Onetravelagencyhascharteredacertainnumberofbusestoaccommodatetheentiregroup oftravelers.Supposethemaximumnumberofpassengerseachbuscanholdis32andthis travelagencywantseachbustoholdexactly[r]
(1)2012WorldMathematicsTeamChampionship JuniorLevel
Team Round·Problems
1.Given
5+
4+
3+
2+
1+a1
=116607.Finda
2.Eachsquarebelowhasawholenumberlessthanorequalto9(includes0).Startingwiththe firstsquareontheleftandtakethesumofthethreenumbersinthethreeadjacentsquares Thenstartwiththesecondsquareontheleftandtakethesumofthethreenumbersonthe threeadjacentsquares.Followthispattern,thesumsareinorder6,5,3,4,6,5,3,4,…
Thenfindthenumberinsidethe1123rdsquarefromtheleft.
3.Base3numberscanonlyhave3possibledigits:0,1,2.Forexamples,123,2013,11223
AnyBase3numbercanbeconvertedintoaBase10numberasfollows: Base3number:123,correspondingBase10number:
1×31+2×30=3+2=5 10
Base3number:2013,correspondingBase10number:
2×32+0×31+1×30=18+0+1=19 10
Base3number:11223,correspondingBase10number:
1×33+1×32+2×31+2×30=27+9+6+2=44 10
Ontheotherhand,anyBase10numbercanalsobeconvertedtoaBase3numberbyusing shortdivisionasfollows:
Therefore, 4410=11223
Considerthe5-digitBase3numberswhichthedigitsreadthesamefromlefttorightorfrom righttoleft.Findthesumofthelargestandsmallestamongallsuch5-digitBase3numbers
Writethesum bothinBase3andinBase10 4.Supposen=222…2︸
20122′s
×333…3︸
20123′s
×9.Findthesumofallthedigitsofn
5.Supposeaandbarenonzeronaturalnumbersanda4+b8 hasanapproximatedvalueof2.6 Whatisitsexactvalueofthissum?
(2)I,andJasshown.Usingthese10points,threesetsofisoscelestriangles(5trianglesineach set)canbeformedsothat:
(a)Thefivetrianglesineachsethavethesameshapeandsizeandtheydonothaveany overlap
(b)Trianglesfromdifferentsethavethesameshapebutmayormaynothavethesamesize (c)Thefivetrianglesfromoneofthese3setscanberearrangedtoforma5-pointedstar Find:(1)Vertexangleinanisoscelestriangleisthenon-equalangle.Fromthesethreesets,
listthethreetrianglesthathaveAasthevertexangle (2)Drawthe5-pointedstarusingfivetrianglesfrom (c)
Fig.1 Fig.2 Fig.3 Fig.4
7.AsintheFig.2,thereare3×4=12smallsquaresandoneofthesquareshasaletterofxinit Usingthesquaresinthisfigure,how manyrectangles(notcountingsquares)canbeformed thatincludetheletterx?
8.TheFig.3showsthreecirclesallwithradius10.Ifeachcirclegoesthroughthecentersofthe othertwocircles,findtheareaoftheshadedarea.(Expressyouranswerintermsofπ.) 9.Amongtheproperfactorsof324×3252,how manyareperfectsquares?
10.GivenasquareABCDasintheFig.4.DrawastraightlinefromDtoABintersectingatEas inthefigure.Iftheratiooftheareaof△ADEtotheareaofquadrilateralEDCBis5∶9,find
BE
EAandexpressyouranswerinsimplestfractionform
11.Supposethe6digitnumber18abc2canbedividedevenlybyboth17and31.Findthelargest possiblevaluefora+b+c
12.SupposethereisapileofBlackandwhiteGopieces.If15 Whitepiecesaretakenaway,the numberofBlackpiecestothenumberofWhitepiecesis2∶1.Ifanother45Blackpiecesare furthertakenaway,thentheresultingratioofremainingBlackpiecestoremaining White
piecesis1∶5.How manyBlackand Whitepieceswerethereinthebeginning?
13.Onetravelagencyhascharteredacertainnumberofbusestoaccommodatetheentiregroup oftravelers.Supposethemaximumnumberofpassengerseachbuscanholdis32andthis travelagencywantseachbustoholdexactlythesamenumberofpassengers.Iftheagency places22passengersineachbus,thenthereisonetravelergotleftoutonthelastbus.Ifthe agencychartersonelessbus,theneachbuswouldholdexactlythesamenumberoftravelers withnooneleftoutafterredistribution.How manybusesdidtheagencyoriginallycharter andhow manytravelersarethere?
