingthesmallestperfectsquarewith  digits.EverytimeBernardowritesanumber，Silviaerasesthelast digitsofit.Bernardothenwritesthenextperfectsquare，Silviaerasesthelast  digitsofit，andthisprocesscontinuesuntilthelasttwonumbersthatremainontheboarddifferbyatleast2.Let  bethesmallestpositiveintegernotwrittenontheboard.Forexample，if ，thenthenumbersthatBernardowritesare ，andthenumbersshowingontheboardafterSilviaerasesare  and ，andthus .Whatisthesumofthedigitsof ?AMC12AAnswerKey1B2C3B4D5E6D7D8D9E10B11E12C13A14C15D16D17B18D19B20A21E22A23C24B25E