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Givenarectangle ABCD with AB =4and BC =6andasquare AEFG oflength 13that sharesavertex A withtherectangle ABCD. Supposethissquare,onthesameplaneas[r]
(1)2010WorldMathematicsTeamChampionship AdvancedLevel
Team Round·Problems
Fig.1
1.GiventwononemptysetsAandB.IntheVenn'sdiagramshowninFig.1, defineA※Btobetheshadedarea.If
M={x|y= -x2+3x+10},andN={y|y=3x-1},
thenM※N=
2.Givenfunctionsf(x)=x2-3andg(x)=m(x-1).Ifforanyx
0∈[-3,3]thereexists
x′∈[-3,3]suchthatg(x′)=f(x0),thenthevaluerangefortherealnumbermis
3.Ifthesolutionsetofxfortheinequalitymx>nis(-∞,3),thenthesolutionsetofxforthe inequality(m-n)x+m+n>0is
4.Use[x]torepresentthelargestintegerthatisnotlargerthanx.Ifarealnumberrsatisfies
[r+101]+[r+102]+…+[r+109]=122,thenthevalueof[10r]is
5.Givenaquadraticequationx2-xsinθ+sinθ-5=0intermsofx.Thenthisequation'slargest
rootis anditssmallestrootis
6.Ifthethreestraightlines2x-y+1=0,x+y+2=0,x+ay=0cannotformatriangle,then
amusttakeonvaluesof
7.Supposethatx,yandzarerealnumbersthatsatisfyx+2y+3z=1andyz+zx+xy=-1, thenthevaluerangeforx+y+zis
8.Supposea,b,c∈R+,a+b+c=1andM= 3a+1+ 3b+1+ 3c+1,thentheintegerpartof
Mis
9.Givenasequence{an}wherea1=3 ,a2=5andan+2a2n=a3n+1.Thentheformulafor
an=
10.Therootoftheequationlogm(x2+1+x)+logm(x2+2+x)=1
2logm2(m>0andm≠1)isx
=
11.Givena=(m+2,n),b=(m-2,n-4),a⊥band|a|+|b|=8,thenm+n= 12.Theareaof△ABCis2.LetAD,BEandCFbethethreemediansintersectingatpointG,
andpointsH,IandJbeonthesethreemedians,respectively,suchthatAH∶HD=1∶1,
BI∶IE=1∶2,andCJ∶JF=1∶3,thentheareaof△HIJis
13.Supposea+b+c=12andab+bc+ca=45wherea,bandcarepositiverealnumbers,then themaximumpossiblevalueofabcis
14.SupposeC:x2+(y-1)2=r2andy=sinxhaveonlyoneintersectionandthexcoordinateof
(2)15.TheareaoftheregiononthexyplaneoverwhichPranges,
P∈ (x,y) (x-cos4θ)2+ỉy-12sinθ
è
ư ø
2
=1,θ∈R
{ },is
16.Suppose[r(cosθ+isinθ)]10=243(1- 3)i-243(1+ 3)
64(1+i) Ifr>0,0<θ<π3, thenr= ,θ=
17.Fromapoint(5, 2)insidetheellipsex9 +2 y5 =12 ,drawtwochordsABandCDwithA,
B,CandDontheellipse.FromAandB,drawtwotangentssothattheyintersectatE
FromCandDdrawanothertwotangentssothattheyintersectatF Thenthelinearequa-tionforthelinethatconnectsEandFis
18.Foreachn=1,2,3,…,straightliney=x+n+1intersectstheparabolax2=1
8y-321 ỉ è
ư øat twopoints.Let|AnBn|denotethelengthofthechordconnectingtheintersecting2pointsfor
eachn.Definean=
n|AnBn|2.LetSnbethesumofthefirstntermsinthesequence{an}
ThenS2010=
19.Consideracubewithedgelengthathatishangingaboveaplaneα.Parallellightraysthatare perpendiculartoplaneαprojectthiscubeontoαtoformashadowregionandthencirclesare drawninsidethatprojectionregion.Thenthediameterofthelargestcircleis
20.Lettheradiusof☉Abe2andpointPisoutsideof☉A.Also,letstraightlinePMtangentcircle ☉AatpointMandthatcos∠MPA=23.NowifweusethelinesegmentPAastheaxisandrotate thelinesegmentPMand☉AaroundPA foronerevolution.Thenthevolumeofthepartofthesol-idformedbyrotatingPAthatisoutsideofthesphereformedby☉Ais
Team RoundAnswers 1.[-2,-1]∪(5,+∞)
2.æè-∞,-32ùû∪[3,+∞) 3.(2,+∞)
4.135
5.1+ 172 ;-3 6.±1or-12 7.3-3
4 ,3+3 34 é
ë
ù û
8.4 9.3· 5ỉ3
è ø
2n-1-1
10.0
11.2 12.1948 13.54 14.-4 15.4π
16.26;12π,1760π
17.95x+52y=1 18.20102011
(3)RelayRound·Problems FirstRound
1A Solvetheinequalityfora,2a2+(4 2-7)a+(3-2 2)<0 1B LetA=theanswerpassedfromyourteammate.Solvetheinequality
2loga(x-2)>loga2+loga(6-x)usingvalueofafromA SecondRound
2A Giveasequence{an}suchthata1=1and2an+1+an=2n+1.Findan
2B LetA=theanswerpassedfromyourteammate.Supposethehypotenuseofarighttriangle
hasalengthof5andthecosineofoneofitsanglesislimn→+∞2An-1.Ifweuseoneofsidesofthis
righttriangleasaxisandrotatethetrianglearoundthisaxistogetasolidofrevolution,find thelargestvolumeofallsuchsolids
ThirdRound
3A Supposethatthedirectrixofaparabolay2=2px(p>0)istangenttothecircle
x2+y2-4x+2y-4=0.Findp.
