AMC12B试题及解答Word文档下载推荐.docx

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AMC12B试题及解答Word文档下载推荐.docx

dragonfly

Sincetheharmonicmeanis

Jtimestheirproductdividedbytheirsum,wegettheequation

2x1x2016

1+2016

whichisthen

4032

（A）2

2017

whichisfinallyclosestto

Problem3

LetX-—2016.WhatisthevalueofIM—力I一I刎—兽?

Firstofall,letspluginallofthe©

sintotheequation.

||-2016|一（-2016）1一|-20161-（-2016）

Thenwesimplifytoget

|2016+20161-20161^2016

whichsimplifiesinto

2016+2016

（D）4032

andfinallyweget

Problem4

Theratioofthemeasuresoftwoacuteanglesis5:

1,andthecomplement

ofoneofthesetwoanglesistwiceaslargeasthecomplementoftheother.Whatisthesumofthedegreemeasuresofthetwoangles?

（A）75（B）90（C）135（D）150（E）270

Wesetupequationstofindeachangle.Thelargeranglewillbe

representedasxandthelargeranglewillwerepresentedas/indegrees.

Thisimpliesthat

4x=5y

and

2x（90—j:

）=90—y

（C）135

sincethelargertheoriginalangle,thesmallerthecomplement.

Wethenfindthat「一.£

and彳—沉丄，andtheirsumis

Problem5

TheWarof1S12startedwithadeclarationofwaronThursday,

June1812.Thepeacetreatytoendthewarwassigned019dayslater,

onDecember24,1814.Onwhatdayoftheweekwasthetreatysigned?

Tofindwhatdayoftheweekitisin919days,wehavetodivide919by7toseetheremainder,andthenaddtheremaindertothecurrentday.Weget

919

that7hasaremainderof2,soweincreasethecurrentdayby2to

（B）SatiLrday

get———I

Problem6

AllthreeverticesofAAJ3Clieontheparaboladefinedby9=T,withAattheoriginandBCparalleltothe工-axis.Theareaofthetriangle

is64.WhatisthelengthofDC?

（A）4（B）6（C）8（D）10（E）16

Albert471

Plottingpoints门and:

onthegraphshowsthattheyareat（—上、止）and■,r”），whichisisosceles.Bysettingupthetrianglearea

Makingx=4,andthelength

（c）s

64=—*2t*rr2=G4formulayouget:

2ofDCissotheansweris

Problem7

Joshwritesthenumbers1’2,3,・•・、99,100.Hemarksout1,skipsthenextnumber

（2）,marksout3,andcontinuesskippingandmarkingoutthenextnumbertotheendofthelist.Thenhegoesbacktothestartofhislist,

marksoutthefirstremainingnumber⑵,skipsthenextnumber⑴,marks

out6,skips8,marksout10,andsoontotheend.Joshcontinuesinthis

（D）64

manneruntilonlyonenumberremains.Whatisthatnumber?

（A）13（B）32（C）56（D）64

ByAlbert471

Followingthepattern,youarecrossingout...

（E）96

Time1:

Everynon-multipleof

Time2:

Time3:

Followingthispattern,youareleftwitheverymultipleofG4whichis

only

Problem8

Athinpieceofwoodofuniformdensityintheshapeofanequilateral

trianglewithsidelength3inchesweighs12ounces.Asecondpieceofthesametypeofwood,withthesamethickness,alsointheshapeofanequilateraltriangle,hassidelengthof5inches.Whichofthefollowingis

closesttotheweight,inounces,ofthesecondpiece?

（A）14,0（B）1G.0（C）20.0（D）33.3（E）55.6

Wecansolvethisproblembyusingsimilartriangles,sincetwoequilateral

trianglesarealwayssimilar.Wecanthenuse

100

；

whichisclosest

Anotherapproachtothisproblem,verysimilartothepreviousonebutperhapsexplainedmorethoroughly,istouseproportions.First,sincethe

thicknessanddensityarethesame,wecansetupaproportionbasedontheprinciplethatV,thusdV=m.

However,sincedensityandthicknessarethesame

and^4ocb3（recognizingthattheareaofanequilateraltriangleiswecansaythatm<

x涉.

Then,byincreasingsbyafactorof3,s2isincreasedbyafactorof

（D）333

TH—12#■—

thus9or

Problem9

Carldecidedtofenceinhisrectangulargarden.Hebought20fenceposts,

placedoneoneachofthefourcorners,andspacedouttherestevenlyalongtheedgesofthegarden,leavingexactly4yardsbetween

neighboringposts.Thelongersideofhisgarden,includingthecorners,hastwiceasmanypostsastheshorterside,includingthecorners.Whatisthearea,insquareyards,ofCarl'

sgarden?

（A）256（B）336（C）384（D）448（E）512

Tostart,usealgebratodeterminethenumberofpostsoneachside.Youhave（thelongsidescountfor2becausetherearetwiceasmany）Gx=20+4（eachcornerisdoublecountedsoyoumustadd4）

Makingtheshorterendhave1,andthelongerend

have8.（（8一1）7"

（（4-1）木1）=28.12=336.Therefore,the

（B）336

answeris

Problem10

AquadrilateralhasverticesP（a、b）,and人一"

），

whereaandbareintegerswithd>

0.TheareaofPQRSis16.What

is«

+b?

