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Useapieceofpaperthatis0.02cmthick,87.92mlong,and0.12m wideto wraptightlyaround a0.12m longcylindricaltubeofdiameter10cm in multiplelayers.Thisformsa papertube wrapping aroundthe original[r]
(1)2011WorldMathematicsTeamChampionship JuniorLevel
Team Round·Problems
1.Givenfournumberswithanaverageof12.Supposethisaverageis9greaterthanthesmallest ofthesefournumbers,5lessthanthesecondlargestnumber,anditssum withthesecond smallestnumberissameasthelargestnumber.Writeoutthesefournumbersinascending order
Fig.1
2.TheHopeCupCompetitionstartedin1990.Thisyearis2011.If wewrite20111990asacontinuedfraction
a+
b+
c+
d +e1
Fig.2
(a)Whatisthelargestnaturalnumberamongtheintegersa,
b,c,d,ande?
(b)Whatisthesumofa,b,c,d,ande
3.Placefourregularhexagons withedge1onadesk without overlappingasintheFig.1.Supposetheyareallowedtotouch eachothermatchingcompletelysidebysideandformdifferent
figures.IfLrepresentstheperimeteroftheformingfigure,findallpossibleL
4.Asinthe Fig.2,given an opentopcubictin box (with only sides)ofdimensions 6cm×6cm×6cm.Supposewewanttochangeitintoanopentopboxwithnaturalnumber edgesandwithonly56 oftheoriginalvolume
(a)Amongallthesepossibleboxes,findthelargestbaseareaincm2?
(b)Whenthebaseareaisthelargest,andtheleastmaterialareused,findthesumoflength widthandheightincm
Fig.3
5.AsintheFig.3,alargerectangleispartitionedinto9smallerrectangles The numbersinside the smallrectangle representtheir areas The numbersoutsidethesmallrectanglesrepresentthelengthsoftheedges Findtheareaoftheoriginallargerectangle
6.Asinthe Fig.4,ABCDEFGH isaregular Octagon.Iftheareaof isoscelestrapezoidABCDis12,findtheareaofthisoctagon
(2)8.Aproperlyworkingclockhasitshourhandandminutehandforminga30°at11o'clockinthe morning.Whenisthenexttimewhenthisclock'shourandminutehandsagainformthatsame angle?
Fig.5
9.Whatisthesmallestprimenumberpsothat29p +3isequaltotheproductoftwo consecutivenon-zeronaturalnumbers?
10.Drawtwoidenticaltrianglestotallyinsideapieceofrectangularpapersothatnovertexor edgeofeithertrianglewouldtouchtheedgeofthepaper.Thesetwotriangleswoulddivide thepaperintonregions
Whatareallthepossiblevalueforn?
11.Leta,b,c,dbenon-zeronaturalnumbersthatarenotmorethan4.Twoofthemare identical,and(a+b)(b+c)(c+d)(d +a)=900.Finda+b+c+d
12.Supposethesumofthedigitsofa4 digitnumberis4andthisnumberisstilla digit numberifwereversetheorderofitsdigits.Amongallsuchpossible4 digitnumbers,find theonethatisclosesttotheperfectsquareofsome2 digitnumber
13.Apieceof13×13squarepaperisbeingcutupintomanysmallerpieces.Ifthesepiecesare puttingbacktogethertoform3squaresofdifferentsizesandeachedgeisintegernumber Findthesumofthese3squares'perimeters
Fig.6
14.AsintheFig.6,ABCDisarectangle,AB =6,AD =10.IfpointEison thediagonalAC,AE =2EC,andS△AEF=1
7SABCD,findAF∶FD
15.How manypairsofnaturalnumbersxandysatisfying3x +5y=121?
16.Acompanysold30% ofacertainmerchandisefromitsinventoryfor$150
each,andeachlost5%.Now,acustomerpaidxdollarseach buyinguptheremaining
inventory.Thistransactionresultsthecompany makingatotalof5% profitonthewhole inventory.Findtheratioofxtothecostofeachitem
Fig.7
17.Letnbeanynonzeronaturalnumberandmistheproductofn-1,n,
n+1,andn×n+1
(a)Ismamultipleof6? (b)Ismamultipleof5?
18.Whatistheremainderwhen82011-32011-62011isdividedby10?
19.AsquarefieldABCDhas32holesasintheFig.7(9holesoneachside) A workercarries32flagsandplacingthemintotheseholesclockwise HestartswithHoleAandskips5holesbeforeheplacesanotherflagin
thenexthole.Afterhehasbeenaroundthesquarefieldanumberoftimes,hediscovers thattherearestillnholeswithoutaflag.Findn
(3)Team RoundAnswers 1.3,8,17,20
2.(a)94;(b)104
3.24,22,20,18,16,14
4.(a)180cm2;(b)28cm 5.77
6.48
7.4
8.11o′clock546
9.3
10.2,3,4,5,6,7,8
11.11
12.1021
13.76
14.3∶4
15.8
16.153140.
17.(a)Yes;(b)Yes
18.9
19.16
(4)RelayRound·Problems
FirstRound
1A.Theedgelengthsofarectangleareprimenumbersanditsperimeteris60
Findthelargestareaofsuchrectangle
Fig.1
1B.LetT=TNYWR (TheNumberYou WillReceive).
AsintheFig.1,DisonthelinesegmentCGandthesumoftheareasof squareABCDandsquareCEFGisT.Ifthesidesofthesetwosquaresdiffer by1,findtheareaofquadrilateralAEFG
SecondRound
2A.Giventhata,b,c,darenaturalnumbersandaisequaltotheareaofasquareP.Ifa<b+
c,b<2c,c<3d,d<500,thenwhichoneofthefournumbers268,272,276,280isthe perimeterofP?
