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2016AMC12AProblem1Whatisthevalueof?SolutionProblem2Forwhatvalueofdoes?SolutionProblem3Theremaindercanbedefinedforallrealnumbersandwithbywheredenotesthegreatestintegerlessthanorequalto.Whatisthevalueof?SolutionProblem4Themean,median,andmodeofthedatavaluesareallequalto.Whatisthevalueof?SolutionProblem5Goldbach'sconjecturestatesthateveryevenintegergreaterthan2canbewrittenasthesumoftwoprimenumbers(forexample,).Sofar,noonehasbeenabletoprovethattheconjectureistrue,andnoonehasfoundacounterexampletoshowthattheconjectureisfalse.Whatwouldacounterexampleconsistof?SolutionProblem6Atriangulararrayofcoinshascoininthefirstrow,coinsinthesecondrow,coinsinthethirdrow,andsoonuptocoinsinthethrow.Whatisthesumofthedigitsof?SolutionProblem7Whichofthesedescribesthegraphof?SolutionProblem8Whatistheareaoftheshadedregionofthegivenrectangle?SolutionProblem9Thefivesmallshadedsquaresinsidethisunitsquarearecongruentandhavedisjointinteriors.Themidpointofeachsideofthemiddlesquarecoincideswithoneoftheverticesoftheotherfoursmallsquaresasshown.Thecommonsidelengthis,whereandarepositiveintegers.Whatis?SolutionProblem10Fivefriendssatinamovietheaterinarowcontainingseats,numberedtofromlefttoright.(Thedirectionsleftandrightarefromthepointofviewofthepeopleastheysitintheseats.)DuringthemovieAdawenttothelobbytogetsomepopcorn.Whenshereturned,shefoundthatBeahadmovedtwoseatstotheright,Cecihadmovedoneseattotheleft,andDeeandEdiehadswitchedseats,leavinganendseatforAda.InwhichseathadAdabeensittingbeforeshegotup?SolutionProblem11Eachofthestudentsinacertainsummercampcaneithersing,dance,oract.Somestudentshavemorethanonetalent,butnostudenthasallthreetalents.Therearestudentswhocannotsing,studentswhocannotdance,andstudentswhocannotact.Howmanystudentshavetwoofthesetalents?SolutionProblem12In,,,and.Pointlieson,andbisects.Pointlieson,andbisects.Thebisectorsintersectat.Whatistheratio:?SolutionProblem13Letbeapositivemultipleof.Oneredballandgreenballsarearrangedinalineinrandomorder.Letbetheprobabilitythatatleastofthegreenballsareonthesamesideoftheredball.Observethatandthatapproachesasgrowslarge.Whatisthesumofthedigitsoftheleastvalueofsuchthat?SolutionProblem14Eachvertexofacubeistobelabeledwithanintegerfromthrough,witheachintegerbeingusedonce,insuchawaythatthesumofthefournumbersontheverticesofafaceisthesameforeachface.Arrangementsthatcanbeobtainedfromeachotherthroughrotationsofthecubeareconsideredtobethesame.Howmanydifferentarrangementsarepossible?SolutionProblem15Circleswithcentersand,havingradiiand,respectively,lieonthesamesideoflineandaretangenttoatand,respectively,withbetweenand.Thecirclewithcenterisexternallytangenttoeachoftheothertwocircles.Whatistheareaoftriangle?SolutionProblem16Thegraphsofandareplottedonthesamesetofaxes.Howmanypointsintheplanewithpositive-coordinateslieontwoormoreofthegraphs?SolutionProblem17Letbeasquare.Letandbethecenters,respectively,ofequilateraltriangleswithbasesandeachexteriortothesquare.Whatistheratiooftheareaofsquaretotheareaofsquare?SolutionProblem18Forsomepositiveintegerthenumberhaspositiveintegerdivisors,includingandthenumberHowmanypositiveintegerdivisorsdoesthenumberhave?SolutionProblem19Jerrystartsatontherealnumberline.Hetossesafaircointimes.Whenhegetsheads,hemovesunitinthepositivedirection;whenhegetstails,hemovesunitinthenegativedirection.Theprobabilitythathereachesatsometimeduringthisprocessiswhereandarerelativelyprimepositiveintegers.Whatis(Forexample,hesucceedsifhissequenceoftossesis)SolutionProblem20Abinaryoperationhasthepropertiesthatandthatforallnonzerorealnumbersand(Herethedotrepresentstheusualmultiplicationoperation.)Thesolutiontotheequationcanbewrittenaswhereandarerelativelyprimepositiveintegers.WhatisSolutionProblem21AquadrilateralisinscribedinacircleofradiusThreeofthesidesofthisquadrilateralhavelengthWhatisthelengthofitsfourthside?SolutionProblem22Howmanyorderedtriplesofpositiveintegerssatisfyand?SolutionProblem23Threenumbersintheintervalarechosenindependentlyandatrandom.Whatistheprobabilitythatthechosennumbersarethesidelengthsofatrianglewithpositivearea?SolutionProblem24Thereisasmallestpositiverealnumbersuchthatthereexistsapositiverealnumbersuchthatalltherootsofthepolynomialarereal.Infact,forthisvalueofthevalueofisunique.WhatisthevalueofSolutionProblem25Letbeapositiveinteger.BernardoandSilviataketurnswritinganderasingnumbersonablackboardasfollows:Bernardostartsbywritingthesmallestperfectsquarewithdigits.EverytimeBernardowritesanumber,Silviaerasesthelastdigitsofit.Bernardothenwritesthenextperfectsquare,Silviaerasesthelastdigitsofit,andthisprocesscontinuesuntilthelasttwonumbersthatremainontheboarddifferbyatleast2.Letbethesmallestpositiveintegernotwrittenontheboard.Forexample,if,thenthenumbersthatBernardowritesare,andthenumbersshowingontheboardafterSilviaerasesareand,andthus.Whatisthesumofthedigitsof?2016AMC12AAnswerKey1B2C3B4D5E6D7D8D9E10B11E12C13A14C15D16D17B18D19B20A21E22A23C24B25E