2022年大联盟(Math League)国际夏季五年级数学挑战活动一（含答案）

PAGE

10

2022MathLeagueInternationalSummerChallenge,Grade5(Unofficialversion,forreferenceonly)

Note:Thereareninequestionsintotal.Sevenquestionsareworth10pointseach.Twoquestionsareworth15pointseach.Thetotalpointsare100.

Question1(10Points)

Objective:

Helpasmanyladybugsaspossiblelandontheleaves.

Rules:

Ladybugsarriveinnumericalorder:Ladybug1,Ladybug2,Ladybug3,etc.

Youhelpeachladybugchoosewhethertolandontheleftleafortherightleaf.

Ifononeleaf,thenumberofdotsontwoladybugsaddsuptothenumberofdotsonathirdladybug,alloftheladybugsflyaway.

Inthefigurebelow,youputLadybug1ontherightleaf.ThenyouputLadybug2,Ladybug3,andLadybug4ontheleftleaf.ThenyouputLadybug5ontherightleaf.

NowthereisnowheretoputLadybug6.

IfLadybug6landsontheleftleaf,alloftheladybugswillflyaway,because2+4=6.IfLadybug6landsontherightleaf,alloftheladybugswillflyaway,because1+5=6.

Yousawhowtohelp5ladybugs.Whatisthelargestnumberofladybugsthatyoucanhelpinthiscase?Theansweris8,figurebelow.

Note:Youcan’tskipanyladybugs.Forexample,thefollowingisnotallowed.YouplaceLadybug1andLadybug2ontheleftleaf.ThenyouskipLadybug3,andplaceLadybug4ontheleftorrightleaf.ThisskippingLadybug3isnotallowed.Ladybugsarriveinnumericalorder,andyoumustplaceeachofthemoneitherleafinnumericalorder.Thisistrueforallthefollowingquestions.

Ifonlyladybugsthatarenumberedwithpowersof2(1,2,4,8,16,32,64,128,…)areoutflying,whatisthelargestnumberofladybugsthatyoucanhelp?Ladybugsstillarriveinincreasingorder.

Note:Pleaseenter0ifyouranswerisinfinitelymany,whichmeansyoucanplaceasmanyladybugsontheleavesasyouwant.Thereisnolimit.

Answer:0

Ifonlyladybugsthatarenumberedwithsquarenumbers(1,4,9,16,25,36,49,64,…)areoutflying,whatisthelargestnumberofladybugsthatyoucanhelp?Ladybugsstillarriveinincreasingorder.

Note:Pleaseenter0ifyouranswerisinfinitelymany,whichmeansyoucanplaceasmanyladybugsontheleavesasyouwant.Thereisnolimit.

Answer:0

Ifonlyladybugsthatarenumberedwithcubenumbers(1,8,27,64,125,216,…)areoutflying,whatisthelargestnumberofladybugsthatyoucanhelp?Ladybugsstillarriveinincreasingorder.

Note:Pleaseenter0ifyouranswerisinfinitelymany,whichmeansyoucanplaceasmanyladybugsontheleavesasyouwant.Thereisnolimit.

Answer:0

Ifonlyladybugsthatarenumberedwithnon-multiplesof5(1,2,3,4,6,7,8,9,11,12,13,14,16,17,18,19,…)areoutflying,whatisthelargestnumberofladybugsthatyoucanhelp?Ladybugsstillarriveinincreasingorder.

Note:Pleaseenter0ifyouranswerisinfinitelymany,whichmeansyoucanplaceasmanyladybugsontheleavesasyouwant.Thereisnolimit.

Answer:0

Question2(15Points)

Objective:

Findthegoldbarinasfewweighingsaspossible.

Rules:

Youcanuseabalancescaletocomparetheweightsoftwogroupsofbars.

Onlyonebarisgold,andallotherbarsarecounterfeits(fakes).

Thegoldbarisalittleheavierthaneachofthecounterfeits.

Allcounterfeitsweighthesame.

Theappearancesofallbarsareidentical.Theonlywaytofindoutthegoldbarisusingabalancescale.

Example1:

Therearetwobars.Oneisgold.Theotherisacounterfeit.Thegoldbarisalittleheavierthanthecounterfeit.Youneedoneweighingtofindoutthegoldbar.Inthefigurebelow,bar2isthegoldbar.

Example2:

Therearefourbars.Oneisgold.Theotherthreearecounterfeits.Thegoldbarisalittleheavierthaneachofthecounterfeits.Allcounterfeitsweighthesame.Theminimumnumberofweighingstoguaranteetofindoutthegoldbaris2,figurebelow.Inthefigurebelow,thegoldbarisbar2.

