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