Bydistaneeformulawehave'

■1'

■'

■1■'

I■?

■■

SImplifyingweget—^）（a+b）=8.ThusU+ftanda—bhavetobea

■-andremainintegersis

factorof8.Theonlywayforthemtobefactorsof

if:

I；

and:

丄Sotheansweris

Solutionbyl_Dont_Do_Math

Solution2

Solutionbye_power_pi_times_i

BytheShoelaceTheorem,theareaofthequadrilateralis2ti2—2b'

soa2—b2—8.Sineeaand&

areintegers,u=3andb=1,

a+b=（A）4I

so.

Problem11

Howmanysquareswhosesidesareparalleltotheaxesandwhoseverticeshavecoordinatesthatareintegerslieentirelywithintheregionboundedbytheline站=齐工，theline~_0.1andtheline龙=5.1?

（A）30（B）41（C）45（D）50（E）57

（Note:

diagramisneeded）

Ifwedrawapictureshowingthetriangle,weseethatitwouldbeeasiertocountthesquaresverticallyandnothorizontally.Theupperbound

is1Gsquares（卩=5」木汗），andthelimitforthe工-valueis5squares.First

wecountthe1*1squares.Inthebackrow,thereare12squareswithlength1becausey=4床ugeneratessquaresfrom@,0）to（蠶仆）,andcontinuingonwehave9,6,and3for©

valuesfor1,2,and3intheequationV—兀卫.Sothereare12+9+6+3=30squareswithlengthIinthefigure.For_一’squares,eachsquaretakesup_unleftand2unup.Squarescanalsooverlap.For2*2squares,thebackrow

stretchesfrom（N°

〉to（'

兀），sothereare8squareswithlength2ina2by0box.Repeatingtheprocess,thenextrowstretchesfrom（2,0）

to（2,2打）

sothereare5squares.Continuingandaddingupin

theend,thereare

”■:

乞一、：

squareswithlength-inthefigure.

Squareswithlength3inthebackrowstartatandendat（2*277）,so

thereare4suchsquaresinthebackrow.Asthefrontrowstarts

at（10andendsat（1"

）thereare

I1squareswithlength■■.As

squareswithlengthJwouldnotfitinthetriangle,theanswer

is；

I丨whichis

D）5（）

Problem12

Allthenumbers1,N3,4,5,6,7*&

0arewrittenina3x3arrayofsquares,

onenumberineachsquare,insuchawaythatiftwonumbersare

consecutivethentheyoccupysquaresthatshareanedge.Thenumbersinthefourcornersaddupto18.Whatisthenumberinthecenter?

（A）5（B）6（C）7（D）8（E）9

SolutionbyMlux:

Drawa3x3matrix.Noticethatnoadjacentnumbers

couldbeinthecornerssincetwoconsecutivenumbersmustsharean

edge.Nowfind4nonconsecutivenumbersthataddupto18.

Trying1+3+5+9—18works.Placeeachoddnumberinthecornerin

aclockwiseorder.Thenfillinthespaces.Therehastobea2inbetween

the1and3.Thereisa4between3and5.Thefinalgridshouldsimilartothis.

1,2.3

8,7A

9.6,5

（C）7

isinthemiddle.

Ifwecolorthesquarelikeachessboard,sincethenumbersaltrenatebetweenevenandodd,andtherearefiveoddnumbersandfourevennumbers,theoddnumbersmustbeinthecorners/center,whiletheevennumbersmustbeontheedges.Sincetheoddnumbersaddupto25,andthenumbersinthecornersaddupto18,thenumberinthecentermustbe25-18=7

Problem13

AliceandBoblive10milesapart.OnedayAlicelooksduenorthfromherhouseandseesanairplane.AtthesametimeBoblooksduewestfromhishouseandseesthesameairplane.Theangleofelevationoftheairplaneis30nfromAlice'

spositionand60^fromBob'

sposition.Whichofthefollowingisclosesttotheairplane'

saltitude,inmiles?

（A）3.5（B）4（C）4*5（D）5（E）5.5

Let'

ssetthealtitude=z,distaneefromAlicetoairplane'

sgroundposition（pointrightbelowairplane）二yanddistaneefromBobtoairplane'

sground

position=x

FromAlice'

spointofview,

y.cos30品

cos

tan（0）=-GO=S1D男=亞工二㊁

FromBob'

T.CUSGO.So,VO

、'

l■'

=about5.5.

Weknowthat*+/=丄°

Solvingtheequation（byplugginginxandy）,wegetz=

So,answeris

solutionbysudeepnarala

Non-trigsolutionbye_power_pi_times_i

SetthedistaneefromAlice'

sandBob'

spositiontothepointdirectlybelowtheairplanetobexand甘,respectively.FromthePythagorean

3or.Solvingthe

Theorem,◎?

+『=100.Asbothare30—60—90triangles,thealtitudeoftheairplanecanbeexpressedas

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