Supposetheareaof△ ABC is6 3, AB =6, A isanobtuseangle, D isapointon BC sothat.. BD =2 DC.?[r]
(1)2011
WorldMathematicsTeamChampionship
AdvancedLevel
Team Round
·
Problems
1.SupposeA={x|x2-x<0}andB=x x+1
3
é ë
ù
û+ x+
2
é ë
ù
û=1,x∈A
{
}
FindA∩BC whereBCisthecomplementofBand [a]representsthelargestintegernot
greaterthana
2.SupposethelengthsofthethreesidesthatareoppositetothethreeinterioranglesA,B,and
Cof△ABCarea,b,andc,respectively,andsatisfy
a2+b2+c2-2 7a-4b-6c+20=0.
Findtheareaof△ABC
3.Supposemax{|a+b|,|a-b|,|2012-a|}≥C,whereC
isaconstant,holdstrueforanyre-alnumbersaandb.FindthelargestvalueofC.(Note:max{x,y,z}representsthelargest ofx,y,z)
4.LetA(-3,2),B(5,6)andC(9,-2)bethreepointsontheplane.IfABCDisasquare, findthecoordinatesforDandfindtheareaofthepartofthesquarethatisinQuadrantII
5.Iftheinequality3t2-2t-1≥ 1
3
ỉ è
ư ø
22sin2x-22sinxcosx-
holdsforanyrealnumberx, findtherangeofvaluesfort
6.Findthesolutionsetforinequality 2x-8+2 4-2x-2≤ 2+π.
7.LetSnandTnbethesumofthefirstntermsofarithmeticsequences{an}and{bn
},respec-tively.Supposeb a5
3+b2n-3+ a2n-5 b7+b2n-7=
n
2n+1foranyintegern.Find
S23 T23
Fig.1 8.Ifa,b,andcarepositiverealnumbersanda+b+c
=1,findthein-tegerportionofthenumber 3a2+1+ 3b2+1+ 3c2+1.
9.AsinFig.1,thereare12pointsAi(i
=1,2,3,…,12)ontheel-lipseE:x2 +
y2
3 =1.IfO istheoriginandtheincludedanglesbe-tweenOA→iandOAi+1→(i<12)areallequaltoπ6,find
∑12
i=1
1 |OA→|i
(2)
totalprofitinmillionfromthesetwoinvestments?
11.LetObethecenterofthecircumscribedsphereofrectangularboxABCD-A1B1C1D1with
volume32π3 LabelAB=a,BC=b,andCC1=c
If94isthesmallestvaluefora12+b42,findtheminimumdistancebetweenAandCalongthe
surfaceofthesphere
12.Supposeθ∈ 0,π
2
ỉ è
ư
øandequationx
2-
sin2θx+1=0hasarealrootsinθ
Thenwhatisthevalueforcosθ?
Fig.2 13.Supposeequationx2-ax+b=0hastwopositiverealroots.Thenwhatis
thevaluerangefora+1-ab?
14.AsitisshowninFig.2,theunitcubeABCD-A1B1C1D1hasedgeof1
LetMbeapointonedgeCDandΩbeaplanesectionthatisinsidethe cubeandpassesthroughpointsM,A,andC1.Whatisthisplanesection 'sareawhenthisplaneformsadihedralangle(ortheanglebetweenthese
twoplanes)of60°withplaneABCD?
15.SupposeM=(11+3)2011and(M)representsthedecimalportionofM
FindthevalueofM·(M)
16.Suppose∠xOy=2α,0°<α<90°,Ozbisects ∠xOy,pointAisonOzandOA=a,and
MANisalinethatpassesthroughAandintersectsOxandOyatMandN,respectively FindthevalueforOM1 +ON1 intermsofαanda
17.Supposetheareaof△ABCis6 3,AB=6,Aisanobtuseangle,DisapointonBCsothat
BD=2DC.IfAB→·AD→=4,findcossinBC.
Fig.3 18.SupposeF1andF2arethetwofociofellipsex
2 a2+
y2
b2=1(a>b>0)
Also,asintheFig.3,supposetherightdirectrixoftheellipsetangentto thecirclewithcenteratIandpassesthroughbothF1andF2andthat
IF1→·IF→2= a
2a2-2b2
Findtheeccentricityoftheellipse
19.Letf0(x)= 11-xandf1(x)=xx -1
Definefn+2(x)=fn+1(fn(x)),n=0,1,2,…
Expressf2011(2011)infractionqpofmostreducedterm.Findp+q
20.Ifpositivenumbersx,y,andzsatisfyx+y+z=1and
(3)Team RoundAnswers
1.
{
x0<x<13 or23≤x<1}
2.32 3
3.1006
4.(1,-6);294
5.t≤-1ort≥3
6.[3,4]
7.1225
8.3
9.3
10.4
11.23π
12.5-12
13.[3,+∞)
14.5-1
15.22011.
