A geometricsolidiscomposed by pastingmanyidenticalsmallcubes togethersidetoside.Thefigure3 show the different views ofthis solid.Let m betheedgelengthof thesecubes.Findthesurfacearea oft[r]
(1)2015WMTCIntermediateLevel Team RoundProblems
1.Givenarectanglewithanareaof910andbothitslengthandwidthareintegerslargerthan 20.Findanintegerthatisclosesttothelengthofitsdiagonal
2.Supposearectanglehasanareaof2016andbothitslengthandwidthareintegers.Findthe perimeterofsuchrectanglewithsmallestdifferencebetweenitslengthandwidth
3.How manypossiblepairsofnonzerorealnumbersaandbsothatthereareexactlytwo differentvaluesamongthefournumbersa+b,a-b,aìb,andaữb?
Fig.1 4.Givena>0andb>0.Ifa+b=3,findthesmallestvaluefor
a2+4
a + b
2
b+3
5.LetIbethecenterofthecircumcircleof △ABCand △DEFisformedby usingtheperpendicularbisectorsofIA,IB,andICasitsthreesidesas showninfigure1.IfIA =6andS△DEF=21,findtheperimeterof△DEF
6.Ifx =316+312+39,findx -x12
Fig.2 7.GivenasemicircleO.LetBC (notitsdiameter)beachordandBC︵beits
minorarc.Asshowninfigure2,usethechordBCasaxisofsymmetryand foldtheminorarcBC︵overuntilitintersectsthediameterACatpointD.If
AD
AC =25 andAC =2015,findthelengthofchordBC
8.Randomlyselectapositivefactorfrom62015.Iftheprobabilityofthisfactorhappenstobea
multipleof61512isn
m wheremandnarerelativelyprimes,findthevalueofm -n
Fig.3 9.Asshowninfigure3,pointEisontheextensionofrectangleABCD's
diagonalDB withDB =2BE.LetF bethe midpointofDCandEF
intersectsBCatG.Iftheareaof△AEBis100,findtheareaof△BEG
10.LetSmbetheareaofthetriangularregionthatisenclosedbystraightlines
(2)Fig.4 11.Aninverseproportionfunctiony=xkhastwopointsAandBontheFirst
Quadrant.DrawalinesegmentADthatisperpendiculartotheyaxisatD
andanothersegmentBCthatisperpendiculartoxaxisatCasshownin figure4.Iftheareaof△OABis56 andtheareaof△OCDis32,findk
12.Supposethesum ofkconsecutive positiveintegersis2015.Findthe smallestnumberamongtheseknumbers
13.SupposeMisapositiveintegerandboth8M +40and8M -40areperfectsquares.Findthe valueofM
Fig.5 14.Considerthefigure5.Supposetheinscribedcircleof△ABChasaradius
of2.LetM andN be pointsonAB andBC sothatthey arethe
intersectionsofthelinethatpassesthroughthecenterandparalleltoAC
IfMN =7andAM =4,findtheareaofthetrapezoidAMNC
15.Supposetheequationax2+bx +c=0hasrealsolutionsanditscoefficientsa,b,andc
satisfythefollowingconditions:
(1)a,b,andcarepositiveintegers;
(2)The6digitnumbera2015bisdivisibleby12;
(3)c3+3isdivisiblebyc+3.
Findthemaximumvaluefora+b+c
Fig.6 16.Asshowninfigure6,arayoflightentersfrompointAofa4×mgrid
graph.ThisraywillreflectwheneverithitsthesidesAB,BC,CD, orAD.However,ifwouldleavethegraph whenithitsthecorner pointsA,B,C,orD.SupposethisrayoflightentersfromAand passesthrough2016gridpoints (includingpointsAandDandeach pointwouldonlycountonce)andthenleavethegraphatpointD.Findm
17.Ifpositiveintegersxandysatisfytheequationx3+5x2y+8xy2+6y3=91,findthevalue
ofx +y
18.Supposeacircleofradius1thatistheinscribedcircleofaregularhexagonandalsothe circumcircleofasquare.Letaandbbetheedgelengthsofthehexagonandsquare,
(3)Fig.7 19.Considerthefigure7.Giventhree pointsA(-3,0),B(3,0),and
C(0,-3).How manypossiblepointsE(x,y)(where0<y<4)that willmake△ABEsimilarto△ABC?
