ofsquare IJHK aretheintersectionsofthetwo main diagonalsof EBFP and MPND , respectively.Findtheareaofsquare IJHK9. Fig.4.[r]
(1)2014
WorldMathematicsTeamChampionship
JuniorLevel
Team Round
·
Problems
1.Observethepatternofthefollowingsequenceofnumbers:
Fig.1
1,12,13,23,14,24,34,15,25,35,45,… Whichtermgivesthenumber1126?
2.How many wayscanthenumber14bewrittenasasum ofprime numbers? (3+11and11+3areconsideredthesame)
3.TheFig.1iscomposedofnineidenticalregularhexagonseachofedge
Fig.2
length1.Threeneighboring(eachhasatleastonecommonedgewithanother one)hexagonsaretakenoutsothattheremainingfigureisnotdisconnected andhasthesameperimeterastheoriginalfigure.Whichthreehexagonscan betakenout?
4.TriangleABCintheFig.2isanequilateraltriangle whereD andEare midpointsofABandAC,respectively.TrapezoidDECBisdividedintofour
Fig.3
smallerisoscelestrapezoidsofidenticalshapeandarea.How manytimesis theperimeterof△ABClargethantheperimeteroftheisoscelestrapezoid
DEQP?
5.How manyintegersacansatisfytheinequality11<4 2014<a 125?
6.Randomlyselectthreedistinctnumbersfrom {1,2,3,4,5,6}toforma digitnumber.How manydifferentquotientsarepossibleifthesumofall
suchpossible3 digitnumbersformedisdividedbythedifferentsumsoftheirthreedigits?
7.AsshownintheFig.3,ABCDEisaregularpentagonandtheareaofquadrilateralACQPis1 Findtheareaofthe5 pointstarAMBNC…PA (theshadedregion)
8.AsshownintheFig.4,ABCDisasquarewheresquareEBFPandsquareMPNDsharea commonvertexPandhaveareasof64and16,respectively.SupposethetwoverticesJandK
ofsquareIJHK aretheintersectionsofthetwo main diagonalsofEBFP andMPND, respectively.FindtheareaofsquareIJHK
Fig.4
9.Supposethesidelengthofasquareisaprimenumberathatislessthan20 andthesidelengthofanequilateraltriangleisanaturalnumberb.Ifthe perimeterofthissquareis10longerthantheperimeterofthisequilateral triangle,findthenumberofpossiblevaluesforb
(2)integer,whatisthemaximumpossiblevaluefory?
11.SupposenumbersAandBhaveonlyprimefactorsof2and5andtheirGreatestCommon
Divisor(GCD)is50.IfAhas6factorsandBhas12factors,findallpossibleA+B
Fig.5
12.AsshownintheFig.5,ABCDEFisaregularhexagon.Theshadedtriangle hasverticesattheintersectionofADandFCandthe midpointsofACand
EC.Theareaoftheshadedtriangleis6.FindtheareaofABCDEF
13.How manyfractionsamong20141 ,2
2014,32014,…,20122014,and20132014aresimplified fractions?
14.Ninepointscanbeusedtoconstruct8straightlineswitheachlinepassingthroughexactly3 points(seeFig.6)or9straightlines (seeFig.7)oreven10straightlineswitheachline passingthroughexactly3points(seeFig.8).Whatisthemaximumnumberofstraightlines canbeconstructedusing10pointssothateachlinepassesthroughexactly3points?
Fig.6 Fig.7 Fig.8
15.Supposethereare1350 passengerbusesforarentalof$800perbusand2040 passenger busesforarentalof$680perbus.Ifthereisatotalof720passengers,inordertominimize thetotaltransportationcost,how many40 passengerbusesshouldbeused?
16.Givenanumberthathasfinitenumberofdecimalsandsatisfiestheconditionthatifits decimalpointismovedtotherightbyacertainnumberofpositions,thenewnumberislar- gerthantheoriginalnumberby9.999.Findthesumofallnumbersthatsatisfythiscondi-tion
17.SupposeNisanaturalnumberwithdistinctdigitsandeachdigitisalsoitsfactor.Findthe valueoflargestsuchnumberN
18.SupposethreepeopleA,B,andCarewalkinginthesamedirection,eachataconstant
speed,alongastraightline.Inthebeginning,theirpositionsareasshowninFig.9.Aftera certainamountoftime,theirpositionsareasshowninFig.10.Astheycontinuewalkingwith theiroriginalconstantspeed,Aisfinallypositionedinthemiddleandisequaldistancedfrom
BandC.FindthedistancebetweenBandCatthattime (alldistancesare measuredin meters)
Fig.9 Fig.10
Fig.11
(3)togethertoformrectangularsolidsofdifferentconfigurations.Foreachconfiguration,add upallthenumbersontheoutsideofthesolid(includingthoseonthebottomsurface).What isthesmallestsumamongalltheconfigurations?
Fig.12
20.TheFig.12has121×1smallsquaresandatotalof20vertices(somesquaresshare commonvertices).Amongthetrianglesthatcanbeformedbyusingthreepoints fromthese20points,how manyofthemhaveanareaof2?