16.2cosa α
17.2 37
18.22
19.4021
(4)RelayRound
·
Problems
FirstRound
Fig.1
1A.TherearethreevectorsOA→,OB→,andOC→onaplaneasinthe Fig.1
SupposeOA→,OB→andOA→,OC→haveincludedanglesof150°and60°, respectively.Also,|OA→|=|OB→|=1and
|OC→|=2.IfOC→=λOA→+μOB→whereλ,μ∈R,findλ2+μ2.
1B.LetT = TNYWR (TheNumberYou WillReceive).
Therealvaluefunctionfisdefinedasf(x+y)=f(x)+yforanyrealnumbersxandy
Iff(T)=2011,findf(2011)
SecondRound
2A.Findthesumofallrealrootsoftheequation(x+1)(x2+1)(x3+1)=30x3.
2B.LetT = TNYWR
LetA(x1,y1)andB(x2,y2)bepointsmovingalongtheparabolay2=4xandellipse x2
4 +y
2
T=1,respectively.LetN(1,0)beafixedpoint.IfAB∥x-axisandx1<x2,findthe
intervalwhere△NABcantakeasitsperimeter
ThirdRound
3A.LetBbethereflectionpointofA(4,1)overtheaxisofsymmetrylinex-y-1=0
Findthesmallestvaluefor9a+27b+1wherethelineax+by
-2=0passesthroughB
3B.LetT=TNYWR
Ifthefunctionf(x)= T+2tx-xtakesonlargestvalueM wheretandMarepositive
naturalnumbers,findM
RelayRoundAnswers
FirstRound 1A.28. 1B.3994
SecondRound 2A.3.
2B.10
3,4
æ è
ö ø
(5)IndividualRound
·
Problems
FirstRound
1.Solvetheinequality x+5>x-1
2.Ifwedefineafunctionf(x+a)=|x-2|-|x+2|andf[f(a)]=3,findthevaluefora
3.Ifthecoefficientsa,b,andcofthequadraticequationax2+bx+c=0(abc
≠0)formageo-metricsequenceandtheratioofitstworootsx1andx2isλ,findλ+λ 1
4.LetP(1,1)beapointinsidecirclex2+y2=4andletABandCDbetwochordsofthecircle
passingthroughP.IfthetangentstoAandBintersectatMandthetangentstoCandD in-tersectatN,findtheequationofthelineMN
SecondRound
5.Supposearectangularboxhasintegeredgelengthsanditsmaindiagonalis25.Whatisthe smallestfaceintermsofareaamongallsuchpossibleboxes?
6.Ifrealnumbersx,y,andzsatisfytheequation
x2+2y2+5z2+2xy+4yz-2x+2y+2z+11=0,
findtherangeofvaluesx+2y+3ztakes
7.LetM(x0,y0)beapointinsidecirclex2+y2=r2(r>0)andx0y0≠0.Findthenumberof
pointsthatthelinex0x+y0y=r2intersectsthecircle
Fig.1 8.AsintheFig.1,AB1C1,C1B2C2andC2B3C3areequilateral
trianglesallwithedgelengthof2sittingonastraightlinenext toeachotherwithcommonverticesatC1andC2.VerticesB1, B2,B3areallonthesamesideofthatline.Supposethereare
10pointsP1,P2,…,P10onsideB3C3anddefine mi=AB→2·AP→i(i=1,2,…,10)
Findm1+m2+…+m10
ThirdRound
9.Ifxisanacuteangleandtanx= 2-1,findx
10.In △ABC,leta,b,andcbetheoppositesidesof∠A,∠B,and∠C,respectively If∠C=3∠B,comparethesizerelationshipbetweencand3b
11.Supposethethreerealrootsfortheequationx3-(4+d)x2+5dx-d2=0,wheredisa
naturalnumber,representthesquareofsomerighttriangle'sthreesides.Findd
12.SupposeF1andF2arebothfocifortheellipsex m +
y2
n =1andfociforthehyperbola x2
p -y2
q=1(m,n,p,q∈R+).IfMisanintersectionoftheellipseandthehyperbolaand
|MF1→|·|MF2→|=1
(6)FourthRound
13.Ifx=sin35°cos65°-cos65°cos5°-cos55°cos5°,whatisthevalueforx?
14.A-BCDisaregulartetrahedronwithedgelengthof24.O isasphereinscribedinsidethetet-rahedron.O1isasmallspherethatistangenttotheupperthreesidesofthetetrahedronand
thelargesphere.Whatisthevolumeofthatsmallsphere? (Expressyouranswerinterms ofπ)
FifthRound
15.Findtheintervalsforxwherethefunctiony=
|
log2|x+1||
ismonotonicallydecreasing16.DetermineS100ifSn+1= S n
1+nSn,n=0,1,2,3,… andS0=
1
100.(Expresstheanswerin fractionoflowestterm)
IndividualRoundAnswers
FirstRound
1.[-5,4)
2.32
3.-1
4.x+y=4
SecondRound
5.108
6.[-1,5]
7.0
8.180
ThirdRound
9.22.5°
10.c<3b
11.4
12.92
FourthRound
13.-34
14.8 6π
FifthRound
15.(-∞,-2]and(-1,0]