2023年AMC12真题及答案

AMC12A

Problem1

Whatisthevalueof

?

HYPERLINK\o"2023AMC12AProblems/Problem1"

Solution

Problem2

Forwhatvalueof

does

?

HYPERLINK\o"2023AMC12AProblems/Problem2"

Solution

Problem3

Theremaindercanbedefinedforallrealnumbers

and

with

bywhere

denotesthegreatestintegerlessthanorequalto

.Whatisthevalueof

?

HYPERLINK\o"2023AMC12AProblems/Problem3"

Solution

Problem4

Themean,median,andmodeofthe

datavalues

areallequalto

.Whatisthevalueof

?

HYPERLINK\o"2023AMC12AProblems/Problem4"

Solution

Problem5

Goldbach'sconjecturestatesthateveryevenintegergreaterthan2canbewrittenasthesumoftwoprimenumbers(forexample,

).Sofar,noonehasbeenabletoprovethattheconjectureistrue,andnoonehasfoundacounterexampletoshowthattheconjectureisfalse.Whatwouldacounterexampleconsistof?

HYPERLINK\o"2023AMC12AProblems/Problem5"

Solution

Problem6

Atriangulararrayof

coinshas

coininthefirstrow,

coinsinthesecondrow,

coinsinthethirdrow,andsoonupto

coinsinthe

throw.Whatisthesumofthedigitsof

?

HYPERLINK\o"2023AMC12AProblems/Problem6"

Solution

Problem7

Whichofthesedescribesthegraphof

?

HYPERLINK\o"2023AMC12AProblems/Problem7"

Solution

Problem8

Whatistheareaoftheshadedregionofthegiven

rectangle?

HYPERLINK\o"2023AMC12AProblems/Problem8"

Solution

Problem9

Thefivesmallshadedsquaresinsidethisunitsquarearecongruentandhavedisjointinteriors.Themidpointofeachsideofthemiddlesquarecoincideswithoneoftheverticesoftheotherfoursmallsquaresasshown.Thecommonsidelengthis

,where

and

arepositiveintegers.Whatis

?

HYPERLINK\o"2023AMC12AProblems/Problem9"

Solution

Problem10

Fivefriendssatinamovietheaterinarowcontaining

seats,numbered

to

fromlefttoright.(Thedirections"left"and"right"arefromthepointofviewofthepeopleastheysitintheseats.)DuringthemovieAdawenttothelobbytogetsomepopcorn.Whenshereturned,shefoundthatBeahadmovedtwoseatstotheright,Cecihadmovedoneseattotheleft,andDeeandEdiehadswitchedseats,leavinganendseatforAda.InwhichseathadAdabeensittingbeforeshegotup?

HYPERLINK\o"2023AMC12AProblems/Problem10"

Solution

Problem11

Eachofthe

studentsinacertainsummercampcaneithersing,dance,oract.Somestudentshavemorethanonetalent,butnostudenthasallthreetalents.Thereare

studentswhocannotsing,

studentswhocannotdance,and

studentswhocannotact.Howmanystudentshavetwoofthesetalents?

HYPERLINK\o"2023AMC12AProblems/Problem11"

Solution

Problem12

In

,

,

,and

.Point

lieson

,and

bisects

.Point

lieson

,and

bisects

.Thebisectorsintersectat

.Whatistheratio

:

?

HYPERLINK\o"2023AMC12AProblems/Problem12"

Solution

Problem13

Let

beapositivemultipleof

.Oneredballand

greenballsarearrangedinalineinrandomorder.Let

betheprobabilitythatatleast

ofthegreenballsareonthesamesideoftheredball.Observethat

andthat

approaches

as

growslarge.Whatisthesumofthedigitsoftheleastvalueof

suchthat

?

HYPERLINK\o"2023AMC12AProblems/Problem13"

Solution

Problem14

Eachvertexofacubeistobelabeledwithanintegerfrom

through

,witheachintegerbeingusedonce,insuchawaythatthesumofthefournumbersontheverticesofafaceisthesameforeachface.Arrangementsthatcanbeobtainedfromeachotherthroughrotationsofthecubeareconsideredtobethesame.Howmanydifferentarrangementsarepossible?

HYPERLINK\o"2023AMC12AProblems/Problem14"

Solution

Problem15

Circleswithcenters

and

,havingradii

and

,respectively,lieonthesamesideofline

andaretangentto

at

and

,respectively,with

between

and

.Thecirclewithcenter

isexternallytangenttoeachoftheothertwocircles.Whatistheareaoftriangle

?

HYPERLINK\o"2023AMC12AProblems/Problem15"

Solution

Problem16

Thegraphsof

and

areplottedonthesamesetofaxes.Howmanypointsintheplanewithpositive

-coordinateslieontwoormoreofthegraphs?

HYPERLINK\o"2023AMC12AProblems/Problem16"

Solution

Problem17

Let

beasquare.Let

and

bethecenters,respectively,ofequilateraltriangleswithbases

and

eachexteriortothesquare.Whatistheratiooftheareaofsquare

totheareaofsquare

?

HYPERLINK\o"2023AMC12AProblems/Problem17"

Solution

Problem18

Forsomepositiveinteger

thenumber

has

positiveintegerdivisors,including

andthenumber

Howmanypositiveintegerdivisorsdoesthenumber

have?

HYPERLINK\o"2023AMC12AProblems/Problem18"

Solution

Problem19

Jerrystartsat

ontherealnumberline.Hetossesafaircoin

times.Whenhegetsheads,hemoves

unitinthepositivedirection;whenhegetstails,hemoves

unitinthenegativedirection.Theprobabilitythathereaches

atsometimeduringthisprocessis

where

and

arerelativelyprimepositiveintegers.Whatis

(Forexample,hesucceedsifhissequenceoftossesis

)

HYPERLINK\o"2023AMC12AProblems/Problem19"

Solution

Problem20

Abinaryoperation

hasthepropertiesthat

andthat

forallnonzerorealnumbers

and

(Herethedot

representstheusualmultiplicationoperation.)Thesolutiontotheequation

canbewrittenas

where

and

arerelativelyprimepositiveintegers.Whatis

HYPERLINK\o"2023AMC12AProblems/Problem20"

Solution

Problem21

Aquadrilateralisinscribedinacircleofradius

Threeofthesidesofthisquadrilateralhavelength

Whatisthelengthofitsfourthside?

HYPERLINK\o"2023AMC12AProblems/Problem21"

Solution

Problem22

Howmanyorderedtriples

ofpositiveintegerssatisfy

and

?

HYPERLINK\o"2023AMC12AProblems/Problem22"

Solution

Problem23

Threenumbersintheinterval

arechosenindependentlyandatrandom.Whatistheprobabilitythatthechosennumbersarethesidelengthsofatrianglewithpositivearea?

HYPERLINK\o"2023AMC12AProblems/Problem23"

Solution

Problem24

Thereisasmallestpositiverealnumber

suchthatthereexistsapositiverealnumber

suchthatalltherootsofthepolynomial

arereal.Infact,forthisvalueof

thevalueof

isunique.Whatisthevalueof

HYPERLINK\o"2023AMC12AProblems/Problem24"

Solution

Problem25

Let

beapositiveinteger.BernardoandSilviataketurnswritinganderasingnumbersona