Team RoundAnswers
1.312
2.10
3.H,G,ForB,C,D
4.2.4
5.107
6.1
7.2
8.36
9.3
10.770
11.6300or550or250
12.144
13.936
14.12
15.3
16.1.213
17.9867312
18.560
19.60
(4)RelayRound
·
Problems
FirstRound
Fig.1
1A.How manytrianglesareintheFig.1?
1B.LetT= TNYWR (TheNumberYou WillReceive)andletS=3T
FindthesumofallpositiveintegerslessthanSthathaveoddnumberoffactors
SecondRound
2A.Thethreenumbers277,362,and515shareacommoncharacteristicinthat theyhaveacommonremainderRwhentheyaredividedbyacommondivisorD
notequalto1.Find(D-R)
Fig.2
2B.LetT=TNYWR (TheNumberYouWillReceive).Supposeacircleofradius8 andcenterOandasquareofsidelengthofTandhasoneofitsverticesatOas shownintheFig.2
LetS1=areaoftheregioninsidethesquarebutoutsidethecircle
S2= areaoftheregioninsidethecirclebutoutsidethesquare
FindS2-S1.(Useπ=3)
ThirdRound
3A.Suppose
1WMTC
× 3
3MTC1 whereeachletterrepresentsadistinctdigit.FindW+M+T+C 3B.LetT=TNYWR (TheNumberYouWillReceive)andletS=T-19.upposethereareN2
digitnumbersabthatsatisfya-b=S
Findtheunitsdigitofthenumber■N————·N·— ————N■·N…—N■
100N's
.(productof100N's)
RelayRoundAnswers
FirstRound
1A.35 1B.385.
SecondRound
2A.12 2B.48.
ThirdRound
(5)IndividualRound
·
Problems
Fig.1
FirstRound
1.AsshownintheFig.1,AE=4,ED=3,BC=6,andDC=9.Findthearea of△ABD
2.Giventhatboth4 digitnumbers10abandba01areperfectsquares
Findab
3.Anoperahousehas27rowsofseats.Eachrowhas2moreseatsthantherowimmediatelyin frontofit.Ifthe14throwhas60seats,how manyseatsdoesthisoperahousehave?
Fig.2
4.Findthelastdigit(unitsdigit)forthenumber
2×2014+3×2013+4×2012+…+1008×1008
SecondRound
5.AsinFig.2ABCDEFbelowisaregularhexagon.Iftheareaofquadrilateral
ACDEis8,findtheareaofthishexagon
6.Divideacubewithintegeredgelengthinto153smallercubesinwhich152of themarecubesofedgelength1.Findtheedgelengthoftheoriginalcube
Fig.3
7.Ifeachofthelettersfroma,b,c,d,anderepresentsadifferentnumberfrom 3,4,5,6,and7andthata+1+b=1+c+e=d+e+2,finde
8.AsshowninFig.3,theareaofisoscelesrighttriangleABC(withrightangleat
BandAB=BC)is1.NowusethethreeedgesAC,AB,andBCassidesto constructexternalsquaresACDE,AFGB,andBHIC.Findtheareaofthe hexagonDEFGHI
ThirdRound
9.How manyquadrilateralsdoestheFig.4have?
Fig.4
10.SupposeA,B,andCareanythreedistinctdigitsfromnumbers1to9and
A+2B+3C=12.IfA,B,andC areusedasdigits,eachusedonceandon-lyonce,toforma3 digitnumber,how manysuch3 digitnumbersare largerthan321?
11.Agroupofstudentslinedupinastraightlinefromlefttorightastheycount offnumbers1,2,1,2,1,2,…asshowninthefirstpictureandthenthey
(6)12.Acertainnumberofstudentsengaginginropejumping (skipping)contestaredividedinto threegroupsA,B,andC.
Everycontestantmustbelongtooneandonlyonegroup.Theav-eragenumbersofropejumpingperformedbystudentsingroupsA,B,andCare100,80, and70,respectively.IfgroupsAandBarecombined,theircombinedaverageis85times.If
groupsBandCarecombined,thecombinedaverageis76times.Whatistheaveragewhen allthreegroupsarecombined?
Fig.5
FourthRound
13.SupposeABCisanisoscelesrighttrianglewithrightangleatAandAB= 4.UseA,B,andCascentersandABastheradiustodrawthreearcs formingafigureasshownbelow.Findtheareaoftheshadedportionof
Fig.5.(Useπ=3)
14.TheFig.6iscomposedof7nsmall1×1squares.Rectangleswithedges
Fig.6
paralleltotheedges ofthese1×1 squarescan beconstructed by connectingverticesofthese squares.Ifthere are exactly 154 such
rectangles(excludingsquares)witharea3,findn
FifthRound
15.ConsiderthetwocirclesasshownintheFig.7.If23 ofthesmallcircle
Fig.7
and45 ofthelargecircleareshadedandifthesmallcirclehasanareaof12, whatistheareaofthelargecircle?
16.AsshowninFig.8,convex4 sidedpolygonshave2diagonals,convex5 sidedpolygonshave5 diagonals,andconvex6 sided polygonshave9
diagonals,andsoon.How manysidesdoconvexpolygonshaveifthereare2015diagonals?
Fig.8
IndividualRoundAnswers
FirstRound
1.27
2.89
3.1620
4.7
SecondRound
5.12
6.6
7.4or7
8.12
ThirdRound
9.60
10.4
11.31
12.80
FourthRound
13.8
14.14
FifthRound
15.20