Anotherwaytodothisisthatyoufirstcomparetwobars.Ifoneisheavier,youaredone.Otherwise,youcomparetheothertwobars.Again,theminimumnumberofweighingstoguaranteetofindoutthegoldbaris2.

Thereare8bars.Oneisgold.Theother7arecounterfeits.Thegoldbarisalittleheavierthaneachofthecounterfeits.Allcounterfeitsweighthesame.Whatistheminimumnumberofweighingstoguaranteetofindoutthegoldbar?

Answer:2

Thereare9bars.Oneisgold.Theother8arecounterfeits.Thegoldbarisalittleheavierthaneachofthecounterfeits.Allcounterfeitsweighthesame.Whatistheminimumnumberofweighingstoguaranteetofindoutthegoldbar?

Answer:2

Thereare10bars.Oneisgold.Theother9arecounterfeits.Thegoldbarisalittleheavierthaneachofthecounterfeits.Allcounterfeitsweighthesame.Whatistheminimumnumberofweighingstoguaranteetofindoutthegoldbar?

Answer:3

Ifyoucandoatmost1weighing,thelargestnumberofbarsyoucanstartwithandstillfindthegoldbaris3.Andhereiswhy.

Placebar1ontheleftpan,andbar2ontherightpan.Ifbar2isheavier,thenthegoldbarisbar2,figurebelow.

Iftheweighingisbalanced,figurebelow,thenthegoldbarisbar3.

Ifyoucandoatmost2weighings,whatisthelargestnumberofbarsyoucanstartwithandstillfindthegoldbar?

Answer:9

Ifyoucandoatmost3weighings,whatisthelargestnumberofbarsyoucanstartwithandstillfindthegoldbar?

Answer:27

Nowwehavenewrules:

Youcanuseabalancescaletocomparetheweightsoftwogroupsofbars.

Onlyonebarisgold,andallotherbarsarecounterfeits(fakes).

Thegoldbariseitherlighterorheavierthaneachofthecounterfeits,butyoudon’tknowwhich.

Allcounterfeitsweighthesame.

Theappearancesofallbarsareidentical.Theonlywaytofindoutthegoldbarisusingabalancescale.

Thereare4bars.Oneisgold.Theother3arecounterfeits.Thegoldbariseitherlighterorheavierthaneachofthecounterfeits,butyoudon’tknowwhich.Allcounterfeitsweighthesame.Whatistheminimumnumberofweighingstoguaranteetofindoutthegoldbar?

Answer:2

Thereare8bars.Oneisgold.Theother7arecounterfeits.Thegoldbariseitherlighterorheavierthaneachofthecounterfeits,butyoudon’tknowwhich.Allcounterfeitsweighthesame.Whatistheminimumnumberofweighingstoguaranteetofindoutthegoldbar?

Answer:3

Thereare12bars.Oneisgold.Theother11arecounterfeits.Thegoldbariseitherlighterorheavierthaneachofthecounterfeits,butyoudon’tknowwhich.Allcounterfeitsweighthesame.Whatistheminimumnumberofweighingstoguaranteetofindoutthegoldbar?

Answer:3

Thereare13bars.Oneisgold.Theother12arecounterfeits.Thegoldbariseitherlighterorheavierthaneachofthecounterfeits,butyoudon’tknowwhich.Allcounterfeitsweighthesame.Whatistheminimumnumberofweighingstoguaranteetofindoutthegoldbar?

Answer:3

Question3(10Points)

Trueorfalse:Wheneverthereare19girlsand19boysareseatedaroundacirculartable,thereisalwaysapersonbothofwhoseneighborsareboys.

Note:Pleaseenter1ifyouranswerisTrue,and0ifyouranswerisFalse.

Answer:1

Trueorfalse:Wheneverthereare20girlsand20boysareseatedaroundacirculartable,thereisalwaysapersonbothofwhoseneighborsareboys.

Note:Pleaseenter1ifyouranswerisTrue,and0ifyouranswerisFalse.

Answer:0

Question4(10Points)

Howmanydiagonalsdoesaregular20-gonhave?

Note:Aregular20-gonisapolygon.Ithas20sides(edges)and20angles.Allsideshavethesamelength,andallanglesareequalinmeasure.

Answer:170

Question5(10Points)

Givenaregular180-gon,weconstruct180isoscelestrianglesontheexteriorofthepolygonsuchthateachisoscelestrianglehasanedgeofthepolygonasitsbaseandhaslegsformedbytheextensionsofthetwoadjacentsides.Computeindegreesthelargestangleofonesuchisoscelestriangle.

