2016年 AMC 12B 試題與解答

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2016AMC12B

Problem1

Whatisthevalueof

when

?

Solution

By:Dragonfly

Wefindthat

isthesameas

,sinceanumbertothepowerof

isjustthereciprocalofthatnumber.Wethengettheequationtobe

Wecanthensimplifytheequationtoget

Problem2

Theharmonicmeanoftwonumberscanbecalculatedastwicetheirproductdividedbytheirsum.Theharmonicmeanof

and

isclosesttowhichinteger?

Solution

By:dragonfly

Sincetheharmonicmeanis

timestheirproductdividedbytheirsum,wegettheequation

whichisthen

whichisfinallyclosestto

.

Problem3

Let

.Whatisthevalueof

?

Solution

By:dragonfly

Firstofall,letspluginallofthe

'sintotheequation.

Thenwesimplifytoget

whichsimplifiesinto

andfinallyweget

Problem4

Theratioofthemeasuresoftwoacuteanglesis

,andthecomplementofoneofthesetwoanglesistwiceaslargeasthecomplementoftheother.Whatisthesumofthedegreemeasuresofthetwoangles?

Solution

By:dragonfly

Wesetupequationstofindeachangle.Thelargeranglewillberepresentedas

andthelargeranglewillwerepresentedas

,indegrees.Thisimpliesthat

and

sincethelargertheoriginalangle,thesmallerthecomplement.

Wethenfindthat

and

,andtheirsumis

Problem5

TheWarof

startedwithadeclarationofwaronThursday,June

,

.Thepeacetreatytoendthewarwassigned

dayslater,onDecember

,

.Onwhatdayoftheweekwasthetreatysigned?

Solution

By:dragonfly

Tofindwhatdayoftheweekitisin

days,wehavetodivide

by

toseetheremainder,andthenaddtheremaindertothecurrentday.Wegetthat

hasaremainderof2,soweincreasethecurrentdayby

toget

Problem6

Allthreeverticesof

lieontheparaboladefinedby

,with

attheoriginand

paralleltothe

-axis.Theareaofthetriangleis

.Whatisthelengthof

?

Solution

By:Albert471

Plottingpoints

and

onthegraphshowsthattheyareat

and

,whichisisosceles.Bysettingupthetriangleareaformulayouget:

Makingx=4,andthelengthof

is

,sotheansweris

.

Problem7

Joshwritesthenumbers

.Hemarksout

,skipsthenextnumber

,marksout

,andcontinuesskippingandmarkingoutthenextnumbertotheendofthelist.Thenhegoesbacktothestartofhislist,marksoutthefirstremainingnumber

,skipsthenextnumber

,marksout

,skips

,marksout

,andsoontotheend.Joshcontinuesinthismanneruntilonlyonenumberremains.Whatisthatnumber?

Solution

ByAlbert471

Followingthepattern,youarecrossingout...

Time1:Everynon-multipleof

Time2:Everynon-multipleof

Time3:Everynon-multipleof

Followingthispattern,youareleftwitheverymultipleof

whichisonly.

Problem8

Athinpieceofwoodofuniformdensityintheshapeofanequilateraltrianglewithsidelength

inchesweighs

ounces.Asecondpieceofthesametypeofwood,withthesamethickness,alsointheshapeofanequilateraltriangle,hassidelengthof

inches.Whichofthefollowingisclosesttotheweight,inounces,ofthesecondpiece?

Solution1

By:dragonfly

Wecansolvethisproblembyusingsimilartriangles,sincetwoequilateraltrianglesarealwayssimilar.Wecanthenuse

.

Wecanthensolvetheequationtoget

whichisclosestto

Solution2

Anotherapproachtothisproblem,verysimilartothepreviousonebutperhapsexplainedmorethoroughly,istouseproportions.First,sincethethicknessanddensityarethesame,wecansetupaproportionbasedontheprinciplethat

,thus

.

However,sincedensityandthicknessarethesameand

(recognizingthattheareaofanequilateraltriangleis

),wecansaythat

.

Then,byincreasingsbyafactorof

,

isincreasedbyafactorof

,thus

or

.

Problem9

Carldecidedtofenceinhisrectangulargarden.Hebought

fenceposts,placedoneoneachofthefourcorners,andspacedouttherestevenlyalongtheedgesofthegarden,leavingexactly

yardsbetweenneighboringposts.Thelongersideofhisgarden,includingthecorners,hastwiceasmanypostsastheshorterside,includingthecorners.Whatisthearea,insquareyards,ofCarl’sgarden?

