2021年大联盟(Math League)国际夏季五年级数学挑战活动一

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2021美国"大联盟"(MathLeague)国际夏季数学挑战活动2021MathLeagueInternationalSummerChallenge

Grade5,IndividualQuestions

Question1:

Sarahisusingpopsiclestickstobuildsomegridsforhercityplanningproject.Sheneeds4popsiclestickstomakea1by1grid,and12stickstomakea2by2gridasshownbelow.Howmanysticksdoessheneedtomakea10by20grid?

Question2:

Thesidesofthelargerectangleare20mand16m,figurebelow,notdrawntoscale.Allsixshadedrectanglesareidentical.Whatisthetotalareaofalltheshadedregions,insquaremeters?

Question3:

Iftwosidesofasquarefieldwereincreasedbyfivefeet,asseeninthediagrambelow,notdrawntoscale,theareaofthefieldwouldincreaseby245squarefeet.Findtheareaoftheoriginalsquare.

Question4:

Duringpractice,Dana’ssixarrowslandonthetargetshown.Eacharrowisinsideoneoftheregionsofthetarget.Whichofthefollowingtotalscoresispossible:44,31,26,16?(Ifanarrowlandsinsidethesmallestcircleinthecenter,thescoreDanagetsis7.Ifanarrowlandsinsidethesecondcircle,butnotinsidethesmallestcircle,thescoreDanagetsis5.Ifanarrowlandsinsidethelargestcircle,butnotinsidethesecondcircleorthesmallestcircle,thescoreDanagetsis3.Assumenoarrowlandsontheboundaryofanyofthethreecircles.)

44

31

26

16

Question5:

Thereare1000peopleinaroom.Eachpaireithershakeshandsordoesnot.

Isitalwaystruethatsometwopeopleshookthesamenumberofhands?(Pleaseenter1ifyouranswerisYes,and0ifyouranswerisNo.)

Isitalwaystruethatsomethreepeopleshookthesamenumberofhands?(Pleaseenter1ifyouranswerisYes,and0ifyouranswerisNo.)

Isitalwaystruethatsomefourpeopleshookthesamenumberofhands?(Pleaseenter1ifyouranswerisYes,and0ifyouranswerisNo.)

Isitalwaystruethatsomefivepeopleshookthesamenumberofhands?(Pleaseenter1ifyouranswerisYes,and0ifyouranswerisNo.)

Note:

Onedoesn’tshakehis/herownhand.

Onedoesn’tshakethesameperson’shandmorethanonce.

Question6:

Thenumbers1through10arewritteninarow.Canthesigns“+”and“–”beplacedbetweenthem,sothatthevalueoftheresultingexpressionis0?(Note:Thereare10numbers,andthereare9placestoplacesigns.)

Question7:

AlbertandBernardjustbecamefriendswithCheryl,andtheywanttoknowwhenherbirthdayis.Cherylgivesthemalistof10possibledates,figurebelow.CherylthentellsAlbertandBernardseparatelythemonthandthedayofherbirthdayrespectively.(ThatisCheryltellsAlbertthemonthofherbirthday,andtellsBernardthedayofherbirthday.)

Conversation:

Albert:Idon’tknowwhenCheryl’sbirthdayis,butIknowthatBernardalsodoesn’tknow.

Bernard:AtfirstIdidn’tknowwhenCheryl’sbirthdayis,butIknownow.

Albert:ThenIalsoknowwhenCheryl’sbirthdayis.WhenisCheryl’sbirthday?

Pleaseenterthemonthfirst,followedbytheday.Whenyouenterthemonth,pleaseenter5forMay,6forJune,7forJuly,and8forAugust.

Question8:

Goal:

Arrangethesixdifferentjuicesintothefollowingorder.

Theorderis:1–red,2–orange,3–yellow,4–green,5–blue,6–violet.

Rules:

Youmayonlypourafilledcupintoanemptycup.

Youmaynotswitchthepositionsofanycups.

Theemptycupmustendupontheright.

Examples:

2Poursarenecessarytosolvethefollowingexamplepuzzle,figurebelow.

PourOne,figurebelow.

PourTwo,figurebelow:

Foranotherexamplebelow,thisexampleneedsfourpours.

PourOne,figurebelow.

PourTwo,figurebelow.