Note:

Aregular180-gonisapolygon.Ithas180sides(edges)and180angles.Allsideshavethesamelength,andallanglesareequalinmeasure.

Figurebelowshowsthebaseandtwolegsofanisoscelestriangle.Theshapeofthisisoscelestriangleinthefigurebelowisnotnecessarilytheshapeoftheisoscelestrianglesinthisquestion.

Enterthenumericvalueofyouransweronly.Forexample,ifyouransweris100degrees,pleaseenter100.

Answer:176

Question6(10Points)

Youareacontestantonagameshow.Therearethreecloseddoorsinfrontofyou.Thegameshowhosttellsyouthatbehindoneofthesedoorsisamilliondollarsincash,andthatbehindtheothertwodoorstherearetrashes.Youdonotknowwhichdoorscontainwhichprizes,butthegameshowhostdoes.

Thegameyouaregoingtoplayisverysimple:youpickoneofthethreedoorsandwintheprizebehindit.Afteryouhavemadeyourselection,thegameshowhostopensoneofthetwodoorsthatyoudidnotchooseandrevealstrash.Atthispoint,youaregiventheoptiontoeitherstickwithyouroriginaldoororswitchyourchoicetotheonlyremainingcloseddoor.

Tomaximizeyourchancetowinamilliondollarsincash,shouldyouswitch?Answer:

Yes

Choices:(a)Yes

No

Itdoesn’tmatterbecausetheprobabilityforyoutowinamilliondollarsincashstaysthesamenomatterifyouswitchornot.

Question7(10Points)

Onehundredextremelyintelligentmaleprisonersareimprisonedinsolitarycellsandondeathrow.Eachcellissoundproofedandcompletelywindowless.Thereisaseparateroomwithonehundredsmallboxesnumberedandlabeledfrom1to100.Insideeachoftheseboxesisaslipofpaperwithoneoftheprisoners’namesonit.Eachprisoner’snameonlyappearsonceandisinonlyoneoftheonehundredboxes.Thewardendecidesheisgoingtoplayagamewithalloftheprisoners.Iftheywin,theywillallbeletfree,butiftheylosethegame,theywillallbeimmediatelyexecuted.

Thehundredprisonersareallowedtoenterthisseparateroomwith100boxesinanypredeterminedordertheywish,buteachcanonlyentertheroomonceandthegameendsassoonasthehundredthpersonenterstheroom.(Atanytime,onlyoneprisonerisallowedtoenterandremaininthisroom.)Onceaprisonerenterstheroom,heisallowedtoopenandlookinsideasmanyasXboxes,whereXisapositiveintegernotgreaterthan100.Afteraprisonerhasopenedtheboxes(notmorethanX)andlookedinside,hemustshutthemandleaveeverythingexactlythewayitwasbeforeheentered.Theprisonersarenotallowedtocommunicatewitheachotherinanyway.Ifeveryprisonerisabletoentertheroomandopentheboxthatcontainshisownname,theywillallbereleasedfromprisonimmediately!However,ifevenjustoneprisonerenterstheroom,opensXboxes,anddoesnotopentheboxcontaininghisownname,theywillallbeexecutedimmediately.Luckilyfortheprisoners,thewardenhasdecidedtoallowthefirstprisonerintheroomtoopenallonehundredboxesifnecessaryandswitchthetwonamesinanytwoboxesifhewouldliketo.Thefirstprisonermustshutalltheboxesheopenedandleave

everythingexactlythewayitwasbeforeheentered,withthepossibleexceptionofthetwonameshechosetoswitch.

Again,inordertowinthisgame,allonehundredprisonersneedtoentertheroomandopentheboxwiththeirnameinit.Thewardenallowstheprisonerstogettogetherinthecourtyardtheweekbeforethisgamebeginstodiscussandcomeupwithaplan.Theprisonerscomeupwithaplantoguaranteethattheywouldwinthegame.Ofcourse,ifXisbigenough,forexampleX=100,thentheywillallbereleasedfromprison.WhatistheleastpossiblevalueofX?

Note:

Notwoormoreprisonerssharethesamename.

Thefirstprisonerintheroomisanexception,meaningthatheisabletoopenallonehundredboxesifnecessaryandswitchthetwonamesinanytwoboxesifhewouldliketo.Eachoftheother99prisonersisallowedtoopenandlookinsideasmanyasXboxes,whereXisapositiveintegernotgreaterthan100.Andtheycan’tswitchanythingordoanythingelse.

Noprisonerisallowedtomakeanykindofmark,sign,orhint.Theprisonersarenotallowedtocommunicatewitheachotherinanyway.

Here’spartoftheirplan:Firsttheprisonerswilldecidetheorderinwhichtheywillentertheroomandnumb