Solution

ByAlbert471

Tostart,usealgebratodeterminethenumberofpostsoneachside.Youhave(thelongsidescountfor

becausetherearetwiceasmany)

(eachcornerisdoublecountedsoyoumustadd

)Makingtheshorterendhave

,andthelongerendhave

.

.Therefore,theansweris

Problem10

Aquadrilateralhasvertices,,,and,whereandareintegerswith.Theareaofis.Whatis?

Solution1

Bydistanceformulawehave.SImplifyingweget.Thusandhavetobeafactorof8.Theonlywayforthemtobefactorsofandremainintegersisifand.Sotheansweris

SolutionbyI_Dont_Do_Math

Solution2

Solutionbye_power_pi_times_i

BytheShoelaceTheorem,theareaofthequadrilateralis,so.Sinceandareintegers,and,so.

Problem11

Howmanysquareswhosesidesareparalleltotheaxesandwhoseverticeshavecoordinatesthatareintegerslieentirelywithintheregionboundedbytheline,thelineandtheline

Solution

Solutionbye_power_pi_times_i

(Note:diagramisneeded)

Ifwedrawapictureshowingthetriangle,weseethatitwouldbeeasiertocountthesquaresverticallyandnothorizontally.Theupperboundissquares,andthelimitforthe-valueissquares.Firstwecountthesquares.Inthebackrow,therearesquareswithlengthbecausegeneratessquaresfromto,andcontinuingonwehave,,andfor-valuesfor,,andintheequation.Sotherearesquareswithlengthinthefigure.Forsquares,eachsquaretakesupunleftandunup.Squarescanalsooverlap.Forsquares,thebackrowstretchesfromto,sotherearesquareswithlengthinabybox.Repeatingtheprocess,thenextrowstretchesfromto,sotherearesquares.Continuingandaddingupintheend,therearesquareswithlengthinthefigure.Squareswithlengthinthebackrowstartatandendat,sotherearesuchsquaresinthebackrow.Asthefrontrowstartsatandendsattherearesquareswithlength.Assquareswithlengthwouldnotfitinthetriangle,theansweriswhichis.

Problem12

Allthenumbersarewritteninaarrayofsquares,onenumberineachsquare,insuchawaythatiftwonumbersareconsecutivethentheyoccupysquaresthatshareanedge.Thenumbersinthefourcornersaddupto.Whatisthenumberinthecenter?

Solution

SolutionbyMlux:Drawamatrix.Noticethatnoadjacentnumberscouldbeinthecornerssincetwoconsecutivenumbersmustshareanedge.Nowfind4nonconsecutivenumbersthataddupto.Tryingworks.Placeeachoddnumberinthecornerinaclockwiseorder.Thenfillinthespaces.Therehastobeainbetweentheand.Thereisabetweenand.Thefinalgridshouldsimilartothis.

isinthemiddle.

Solution2

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

Problem13

AliceandBoblivemilesapart.OnedayAlicelooksduenorthfromherhouseandseesanairplane.AtthesametimeBoblooksduewestfromhishouseandseesthesameairplane.TheangleofelevationoftheairplaneisfromAlice'spositionandfromBob'sposition.Whichofthefollowingisclosesttotheairplane'saltitude,inmiles?

Solution

Let'ssetthealtitude=z,distancefromAlicetoairplane'sgroundposition(pointrightbelowairplane)=yanddistancefromBobtoairplane'sgroundposition=x

FromAlice'spointofview,..So,

FromBob'spointofview,..So,

Weknowthat+=

Solvingtheequation(byplugginginxandy),wegetz==about5.5.

So,answeris

solutionbysudeepnarala

Solution2

Non-trigsolutionbye_power_pi_times_i

SetthedistancefromAlice'sandBob'spositiontothepointdirectlybelowtheairplanetobeand,respectively.FromthePythagoreanTheorem,.Asbotharetriangles,thealtitudeoftheairplanecanbeexpressedasor.Solvingtheequation,weget.Pluggingthisintotheequation,weget,or(cannotbenegative),sothealtitudeis,whichisclosestto

Problem14

Thesumofaninfinitegeometricseriesisapositivenumber,andthesecondtermintheseriesis.Whatisthesmallestpossiblevalueof

Solution

Thesecondterminageometricseriesis,whereisthecommonratiofortheseriesandisthefirsttermoftheseries.Soweknowthatandwewishtofindtheminimumvalueoftheinfinitesumoftheseries.Weknowthat:andsubstitutingin,wegetthat.Fromhere,youcaneitherusecalculusorAM-GM.