PourThree,figurebelow.

PourFour,figurebelow.

For6differentjuicesandoneemptycup,thereare5040(=7!=7×6×5×4×3×2×1)possiblepermutations.Twoofthemwerelistedabove.Oneneeds2pours.Theotherneeds4pours.

Ofallthe5040possiblepermutations,howmanypermutationsareimpossibletosolve?Thatisnomatterhowyoupourthejuices,youcan’tarrangethejuicesinorder.

Whatisthelargestnumberofpoursyouneedtosolveanysolvablepermutation?

For1000001differentjuicesandoneemptycup,thereare1000002!possiblepermutations.Thereisapre-definedorderofthe1000001juices,justastheorderforthe6juicesabove.

Ofallthe1000002!possiblepermutations,howmanypermutationsareimpossibletosolve?Thatisnomatterhowyoupourthejuices,youcan’tarrangethejuicesinorder.

Whatisthelargestnumberofpoursyouneedtosolveanysolvablepermutation(for1000001differentjuicesandoneemptycup)?

Question9:

Twelvestraightjacketedprisonersareonthedeathrow.Tomorrowtheywillbearrangedinasingle-fileline,positionsdeterminedrandomly,allfacingthesamedirection.Thepersoninthebackoftheline,whowillbereferredtoasthetwelfthperson,canseetheelevenpeoplestandinginfrontofhim;theeleventhpersoncanseethetenpeoplestandinginfrontofhim,butnotthetwelfthpersonwhoisstandingbehindhim;soonandsoforthuntilyougettothefirstpersoninlinewhocannotseeanyone.Thewardenofthisprisonlikestoplaygameswithhisprisonersandhaslinedeveryoneupinthisfashionsothathecangivethemanopportunitytogaintheirfreedom.

Thewardenwillplace,atrandom,ablackorwhitehatoneachprisoner’shead.Theprisonerscannotseethecolorofthehattheyarewearing,buttheycanseethecolorofthehatsofeachprisonerstandinginfrontoftheminline.Afterallofthehatshavebeenplacedontheprisoners’heads,thewardenasksthepersoninthebackoftheline,thetwelfthperson,“whatisthecolorofthehatonyourhead?”Theprisonercanreplybysayingeither“black”or“white”,butnothingelse.Whentheprisoneranswersthewarden,alltheprisonersinlinecanhearthereply.Iftheprisonerrepliedwiththecorrectcolorofhishat,hewillbefreedfromtheprison.Ifheiswrong,hewillbeexecutedthatnight.Thisgamecontinuesuntilalltwelveoftheprisonersinlinehavebeenaskedaboutthecoloroftheirhat.Whilethegameisinprogress,noprisonerisallowedtospeak,move,ordoanythinguntilitistheirturntoanswerthequestion.Thewardenallowstheprisonerstogettogetherinthecourtyardthedaybeforethisgametocomeupwithaplantomaximizethenumberoflivestheycansave.Theprisonerscomeupwithaplanthatguaranteesxprisoner(s)willsurvive,and12–xprisoner(s)willeachhavea50%chancetosurvive.Whatisthelargestpossiblevalueofx?

Question10:

Somewhereintheworldthereisasmallneighborhoodfullofveryeccentricandintelligentpeople.Therearefivehousesthatareeachadifferentcolor.Thereisapersonofadifferentnationalitywholivesineachhouse.Eachofthesefivepeopledrinkstheirownspecialdrink,smokestheirownspecialbrandofcigarettes,andhastheirownspecialpet.Noonehasthesamepet,smokesthesamebrandofcigarettes,ordrinksthesamedrink.Oneofthesefivepeopleownsapetfish.Withjustthefollowinginformation,figureoutwhoownsthecatandwhoownsthefish:

TheBritishmanlivesintheredhouse.

TheSwedishmanhasadogforapet.

TheDanishmandrinkstea.

Thegreenhouseandthewhitehousearenexttoeachother,andthegreenhouseistotheleftofthewhitehouse.

Theownerofthegreenhousedrinkscoffee.

ThepersonthatsmokesPallMallcigaretteshasabirdforapet.

TheowneroftheyellowhousesmokesDunhillcigarettes.

Thepersonwholivesinthemiddlehousedrinksmilk.

TheNorwegianlivesinthefirsthouse,