2B.LetT=TNYWR (TheNumberYou WillReceive)
IfthesumofthesquaresofthreeconsecutivenaturalnumbersisT-23,thenwhatisthesum ofthesethreeconsecutivenaturalnumbers?
ThirdRound
3A.Ateamof5000some(morethan5000butlessthan6000)athletesisgatheredinafootball
field.Theseathletescaneitherbegroupedintoseveralarraysof12×12orseveralarraysof 18×18.How manyathletesarethere?
3B.LetT=TNYWR (TheNumberYou WillReceive)
SupposeT′isthenumberderivedbyreversingthedigitsofTandT+T′+1isthesquareof acertain naturalnumber.Ifthis naturalnumberisthesum of5 consecutive natural numbers,findthese5consecutivenaturalnumbers
RelayRoundAnswers
1A.221 1B.121.
2A.268 2B.27.
3A.5184
(5)IndividualRound·Problems
FirstRound
1.Inacertainyear,thereareexactly4Wednesdaysand4SundaysinJanuary.Whichdayofthe weekisFebruary14thinthatyear?
2.BothaeroplaneAandaeroplaneBcanflyfor6000kilometerswhentheyhaveafulltankof
fuel.Ifbothoftheaeroplanesdepartfrom AirportCwithafulltankandaeroplaneAcan refuelaeroplaneBfromitsowntankatsomepointintheair.Inorderto makesureboth aeroplaneAandBcanreturnto AirportCbeforetheir fuelrunout,whatisthelongest distanceaeroplaneBcanreach (inkilometers)?
Fig.1
3.ConsiderthelinesegmentABasintheFig.1
IfCD =25AD,DB =25AB -1,andAB =12,findCB 4.Ifthedigitsina3-digitnumberabcsatisfyc(a+c)=40and
a(a+b)=36,findallpossibleabc
SecondRound
Fig.2
5.From130to1300,inclusive,therearennaturalnumbersthataremultiples of17or71.Findn
6.Acompanyhadx college-graduate-employees,whichis45% ofthetotal employees.Now,another120college-graduate-employeesarehired, makingthepercentageofcollege-graduate-employeesreaches75% How manyemployeesthecompanyhadbeforethislatesthiring?
7.AsintheFig.2,△ABCiscomposedwithidenticalrighttriangles and
rectangles Fromtopdown,itsfirstlayeriscomposedwith2righttriangles.Its2ndlayeris
composedwith2righttrianglesand2rectangles,andsoon.Eachrighttrianglehasanareaof 1andeachrectanglehasanareaof2.Whichlayerhasanareaof8042?
Fig.3
8.AsintheFig.3,supposethebaseABofaparallelogramABCDis8cm longanditsheightis4cm.IftheareaoftriangleBEFis6cm2largerthan
theareaofCDF,findthelengthofBEincm
ThirdRound
9.Findthenumberoffactorsofthenaturalnumber371250thatarenotlargerthan11.Also, how manyofthesefactorsareprimenumbers?
(6)11.Place2011ofthenumber2011sidebysideandforma(4×2011)-digitnumber.Findthe
Fig.4
remainderwhenthisnumberisdividedby8
12.AsintheFig.4,ABCDisasquarewithedgeas2a.Let ☉Obethe largestcircleinsidesquareABCDandletEFGH bethelargestsquare
inside☉O.IfS1,S2,andS3areusedtorepresenttheareaof△EOF,
theareaofthe arch-shapedregionEmF,andtheareaofcurvedregion
EBF,respectively.FindS1∶S2∶S3(Expresstheanswerintermsofπ)
FourthRound
13.StevenstandsonabridgeAB.HisdistancefromAis716ofthedistance
AB.AtrainiscomingtowardAataspeedof80km/h.Stevenhastwochoices.Hecan eitherruntowardAandhewillmeetthetrainatA,orhecanruntowardBatthesame speedthenthetrainwillcatchuphimatB.Findhisspeedinkmperhour
14.Ina12-hourclock,whattimebetween1∶00amand2∶00am whenthehourhandandthe minutehandformastraightline?
FifthRound
Fig.5
15.As in the Fig.5, △A1B1C1, △A2B2C2 and △A3B3C3 are
equilateraltriangles.LetD1trisectsA1C1andbisectsA2B2.And
letD2trisectsA2C2andbisectsA3B3.Also,thebasesofallthree
trianglesareallonastraightline.Iftheareaof △A1B1C1is
54cm2,thenfindtheareaofthefigurethatisformedbythezigzag
linesegmentB1A1D1A2D2A3C3B1
16.Awebcam,aDVDdrive,andaharddiskdrivetogethercost$100
SupposeoneDVDdrivecostsmorethantwowebcams,twoharddiskdrivescostmorethan sevenDVDdrives,andeight webcamscostmorethanoneharddiskdrive.Assumingthe costofeachoftheseitemsisawholenumberofdollars,how muchdoeseachitemcost?
IndividualRoundAnswers
FirstRound
1.Saturday 2.4000 kilometers
3.7.08 4.604or395
SecondRound
5.85 6.100 7.2011 8.11mm
ThirdRound
9.8,4 10.12 11.3
12.2∶(π -2)∶(4-π)
FourthRound
13.10km/h
14.1∶38211am.and1∶0511am5
FifthRound
15.80cm2.