2022年大联盟(Math League)国际夏季四年级数学挑战活动一（含答案）

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2022MathLeagueInternationalSummerChallenge,Grade4(Unofficialversion,forreferenceonly)

Note:Thereareeightquestionsintotal.Fivequestionsareworth10pointseach.Twoquestionsareworth15pointseach.Onequestionisworth20points.Thetotalpointsare100.

Question1(15Points)

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.

IfyoustartedwithLadybug2insteadofLadybug1,whatisthelargestnumberofladybugsthatyoucanhelp?Ladybugsstillarriveinincreasingorder:Ladybug2,Ladybug3,Ladybug4,etc.

Note:Pleaseenter0ifyouranswerisinfinitelymany,whichmeansyoucanplaceasmanyladybugsontheleavesasyouwant.Thereisnolimit.

Answer:11

Ifonlytheodd-numberedladybugsareoutflying,whatisthelargestnumberofladybugsthatyoucanhelp?Ladybugsstillarriveinincreasingorder:Ladybug1,Ladybug3,Ladybug5,etc.

Note:Pleaseenter0ifyouranswerisinfinitelymany,whichmeansyoucanplaceasmanyladybugsontheleavesasyouwant.Thereisnolimit.

Answer:0

Ifonlytheeven-numberedladybugsareoutflying,whatisthelargestnumberofladybugsthatyoucanhelp?Ladybugsstillarriveinincreasingorder:Ladybug2,Ladybug4,Ladybug6,etc.

Note:Pleaseenter0ifyouranswerisinfinitelymany,whichmeansyoucanplaceasmanyladybugsontheleavesasyouwant.Thereisnolimit.

Answer:8

Ifonlyladybugsthatarenumberedwithmultiplesof3(3,6,9,12,15,…)areoutflying,whatisthelargestnumberofladybugsthatyoucanhelp?Ladybugsstillarriveinincreasingorder.

Note:Pleaseenter0ifyouranswerisinfinitelymany,whichmeansyoucanplaceasmanyladybugsontheleavesasyouwant.Thereisnolimit.

Answer:8

Ifonlyladybugsthatarenumberedwithprimenumbers(2,3,5,7,11,13,17,19,…)areoutflying,whatisthelargestnumberofladybugsthatyoucanhelp?Ladybugsstillarriveinincreasingorder.

Note:Pleaseenter0ifyouranswerisinfinitelymany,whichmeansyoucanplaceasmanyladybugsontheleavesasyouwant.Thereisnolimit.

Answer:0

Ifonlyladybugsthatarenumberedwithnon-multiplesof3(1,2,4,5,7,8,10,11,13,

14,…)areoutflying,whatisthelargestnumberofladybugsthatyoucanhelp?Ladybugsstillarriveinincreasingorder.

Note:Pleaseenter0ifyouranswerisinfinitelymany,whichmeansyoucanplaceasmanyladybugsontheleavesasyouwant.Thereisnolimit.

Answer:0

Ifonlyladybugsthatarenumberedwithnon-multiplesof4(1,2,3,5,6,7,9,10,11,13,14,15,…)areoutflying,whatisthelargestnumberofladybugsthatyoucanhelp?Ladybugsstillarriveinincreasingorder.

Note:Pleaseenter0ifyouranswerisinfinitelymany,whichmeansyoucanplaceasmanyladybugsontheleavesasyouwant.Thereisnolimit.

Answer:0

IfonlyladybugsthatarenumberedwithFibonaccinumbers(1,2,3,5,8,13,21,34,55,89,144,…)areoutflying,whatisthelargestnumberofladybugsthatyoucanhelp?Ladybugsstillarriveinincreasingorder.

Fibonaccinumbers:startingfromthethirdnumber,eachnumberisthesumofthetwoprecedingnumbers.

Note:Pleaseenter0ifyouranswerisinfinitelymany,whichmeansyoucanplaceasmanyladybugsontheleavesasyouwant.Thereisnolimit.

Answer:0

Question2(15Points)

Objective:

Fillyourentiregardenwithplotsofcarrots.

Rule:

Aplotofcarrotsconsistsoftwosquaresofyourgardenthatareeitherverticallyorhorizontallynexttoeachother.

