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2023AMC12AProblem1Whatisthevalueof
?SolutionProblem2Forwhatvalueof
does
?SolutionProblem3Theremaindercanbedefinedforallrealnumbers
and
with
bywhere
denotesthegreatestintegerlessthanorequalto
.Whatisthevalueof
?SolutionProblem4Themean,median,andmodeofthe
datavalues
areallequalto
.Whatisthevalueof
?SolutionProblem5Goldbach'sconjecturestatesthateveryevenintegergreaterthan2canbewrittenasthesumoftwoprimenumbers(forexample,
).Sofar,noonehasbeenabletoprovethattheconjectureistrue,andnoonehasfoundacounterexampletoshowthattheconjectureisfalse.Whatwouldacounterexampleconsistof?SolutionProblem6Atriangulararrayof
coinshas
coininthefirstrow,
coinsinthesecondrow,
coinsinthethirdrow,andsoonupto
coinsinthe
throw.Whatisthesumofthedigitsof
?SolutionProblem7Whichofthesedescribesthegraphof
?SolutionProblem8Whatistheareaoftheshadedregionofthegiven
rectangle?SolutionProblem9Thefivesmallshadedsquaresinsidethisunitsquarearecongruentandhavedisjointinteriors.Themidpointofeachsideofthemiddlesquarecoincideswithoneoftheverticesoftheotherfoursmallsquaresasshown.Thecommonsidelengthis
,where
and
arepositiveintegers.Whatis
?SolutionProblem10Fivefriendssatinamovietheaterinarowcontaining
seats,numbered
to
fromlefttoright.(Thedirections"left"and"right"arefromthepointofviewofthepeopleastheysitintheseats.)DuringthemovieAdawenttothelobbytogetsomepopcorn.Whenshereturned,shefoundthatBeahadmovedtwoseatstotheright,Cecihadmovedoneseattotheleft,andDeeandEdiehadswitchedseats,leavinganendseatforAda.InwhichseathadAdabeensittingbeforeshegotup?SolutionProblem11Eachofthe
studentsinacertainsummercampcaneithersing,dance,oract.Somestudentshavemorethanonetalent,butnostudenthasallthreetalents.Thereare
studentswhocannotsing,
studentswhocannotdance,and
studentswhocannotact.Howmanystudentshavetwoofthesetalents?SolutionProblem12In
,
,
,and
.Point
lieson
,and
bisects
.Point
lieson
,and
bisects
.Thebisectorsintersectat
.Whatistheratio
:
?SolutionProblem13Let
beapositivemultipleof
.Oneredballand
greenballsarearrangedinalineinrandomorder.Let
betheprobabilitythatatleast
ofthegreenballsareonthesamesideoftheredball.Observethat
andthat
approaches
as
growslarge.Whatisthesumofthedigitsoftheleastvalueof
suchthat
?SolutionProblem14Eachvertexofacubeistobelabeledwithanintegerfrom
through
,witheachintegerbeingusedonce,insuchawaythatthesumofthefournumbersontheverticesofafaceisthesameforeachface.Arrangementsthatcanbeobtainedfromeachotherthroughrotationsofthecubeareconsideredtobethesame.Howmanydifferentarrangementsarepossible?SolutionProblem15Circleswithcenters
and
,havingradii
and
,respectively,lieonthesamesideofline
andaretangentto
at
and
,respectively,with
between
and
.Thecirclewithcenter
isexternallytangenttoeachoftheothertwocircles.Whatistheareaoftriangle
?SolutionProblem16Thegraphsof
and
areplottedonthesamesetofaxes.Howmanypointsintheplanewithpositive
-coordinateslieontwoormoreofthegraphs?SolutionProblem17Let
beasquare.Let
and
bethecenters,respectively,ofequilateraltriangleswithbases
and
eachexteriortothesquare.Whatistheratiooftheareaofsquare
totheareaofsquare
?SolutionProblem18Forsomepositiveinteger
thenumber
has
positiveintegerdivisors,including
andthenumber
Howmanypositiveintegerdivisorsdoesthenumber
have?SolutionProblem19Jerrystartsat
ontherealnumberline.Hetossesafaircoin
times.Whenhegetsheads,hemoves
unitinthepositivedirection;whenhegetstails,hemoves
unitinthenegativedirection.Theprobabilitythathereaches
atsometimeduringthisprocessis
where
and
arerelativelyprimepositiveintegers.Whatis
(Forexample,hesucceedsifhissequenceoftossesis
)SolutionProblem20Abinaryoperation
hasthepropertiesthat
andthat
forallnonzerorealnumbers
and
(Herethedot
representstheusualmultiplicationoperation.)Thesolutiontotheequation
canbewrittenas
where
and
arerelativelyprimepositiveintegers.Whatis
SolutionProblem21Aquadrilateralisinscribedinacircleofradius
Threeofthesidesofthisquadrilateralhavelength
Whatisthelengthofitsfourthside?SolutionProblem22Howmanyorderedtriples
ofpositiveintegerssatisfy
and
?SolutionProblem23Threenumbersintheinterval
arechosenindependentlyandatrandom.Whatistheprobabilitythatthechosennumbersarethesidelengthsofatrianglewithpositivearea?SolutionProblem24Thereisasmallestpositiverealnumber
suchthatthereexistsapositiverealnumber
suchthatalltherootsofthepolynomial
arereal.Infact,forthisvalueof
thevalueof
isunique.Whatisthevalueof
SolutionProblem25Let
beapositiveinteger.BernardoandSilviataketurnswritinganderasingnumbersonablackboardasfollows:Bernardostartsbywritingthesmallestperfectsquarewith
digits.EverytimeB
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