AMC数学竞赛12A组真题及答案解析资料Word格式.docx

AMC数学竞赛12A组真题及答案解析资料Word格式.docx

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A.48

B.49

C.50

D.51

E.52

Problem3

Howmanywayscanastudentschedule3mathematicscourses—algebra,geometry,andnumbertheory—ina6-perioddayifnotwomathematicscoursescanbetakeninconsecutiveperiods?

（Whatcoursesthestudenttakesduringtheother3periodsisofnoconcernhere.）

A.3

B.6

C.12

D.18

E.24

Problem4

Alice,Bob,andCharliewereonahikeandwerewonderinghowfarawaythenearesttownwas.WhenAlicesaid,“Weareatleast6milesaway,”Bobreplied,“Weareatmost5milesaway.”Charliethenremarked,“Actuallythenearesttownisatmost4milesaway.”Itturnedoutthatnoneofthethreestatementsweretrue.Let 

d 

bethedistanceinmilestothenearesttown.Whichofthefollowingintervalsisthesetofallpossiblevaluesof 

d?

A.（0,4）

B.（4,5）

C.（4,6）

D.（5,6）

E.（5,+∞）

Problem5

Whatisthesumofallpossiblevaluesof 

k 

forwhichthepolynomials 

x^2-3x+2 

and 

x^2-5x+k 

havearootincommon?

B.4

C.5

D.6

E.10

Problem6

Forpositiveintegers 

m 

n 

suchthat 

m+10<

n+1,boththemeanandthemedianoftheset 

{m,m+4,m+10,n+1,n+2,2n} 

areequalto 

n.Whatis 

m+n?

A.20

B.21

C.22

D.23

Problem7

Forhowmany（notnecessarilypositive）integervaluesof 

isthevalueof 

4000·

（2/5）^naninteger?

C.6

D.8

E.9

Problem8

Allofthetrianglesinthediagrambelowaresimilartoiscocelestriangle 

ABC,inwhich 

AB=AC.Eachofthe7smallesttriangleshasarea1,and 

△ABC 

hasarea40.Whatistheareaoftrapezoid 

DBCE?

A.16

B.18

C.20

D.22

Problem9

Whichofthefollowingdescribesthelargestsubsetofvaluesof 

y 

withintheclosedinterval 

[0,] 

forwhich（x,y）<

=sin（x）+sin（y）forevery 

x 

between 

0 

inclusive?

Problem10

Howmanyorderedpairsofrealnumbers 

（x,y） 

satisfythefollowingsystemofequations?

A.1

B.2

C.3

D.4

E.8

Problem11

Apapertrianglewithsidesoflengths3,4,and5inches,asshown,isfoldedsothatpoint 

A 

fallsonpoint 

B.Whatisthelengthininchesofthecrease?

Problem12

Let 

S 

beasetof6integerstakenfrom 

{1,2,…,12} 

withthepropertythatif 

a 

b 

areelementsof 

with 

a<

b,then 

isnotamultipleof 

a.Whatistheleastpossiblevalueofanelementin 

S？

A.2

B.3

C.4

D.5

E.7

Problem13

Howmanynonnegativeintegerscanbewrittenintheform

where 

for 

?

A.512

B.729

C.1094

D.3281

E.59048

Problem14

Thesolutionstotheequation 

where 

isapositiverealnumberotherthan 

1/3 

or 

1/2,canbewrittenas 

p/q 

p 

q 

arerelativelyprimepositiveintegers.Whatis 

p+q?

A.5

B.13

C.17

D.31

E.35

Problem15

Ascanningcodeconsistsofa 

7x7 

gridofsquares,withsomeofitssquarescoloredblackandtherestcoloredwhite.Theremustbeatleastonesquareofeachcolorinthisgridof 

49 

squares.Ascanningcodeiscalled[i]symmetric[/i]ifitslookdoesnotchangewhentheentiresquareisrotatedbyamultipleof 

90°

counterclockwisearounditscenter,norwhenitisreflectedacrossalinejoiningoppositecornersoralinejoiningmidpointsofoppositesides.Whatisthetotalnumberofpossiblesymmetricscanningcodes?

A.510

B.1022

C.8190

D.8192

E.65534

Problem16

Whichofthefollowingdescribesthesetofvaluesof 

forwhichthecurvesx^2=y^2=a^2 

y=x^2-a 

inthereal 

xy-planeintersectatexactly 

3 

points?

A.a=1/4

B.1/4<

1/2

C.a>

1/4

D.a=1/2

E.a>

Problem17

FarmerPythagorashasafieldintheshapeofarighttriangle.Therighttriangle’slegshavelengths3and4units.Inthecornerwherethosesidesmeetatarightangle,heleavesasmallunplantedsquare 

sothatfromtheairitlooksliketherightanglesymbol.Therestofthefieldisplanted.Theshortestdistancefrom 

tothehypotenuseis2units.Whatfractionofthefieldisplanted?

A.25/27

B.26/27

C.73/75

D.145/147

E.74/75

Problem18

Triangle 

ABC 

AB=50 

AC=10 

hasarea 

120.Let 

D 

bethemidpointof 

andlet 

E 

.Theanglebisectorof 

intersects 

at 

F 

G,respectively.Whatistheareaofquadrilateral 

FDBG?

A.60

B.65

C.70

D.75

E.80

Problem19

bethesetofpositiveintegersthathavenoprimefactorsotherthan 

2, 

3,or 

5.Theinfinitesum1/1+1/2+1/3+1/4+1/5+1/6+1/8+1/9+1/10+1/12+1/15+1/16+1/18+1/20+…ofthereciprocalsoftheelementsof 

c