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2011

WorldMathematicsTeamChampionship

JuniorLevel

Team Round

·

Problems

1.Givenfournumberswithanaverageof12.Supposethisaverageis9greaterthanthesmallest
ofthesefournumbers,5lessthanthesecondlargestnumber,anditssum withthesecond
smallestnumberissameasthelargestnumber.Writeoutthesefournumbersinascending
order.

Fig.1

2.TheHopeCupCompetitionstartedin1990.Thisyearis2011.If
wewrite2011_{1990asacontinuedfraction}

a+ 1

b+ 1

c+ 1

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 5 sides)ofdimensions
6cm×6cm×6cm.Supposewewanttochangeitintoanopentopboxwithnaturalnumber
edgesandwithonly5_{6 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.

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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 4 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,acustomerpaidx_{dollarseach 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_-62011_{isdividedby10?}

19.AsquarefieldABCDhas32holesasintheFig.7(9holesoneachside).
A workercarries32flagsandplacingthemintotheseholesclockwise.
HestartswithHoleAandskips5holesbeforeheplacesanotherflagin

thenexthole.Afterhehasbeenaroundthesquarefieldanumberoftimes,hediscovers
thattherearestillnholeswithoutaflag.Findn.

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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.

16.153_140.

17.(a)Yes;(b)Yes.

18.9.

19.16.

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RelayRound

·

Problems

FirstRound

1A.Theedgelengthsofarectangleareprimenumbersanditsperimeteris60.

Findthelargestareaofsuchrectangle.

Fig.1

1B._LetT₌_{TNYWR (TheNumberYou WillReceive)}_.

AsintheFig.1,DisonthelinesegmentCGandthesumoftheareasof
squareABCD_andsquareCEFGisT.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.

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IndividualRound

·

Problems

FirstRound

1.Inacertainyear,thereareexactly4Wednesdaysand4SundaysinJanuary.Whichdayofthe

weekisFebruary14th_inthatyear?

2.BothaeroplaneA_andaeroplaneB_{canflyfor6000kilometerswhentheyhaveafulltankof}

fuel.Ifbothoftheaeroplanesdepartfrom AirportCwithafulltankandaeroplaneAcan
refuelaeroplaneB_{fromitsowntankatsomepointintheair.}Inorderto makesureboth
aeroplaneAandBcanreturnto AirportCbeforetheir fuelrunout,whatisthelongest
distanceaeroplaneBcanreach (inkilometers)?

Fig.1

3.ConsiderthelinesegmentABasintheFig.1.

IfCD =2₅AD,DB =2₅AB -1,andAB =12,findCB.
4.Ifthedigitsina3-digitnumberabc_satisfyc(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.Its2nd_layeris

composedwith2righttrianglesand2rectangles,andsoon.Eachrighttrianglehasanareaof
1andeachrectanglehasanareaof2.Whichlayerhasanareaof8042?

Fig.3

8.AsintheFig.3,supposethebaseABofaparallelogramABCDis8cm
longanditsheightis4cm.IftheareaoftriangleBEFis6cm2_largerthan

theareaofCDF,findthelengthofBEincm.

ThirdRound

9.Findthenumberoffactorsofthenaturalnumber371250thatarenotlargerthan11.Also,
how manyofthesefactorsareprimenumbers?

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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.HisdistancefromAis7_{16ofthedistance}

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∶382_{11am.and1∶05}_11am5 .

FifthRound

15.80cm2_.

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