Calculus:Let,then.Sinceandareundefined.Thismeansthatweonlyneedtofindwherethederivativeequals,meaning.So,meaningthat

AM-GMFor2positiverealnumbersand,.Letand.Then:.Thisimpliesthat.or.Rearranging

:.Thus,thesmallestvalueis.

Solution2

Asimpleapproachistoinitiallyrecognizethatand.Weknowthat,sincetheseriesmustconverge.Wecanstartbyobservingthegreatestanswerchoice,4.Therefore,,becausethatwouldmake,whichwouldmaketheseriesexceed4.Inordertominimizeboththeinitialtermandtherestoftheseries,wecanrecognizethatistheopitimalratio,thustheansweris.

Problem15

Allthenumbersareassignedtothesixfacesofacube,onenumbertoeachface.Foreachoftheeightverticesofthecube,aproductofthreenumbersiscomputed,wherethethreenumbersarethenumbersassignedtothethreefacesthatincludethatvertex.Whatisthegreatestpossiblevalueofthesumoftheseeightproducts?

Solution

Firstassigneachfacetheletters.Thesumoftheproductofthefacesis.Wecanfactorthisintowhichistheproductofthesumofeachpairofoppositefaces.Inordertomaximizeweusethenumbersor.

Problem16

Inhowmanywayscanbewrittenasthesumofanincreasingsequenceoftwoormoreconsecutivepositiveintegers?

Solution

Weproceedwiththisproblembyconsideringtwocases,when:1)Thereareanoddnumberofconsecutivenumbers,2)Thereareanevennumberofconsecutivenumbers.

Forthefirstcase,wecancleverlychoosetheconvenientformofoursequencetobe

becausethenoursumwilljustbe.Wenowhaveandwillhaveasolutionwhenisaninteger,namelywhenisadivisorof345.Wecheckthatwork,andnomore,becausedoesnotsatisfytherequirementsoftwoormoreconsecutiveintegers,andwhenequalsthenextbiggestfactor,,theremustbenegativeintegersinthesequence.Oursolutionsare.

Fortheevencases,wechooseoursequencetobeoftheform:sothesumis.Inthiscase,wefindoursolutionstobe.

Wehavefoundall7solutionsandouransweris.

Solution2

Thesumfromtowhereandareintegersandis

Letand

Ifwefactorintoallofitsfactorgroupswewillhaveseveralorderedpairswhere

Thenumberofpossiblevaluesforishalfthenumberoffactorsofwhichis

However,wehaveoneextraneouscaseofbecausehere,andwehavethesumofoneconsecutivenumberwhichisnotallowedbythequestion.

Thustheansweris

.

Problem17

Inshowninthefigure,,,,andisanaltitude.Pointsandlieonsidesand,respectively,sothatandareanglebisectors,intersectingatand,respectively.Whatis?

Solution

Gettheareaofthetrianglebyheron'sformula:UsetheareatofindtheheightAHwithknownbaseBC:Applyanglebisectortheoremontriangleandtriangle,wegetand,respectively.Fromnow,youcansimplyusetheanswerchoicesbecauseonlychoicehasinitandweknowthatthesegmentsonitallhaveintegrallengthssothatwillremainthere.However,byscalingupthelengthratio:and.weget.

Problem18

Whatistheareaoftheregionenclosedbythegraphoftheequation

Solution

Considerthecasewhen,.Noticethecircleintersecttheaxesatpointsand.Findtheareaofthiscircleinthefirstquadrant.Theareaismadeofasemi-circlewithradiusofandatriangle:Becauseofsymmetry,theareaisthesameinallfourquadrants.Theansweris

Problem19

Tom,Dick,andHarryareplayingagame.Startingatthesametime,eachofthemflipsafaircoinrepeatedlyuntilhegetshisfirsthead,atwhichpointhestops.Whatistheprobabilitythatallthreefliptheircoinsthesamenumberoftimes?

Solution1

By:dragonfly

Wecansolvethisproblembylistingitasaninfinitegeometricequation.Wegetthattohavethesameamountoftosses,theyhaveachanceofgettingallheads.Thenthenextprobabilityisallofthemgettingtailsandthenonthesecondtry,theyallgetheads.Theprobabilityofthathappeningis.Wethengetthegeometricequation

Andthenwefindthatequalstobecauseoftheformulaofthesumforaninfiniteseries,.