Example:

Isitpossibletofilla3x3gardenwithplotsofcarrots,figurebelow?

Note:Pleaseenter1ifyouranswerisYes,and0ifyouranswerisNo.

Answer:0

Youhavea3x3gardenthathasabigboulderblockingoneofthecornersquares,figurebelow.Isitpossibletofillyourremaininggardenwithplotsofcarrots?

Note:Pleaseenter1ifyouranswerisYes,and0ifyouranswerisNo.

Answer:1

Youhavea3x3gardenthathasabigboulderblockingthecentersquare,figurebelow.Isitpossibletofillyourremaininggardenwithplotsofcarrots?

Note:Pleaseenter1ifyouranswerisYes,and0ifyouranswerisNo.

Answer:1

Youhavea3x3gardenthathasabigboulderblockingoneofthesidesquares,figurebelow.Isitpossibletofillyourremaininggardenwithplotsofcarrots?

Note:Pleaseenter1ifyouranswerisYes,and0ifyouranswerisNo.Answer:0

Youhavea19x19gardenthathasabigboulder(representedbyaredX)blockingthetopleftsquare,figurebelow.Isitpossibletofillyourremaininggardenwithplotsofcarrots?

Note:Pleaseenter1ifyouranswerisYes,and0ifyouranswerisNo.

Answer:1

Youhavea19x19gardenthathasabigboulder(representedbyaredX)blockingoneofthesquares,figurebelow.Isitpossibletofillyourremaininggardenwithplotsofcarrots?

Note:Pleaseenter1ifyouranswerisYes,and0ifyouranswerisNo.

Answer:0

Youhavea19x19gardenthathasabigboulder(representedbyaredX)blockingoneofthesquares,figurebelow.Isitpossibletofillyourremaininggardenwithplotsofcarrots?

Note:Pleaseenter1ifyouranswerisYes,and0ifyouranswerisNo.

Answer:1

Youhavea19x19gardenthathasabigboulder(representedbyaredX)blockingoneofthesquares,figurebelow.Isitpossibletofillyourremaininggardenwithplotsofcarrots?

Note:Pleaseenter1ifyouranswerisYes,and0ifyouranswerisNo.

Answer:0

Ifa20x20gardenhasoneofitscornersblockedbyabigboulder(representedbyaredX),figurebelow,isitpossibletofillyourremaininggardenwithplotsofcarrots?

Note:Pleaseenter1ifyouranswerisYes,and0ifyouranswerisNo.

Answer:0

Ifa20x20gardenhastwoboulders(representedbytworedX’s),figurebelow,isitpossibletofillyourremaininggardenwithplotsofcarrots?

Note:Pleaseenter1ifyouranswerisYes,and0ifyouranswerisNo.

Answer:1

Ifa20x20gardenhastwoboulders(representedbytworedX’s),figurebelow,isitpossibletofillyourremaininggardenwithplotsofcarrots?

Note:Pleaseenter1ifyouranswerisYes,and0ifyouranswerisNo.

Answer:0

Ifa20x20gardenhastwoboulders(representedbytworedX’s),figurebelow,isitpossibletofillyourremaininggardenwithplotsofcarrots?

Note:Pleaseenter1ifyouranswerisYes,and0ifyouranswerisNo.Answer:0

Question3(10Points)

Howmanydiagonalsdoesaregulardecagonhave?

Note:Aregulardecagonisapolygon.Ithas10sides(edges)and10angles.Allsideshavethesamelength,andallanglesareequalinmeasure.

Answer:35

Question4(10Points)

Whatistheonesdigitofthesum1992939202292023?

Answer:0

Question5(10Points)

Howmanywaysarethereforahorseracewiththreehorsestofinishiftiesarepossible?Note:Twoorthreehorsesmaytie.

Supposetherearethreehorses,A,B,andC.OnepossiblewaytofinishthehorseraceisthatAfinishedthefirst,Bfinishedthesecond,andCfinishedthelast.AnotherpossiblewaytofinishthehorseraceisA,B,andCallfinishedatthesametime,i.e.allthreehorsestie.Thequestionisaskingthetotalnumberofallpossiblewaysforahorseracewiththreehorsestofinishiftiesarepossible.