20.Asshowninfigure8,pointBisthemidpointofarcAC︵,pointEisonthe
Fig.8
chordAC,andFisonarcAC︵.IfarcAC︵=120°,∠B =∠FEC=90°,and theradiusofthecircleforthearcAC︵is 3,findthelengthofEF
Team RoundAnswers 1.44
2.180
3.2
4.256
5.14
6.3312
7.806
8.15
9.25
10.20151008
11.2
12.2
13.13
14.14+2 3+
15.16
16.1343
17.3
18.3+ 62+ 15
19.4
(4)RelayRoundProblems
1A.Ifa
b =a =c 2,find33b-da-c
1B.LetT =TR (ThenumberyouwillReceive).Leta,b,andcbethesidesthatareopposite
toanglesA,B,andC,respectively,of△ABC.Ifb =a a+cb ,∠A =30°,anda=T,find theareaof△ABC
Fig.1
2A.Asshowninfigure1,let∠A +∠B+∠C+∠D +∠E+∠F+∠G=x°
Findx
2B.LetT =TR.Supposexandyareintegerthatsatisfythesetofequations
{x +3y2+2xy=18,
y+3x2+4xy=6,
① ② FindthevalueforT(x +y)+2015
3A.Thecubeofanaturalnumbercanbewrittenasthesumoftwoormoreconsecutiveodd
numbers.Forexamples,23=3+5,33=7+9+11,and43=13+15+17+19.If93iswritten
asthesumoftwoormoreconsecutiveoddnumbers,whatisthelargestoddnumberinthis sum?
3B.LetT =TR.Supposepandqarenonzeronaturalnumbersandthatp<q.Ifp
q=0.18…
andq=110,find(p +T)
RelayRoundAnswers
1A.2. 2A.540.
.395or3455
3A.89.
(5)
IndividualRoundProblems
1.Supposerealnumberaanditsreciprocal(multiplicativeinverse)havethesamevalueand
realnumberbanditsadditiveinversehavethesamevalue,findallpossiblevaluesfora-b
2.Supposerealnumbersxandysatisfyingtheequation|x+y|+ x -1=0.Findthevalue forx2015+y2016.
Fig.1 3.Asshowninfigure1,quadrilateralABCDisinscribedincircleO.If∠DOB=
100°,findthedegreemeasurementfor∠BCD
4.Iftheproductoftwo2digitnumbersx2and2yis736,findx +y
5.Ifthesumofallinterioranglesofaconvexnsidedpolygonis7timesthesum ofallitsexteriorangles,findthevalueofn
6.Findthelargestintegernsuchthatn300<7200.
7.Supposethegraphofaninverseproportionfunctiony=xkpassesthroughpoint(1,8).Let
ABbethechordofCircleOwithlengthkandthedistancefromcenterOtoABis3.Find theradiusofCircleO
8.Whatistheprobabilityofselectinganythreeconsecutiveprimesfrom alistofprime numbersthatarenotlargerthan50sothatthesumofthesethreenumberisnotaprime?
9.How manyvaluescanxtakeonsothatthepointsM(1,2),N(6,2),andP(x,0)forma righttriangle△MNP?
10.Iftheradiusoftheinscribedcircleofaregularhexagonis1,findtheareaofthishexagon's circumcircle
11.Findtheroot(s)forequation(x -1-x1 )
2
÷xx22--x +2x +11=1
Fig.2 12.RotaterectangleABCDclockwise90°aroundpointAtoAEFGpositionas
showninfigure2.Iftheareaof △BCDis6+2 5,findtheareaof △DEF
13.Supposestraightliney =kx +bintersectshyperbolay =xk atpoints
(6)TopView FrontViewRight SideView Fig.3
14.A geometricsolidiscomposed by pastingmanyidenticalsmallcubes togethersidetoside.Thefigure3 show the different views ofthis solid.Letmbetheedgelengthof thesecubes.Findthesurfacearea ofthissolid
15.If[2x +1]=3x -12,findx
(Note:[x]representsthelargestintegerthatisnotgreaterthanx.)
16.Asshowninfigure4,theoutsidecircleisthecircumcircleofradius2ofaregularhexagon Theshaded6pointedstarregionisformedbyusingeachsideofthishexagonasaxisof symmetryandflipthecorrespondingarc180°.Findtheproportionoftheareaoftheshaded regiontotheareaofthecircumcircle
Fig.4 Fig.5 Fig.6
17.Asshowninfigure5,pointsDandEareonsidesACandABof△ABC,respectively,and straightlinesDBandECintersectatpointF.Iftheareasof△CDF,△BEF,andAEFD
are3,4,and20413,respectively,findtheareaof△BCF
(7)Fig.7 19.Asshowninfigure7,△ABCand △ACDareboth
equilateraltrianglesofsidelength1.Fixtheposition of △ABCandrotate △ACD onerevolutionaround △ABCusingpointCasanchor.Dothesamething usingBandAasanchor.Findthetotalareathatis sweptbylinesegmentACafterallthreerevolutions
20.Supposetheparabolay =-x2+bxpassesthroughB(2,0).LetCbeapointonthe
parabola'saxisofsymmetryandA=(4,4).FindthecoordinatesofCsothatthelengthAC +BCissmallest
IndividualRoundAnswers 1.±1
2.2
3.130°
4.6
5.16
6.3
7.5
8.134
9.4
10.43π
11.0
12.4
13.0
14.30
15.56,76,32
16.3-1
17.5
18.23
19.π +23