(3)15.Supposea,b,andcarenaturalnumbersrepresentingthreesidesofatriangleandtwoof thesenumbers are different prime numbers.Ifeach ofthesethree numberssatisfies conditions
a+b+c=99,a+c=2b,anda<c,how manysuchtrianglesarethere?
16.Giventherearenconsecutivenaturalnumbersstarting with1.Removethe4largest numbersinthisgroupofnnumbers.Iftheaveragevalueoftheremainingnumbersis49, findn
17.Ifthesumoftheareasoftwosquaresequalstotheareaofathirdsquare,thenwecallthese threesquaresformingan“elegant”set.Considerthereare20squareseachwithdifferentedge lengthsequaltonaturalnumbersfrom1to20.How many“elegant”setsarethereinthese20 squares?
18.Thereare5circularpieceseachhasanumberfrom1to5placedasintheFig.5inclockwise order.Thesepiecesaremovedaccordingtothefollowinginstruction:
(1)Eachpiecemustbemovedintheclockwisedirection
Fig.5
(2)Move①firstandthenmove②,③,④,⑤ Thenrepeattheprocessbymoving①,②,… Move① passnextonepiece
Move② passnexttwopieces Move③ passnextthreepieces Move④ passnextfourpieces Move⑤ passnextfivepieces
Andthenrepeattheprocessbymoving① passnextonepieceandsoon
Whatistheorderofthesefivepiecesafter2012moves? (Writetheorderwith①first.)
Fig.6
19.TheFig.6showsawoodencubethatisconsistedof5×5×5small1cm3
cubes.Theshadedareaontop,front,andsiderepresentsmallcubesthat aretakenoutallthewaythroughtotheoppositesideofthewoodencube Findthevolumeoftheremainingpartofthiscube
20.SupposenisanaturalnumberandS(n)representsthesumofthedigitsof n.Ifn+S(n)=2013,findn
Team RoundAnswers 1.2
2.2
3.1100003,32410
4.18108
5.2.625
Fig.1
6.(1)△AEI,△AJF,△AGB
(2)Fig.1
7.17
8.50π
9.36
10.25
11.14
12.50Blackpiecesand40 Whitepieces
13.24busesand529travelers 14.49
15.3
16.101
17.6
18.①⑤④③②
19.79
(4)RelayRound·Problems
FirstRound
1A.Supposeaboxcontainssomeredandwhiteballs.Theproportionofnumberofredtowhite
ballsis12∶35.Ifwetakeout13 oftheredballsand47 ofthewhiteballsinthebox,findthe proportionofnumberofremainingredballstoremainingwhiteballs.(Expresstheanswerin simplestproperfractionform.)
1B.Letn
m= TNYWR (TheNumberYou WillReceive)
Arectangularpieceofpaperofsizen×mcanberolledintovariousshapesofcylinderswith
nobases.Whatisthelargestvolumeamongthesecylinders? (ExpressAnswerintermsof π.)
SecondRound
2A.Considertwosetsofnumbers:SetA={1,3,5,7,…,97,99}andSetB={2,4,6,8,…,98,
100}.IfanumberisrandomlytakenoutofsetAandanothernumberfromsetBandcalculate theirsum,how manydifferentsumsarepossible?
2B.LetT= TNYWR (TheNumberYou WillReceive)
Supposeanexam has3problemsandthepercentagesofdoingthefirst,second,andthird problemcorrectlyareT%,80%,75%,respectively.Ifastudent musthaveallthese3 problemscorrecttogetan“A”,whatistheminimumpercentageofstudentswhoget“A”in thisexam?
ThirdRound
3A.Ifthesumoftwo2-digitnumbersabandacanda1-digitnumberdis30,andd>b>c>
a,findab
3B.LetT= TNYWR (TheNumberYou WillReceive)
Letabetheaverageofalltheprimenumbersthatarelargerthan10andsmallerthan20 Findthelastdigitoftheproductof(a-T)byitself2012times
RelayRoundAnswers
FirstRound
1A
15
1B.450
π
SecondRound
2A.99. 2B.54%.