Solution2

Callita"win"iftheboysallfliptheircoinsthesamenumberoftimes,andtheprobabilitythattheywinis.Theprobabilitythattheywinontheirfirstflipis.Iftheydon'twinontheirfirstflip,thatmeanstheyallflippedtails(whichalsohappenswithprobability)andthattheirchancesofwinninghavereturnedtowhattheywereatthebeginning.Thiscoversallpossiblesequencesofwinningflips.Sowehave

Solvingforgives.

Problem20

Asetofteamsheldaround-robintournamentinwhicheveryteamplayedeveryotherteamexactlyonce.Everyteamwongamesandlostgames;therewerenoties.Howmanysetsofthreeteamswerethereinwhichbeat,beat,andbeat

Solution

Weusecomplementarycounting.Firstly,becauseeachteamplayedotherteams,thereareteamstotal.Allsetsthatdonothavebeat,beat,andbeathaveoneteamthatbeatsboththeotherteams.Thuswemustcountthenumberofsetsofthreeteamssuchthatoneteambeatsthetwootherteamsandsubtractthatnumberfromthetotalnumberofwaystochoosethreeteams.

Therearewaystochoosetheteamthatbeatthetwootherteams,andtochoosetwoteamsthatthefirstteambothbeat.Thisissets.Therearesetsofthreeteamstotal.Subtracting,weobtainasouranswer.

Problem21

Letbeaunitsquare.Letbethemidpointof.Forletbetheintersectionofand,andletbethefootoftheperpendicularfromto.Whatis

Solution

(ByQwertazertl)

Wearetaskedwithfindingthesumoftheareasofeverywhereisapositiveinteger.Wecanstartbyfindingtheareaofthefirsttriangle,.Thisisequalto??.Noticethatsincetriangleissimilartotriangleina1

:2ratio,mustequal(sincewearedealingwithaunitsquarewhosesidelengthsare1).isofcourseequaltoasitisthemid-pointofCD.Thus,theareaofthefirsttriangleis??.

Thesecondtrianglehasabaseequaltothatof(seethat~)andusingthesamesimilartrianglelogicaswiththefirsttriangle,wefindtheareatobe??.Ifwecontinueandtestthenextfewtriangles,wewillfindthatthesumofallisequaltoor

Thisisknownasatelescopingseriesbecausewecanseethateverytermafterthefirstisgoingtocancelout.Thus,thethesummationisequaltoandaftermultiplyingbythehalfoutinfront,wefindthattheansweris.

Problem22

Foracertainpositiveintegerlessthan,thedecimalequivalentofis,arepeatingdecimalofperiodof,andthedecimalequivalentofis,arepeatingdecimalofperiod.Inwhichintervaldoeslie?

Solution

Solutionbye_power_pi_times_i

If,mustbeafactorof.Also,bythesameprocedure,mustbeafactorof.Checkingthroughallthefactorsofandthatarelessthan,weseethatisasolution,sotheansweris.

Note:isalsoasolution,whichinvalidatesthismethod.However,weneedtoexamineallfactorsofthatarenotfactorsof,,or,or.Additionally,weneedtobeafactorofbutnot,,or.Indeed,satisfiestheserequirements.

Foranyonewhowantsmoreinformationaboutrepeatingdecimals,visit:

s:///wiki/Repeating_decimal

Problem23

Whatisthevolumeoftheregioninthree-dimensionalspacedefinedbytheinequalitiesand

Solution1(NonCalculus)

Thefirstinequalityreferstotheinteriorofaregularoctahedronwithtopandbottomvertices.Itsvolumeis.Thesecondinequalitydescribesanidenticalshape,shiftedupwardsalongtheaxis.Theintersectionwillbeasimilaroctahedron,linearlyscaleddownbyhalf.Thusthevolumeoftheintersectionisone-eighthofthevolumeofthefirstoctahedron,givingananswerof.

Solution2(Calculus)

Let,thenwecantransformthetwoinequalitiestoand.Thenit'sclearthat,consider,,thensincetheareaofis,thevolumeis.Bysymmetry,thecasewhenisthesame.Thustheansweris.

Problem24

Thereareexactlyorderedquadrupletssuchthatand.Whatisthesmallestpossiblevaluefor?

Solution

Let,etc.,sothat.Thenforeachprimepowerintheprimefactorizationof,atleastoneoftheprimefactorizationsofhas,atleastonehas,andallmusthavewith.

Letbethenumberoforderedquadrupletsofintegerssuchthatforall,thelargestis,andthesmallestis.ThenfortheprimefactorizationwemusthaveSolet'stakealookatth