3B LetA=theanswerpassedfromyourteammate.Abaghasxblackand(15-x)whiteballs
thatareidenticalineverywayexceptfortheircolors.Supposethatonetakesout(A+1)balls randomly.UseP(x)torepresenttheprobabilityofgettingAblackballsand1whiteball
ThenP(x)hasthelargestvaluewhenx= RelayRoundAnswers FirstRound
1A 3-2 2,1
2 ỉ
è
ư ø
1B (2,4)
SecondRound
2A
5[2n-(-1)n·2-n]
2B.16π.
ThirdRound
(4)IndividualRound·Problems FirstRound
1.Iff(x)= 11+2lgx+1+41lgx+1+81lgx,thenf(x)+f x1
ỉ è
ư
ø=
2.Whenx→+∞,thegraphoffunctionf(x)=x3(3xx+1 isapproachingthegraphofwhichof-5)
thefollowingfunction ? Answer:
(A)y=x. (B)y=x3. (C)y=x 1 (D)y=x2 3.Thesolutionto2|x|-2|x|=22isx= .
4.Thenumberofnon-emptysubsetsofsetA={x x|x2|-2<0-30 ,x∈Z}is whereZis thesetofintegers
SecondRound
5.IfM=7×10753+2×10573+10372+5×10357+4×102+2×10,thenthesumoftheelementsin
{3,4,5,6,8,9,10,11}thatarefactorsofMis
6.Supposethatrealnumbersxandysatisfyx2+y2=2,andx+ 3y≥ 6.Thenthemaximum
valueofx+yis
7.Ifasequence{an}isdefinedasa1=2,a2=5andan+1=an+an+2,thena2010=
8.If{x|-1≤x≤3}isthesolutionsetforinequality -x2+2x+3-a(x-4)>0,thenthe
valuerangeforais
ThirdRound
9.Ifxandyarepositiverealnumbersthatsatisfy(1+x2-x+1)(1+y2-y+1)=2,
thenxy=
10.TheedgesAA1,ABandADinaparallelepipedABCD-A1B1C1D1havelengthsof2,3and
4,respectively.Ifbothoftheiranglesare60°,thenthelengthofthisparallelepiped'sdiagonal
AC1is
11.Defineasequence{an}tobea1=2,an=4
5an-1whenn≥2.StartingfrompointM,point
Mnmovesrightbya1unitsarrivingatpointM1.Thenthepointmakesaleftturnfor90°and
movesforwardbya2unitsarrivingatpointM2.Makeanother90°leftturnandmoveforward
bya3unitsarrivingatpointM3.Keepingthisprocess,thepointMnapproachestoafixed
pointNinfinitely.UseMasthecoordinateoriginandthestraightlinethroughMtoward rightas
thexaxistoformarectangularCartesiancoordinate,thenthecoordinateofthepointNis
12.Suppose∠B=15°and ∠C=30°in △ABC.LetDbeapointonBCsothatADistheangle bisectorfor∠A.IfAD→=λAB→+μAC→(λ,μ∈R),thenthevalueofλ
(5)FourthRound
13.Foranyrealnumberθ,movethestraightlinel:xcosθ+ysinθ=2andformaregion.Thearea oftheregionis
14.GivenarectangleABCD withAB=4andBC=6andasquareAEFGoflength 13that sharesavertexAwiththerectangleABCD.Supposethissquare,onthesameplaneas
ABCD,rotatesaroundpointAforonerevolution,thenthevaluerangeforthelengthCEis
FifthRound
Fig.1
15.AsinFig.1ontheright,theedgeofthiscubeisa,pointMisonCDso thatCM=a4,andpointNisonGHsothatGN=23a.Supposeaplanethat passesthroughpointsB,MandNanddividesthiscubeintoupperand lowerparts,thenthevolumeofthelowerpartis
16.Giventhefollowingconditions:straightlinesl1:x+3y=2,l2:y=kx(k> 0),andellipseC:x2+4y2=4.l
2intersectsCatpointsMandNwhereMis
inQuadrantⅢ.Supposethatl1intersectsl2atpointP,Oistheorigin,and|MO|,|OP|and
|PN|formanarithmeticsequence.Thenk=
IndividualRoundAnswers FirstRound
1.3 2.(A) 3.±5 4.63 SecondRound 5.39 6.2 7.-3
8.(0,+∞) ThirdRound 9.1 10.55 11.ỉ5041,4041
è ø
12.6- 22
FourthRound 13.4π
14.[13,3 13] FifthRound 15.13a3.