Answer:13

Question6(10Points)

Eachofthe4cardsshownatthebelowhasaletterononesideandadigitontheotherside.

Nowreadthetestsentencebelow:

Wheneverthereisavowelononesideofacard,thereisanevennumberontheothersideofthatcard.

Whatistheminimumnumberofcardsthatyoumustturnovertodetermineifthetestsentence

aboveistrueorfalseforthissetof4cards?

Note:

TherearefivevowelsinEnglishalphabet:A,E,I,O,U.Answer:2

Question7(10Points)

Therearefivemeninfrontofyou.Oneofthemisatruth-teller,amanwhoalwaystellsthetruth,andfourofthemarewhatarecalled“switchers”thatbehaveinthefollowingway:Thefirsttimeyouaskthemaquestiontheywillanswerwitheitherthetruthoralie,randomly.Thesecondtimeyouaskthemaquestion,theywillanswerintheoppositewayofhowtheyansweredforthefirstquestion.Soiftheyanswerthefirstquestiontruthfully,theywillanswerthesecondquestionwithalie,thethirdquestiontruthfully,thefourthquestionwithalie,andsoonandsoforth.

Whatistheminimumnumberofquestionsyouneedtoasksothatyoucandeterminewhichofthefivemeninfrontofyouisthetruth-teller?

Note:

Eachquestioncanonlybedirectedtoonepersonatatime.Andonlythepersonwhomyouaskthequestioncananswerthequestion.

Whenyouaskapersonaquestion,noneoftheotherfourmencanhearyourquestionortheanswer.

Nocommunicationofanykindisallowedamongthefivemen.

Nomancansayanythingexceptwhenansweringyourquestions.

Itisnotnecessarythatyourquestionhastobeayes/noquestion.Youcanasksomethinglike“Whoisthetruth-teller?”,“Whoaretheswitchers?”,etc.

Answer:2

Question8(20Points)

TrueorFalse:Amongany225consecutivepositiveintegers(eachofthemgreaterthan225),thereisexactlyonepositiveintegerdivisibleby225.

Note:Pleaseenter1ifyouranswerisTrue,and0ifyouranswerisFalse.

TrueorFalse:Amongany225consecutivepositiveintegers(eachofthemgreaterthan225),theremaybemorethanonepositiveintegereachdivisibleby225.

Note:Pleaseenter1ifyouranswerisTrue,and0ifyouranswerisFalse.

TrueorFalse:Amongany225consecutivepositiveintegers(eachofthemgreaterthan225),theremaybenoonedivisibleby225.

Note:Pleaseenter1ifyouranswerisTrue,and0ifyouranswerisFalse.

TrueorFalse:If10integersareselectedfromthefirst18positiveintegers,theremustbeapairoftheseintegerswithasumequal19.

Note:Thefirst18positiveintegersare1,2,3,4,…,17,18.

Note:Pleaseenter1ifyouranswerisTrue,and0ifyouranswerisFalse.

TrueorFalse:If11integersareselectedfromthefirst18positiveintegers,theremustbeapairoftheseintegerswithasumequal19.

Note:Pleaseenter1ifyouranswerisTrue,and0ifyouranswerisFalse.

TrueorFalse:If12integersareselectedfromthefirst18positiveintegers,theremustbeapairoftheseintegerswithasumequal19.

Note:Pleaseenter1ifyouranswerisTrue,and0ifyouranswerisFalse.

TrueorFalse:Theminimumnumberofintegersthatmustbeselectedfromthefollowingfourteenintegers,1,3,5,7,9,11,13,15,17,19,21,23,25,27,toguaranteethatatleastonepairoftheseintegersaddupto28is7.

Note:Pleaseenter1ifyouranswerisTrue,and0ifyouranswerisFalse.

TrueorFalse:Theminimumnumberofintegersthatmustbeselectedfromthefollowingfourteenintegers,1,3,5,7,9,11,13,15,17,19,21,23,25,27,toguaranteethatatleastonepairoftheseintegersaddupto28is8.

Note:Pleaseenter1ifyouranswerisTrue,and0ifyouranswerisFalse.

TrueorFalse:Theminimumnumberofintegersthatmustbeselectedfromthefollowingfourteenintegers,1,3,5,7,9,11,13,15,17,19,21,23,25,27,toguarantee