ThirdRound
(5)IndividualRound·Problems
FirstRound
Fig.1
1.SupposethethreesquaresintheFig.1aresamesize withedgelength ofa.Asinthefigures,thefirst squarehasoneshadedincircle,thesecondsquarehas9 equalsizeshadedincircles,andthethirdsquarehas16 equalsizeshadedincircles.Findthesumofallareasin thosethreesquaresthatarenotshaded
2.Supposea=17+20+23+…+170,
b=23+28+33+…+218,
c=12+22+32+…+262.
Rankthelettersa,b,andcfromthesmallesttolargest
(Note:Theformula12+22+32+…+n2=n(n+1)(2n+1)
6 canbeusedtofindc.)
3.LetMbeasetthatconsistsofthefirst2012positiveintegersandSisasubsetofM Atleast how manyelementsS musthavetoguaranteethereexistsonepairofnumbersinShaving theirdifferencethatisamultipleof5?
4.If2redlightbulbsand4yellowlightbulbsarestringedtogetherverticallyto makealight signalsof6lightbulbs(differentcombinationmakesdifferentsignal),how manywayscan theybestackedtoformdifferentlightsignals?
SecondRound
5.Ifyourwatchshowsitis9∶20inthemorningnow,whattime(hoursandminutes)wouldit beafter17999998minutes?
6.ConsiderthreeworkersA,B,andCallmakingthesameproduct.Itisknownthatitwould
takeA4hoursandB5hoursto makethesamenumberofitems.Also,itwouldtakeB4 hoursandC3hourstomakethesamenumberofitems.FindhowlongittakesAtoproduce thesamenumberitemsaswhatCproducesin15hours?
7.Ifthesum ofthefirstnnaturalnumbersstartingwith1isa3-digitnumberwithidentical digits,findn
8.Findthesmallestnaturalnumberasothattheremainderis8whenthenumberof43a isdi-videdby9
ThirdRound
9.Whatisthemaximumnumberofprimenumbersthatcanbeformedbyusingthefirst9natural numbers(from1to9)asdigit(s)? (Eachnumbercanbeusedonlyonce.)
(6)Fig.2
11.Supposetherearesixgoodswithweights7.5kilograms,5kilograms,4.5 kilograms,4kilograms,4kilograms,and2.5kilograms.Theyaretobe placedinthreeboxesinawaythatmakestheheaviestboxaslightaspossi-ble.Thisway,whatistheweightintheheaviestbox?
12.Giventhatamousecanrun5stepsinthetimewhichacatcanrun3stepsbut thedistanceforacat′s4stepsisthesameasamouse′s7steps.Nowsuppose themousehasa3 metersheadstartofthecat.Whatisthedistancethecat
mustrunbeforeitcancatchuptothemouse?
FourthRound
13.AsintheFig.2,26Englishalphabetsarewritten,inorder,insidethe26 squaresthatarrangedintheSshape.Considerawoodencubewith6faces
Fig.3
eachiswrittenanumberfrom1to6 where1and6areontheopposite sides,2and5areopposite,and3and4areopposite.Inthebeginning, placethe woodencubeonletterA withthenumber1ontopandthe
number2facingletterBasshowninthefigure.Nowrollthecubealong theSshapefigurefollowingtheletterinorder.Whatisthenumberontop
whenthecubeisontheletterZ?
14.AsintheFig.3,pointEbisectsAB,pointDisonACsuchthatAD = 2DC,andFistheintersectionofCEandBD.IfS△BEF=35,andS△BFC=
Fig.4
28,findS△DCF
FIfthRound
15.How many3-digitoddnumbersabc,suchthata<b≤canda+b+c=21? 16.TheFig.4isaregularpentagonwithverticesA,B,C,D,andE.Each
ofitsinterioranglesis108°.Connecttheverticesandform 5diagonals ThesediagonalsintersectatF,G,H,I,andJ asshown.Isoscelestrian-glescanbeformedbyusingAasavertexandtwopointsfromtheother9
points.How manysuchisoscelestrianglescanbeformed? IndividualRoundAnswers
FirstRound
1.3a2-3πa2
4 2.b<a<c
3.6
4.15
SecondRound
5.9:18inthemorning
6.16
7.36
8.5
ThirdRound
9.6
10.110
11.10
12.63
FourthRound
13.4
14.283,14,ornosolution
FifthRound
15.3