trường thcs hoàng xuân hãn

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2010

WorldMathematicsTeamChampionship

AdvancedLevel

Team Round

·

Problems

Fig.1

1.GiventwononemptysetsAandB.IntheVenn'sdiagramshowninFig.1,
defineA※Btobetheshadedarea.If

M={x|y= -x2₊₃_x_+10},and_N_={_y_|_y₌₃x_-1},

thenM※N= .

2.Givenfunctionsf(x)=x2_-3and_g(_x)=_m(_x_-1)_._Ifforany_x

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_-_x_sin_θ_+sin_θ_{-5=0intermsof}_x._{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_=1and_M_{= 3}_a_{+1+ 3}_b_{+1+ 3}_c_{+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,BEandCF_{bethethreemediansintersectingatpoint}G,

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₌_r2_and_y_=sin_x_{haveonlyoneintersectionandthe}_x_coordinateof

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15.TheareaoftheregiononthexyplaneoverwhichPranges,

P∈ (x,y) (x-cos₄θ)2+ỉy-1_2sinθ

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)insidetheellipsex_{9 +}2 y_{5 =1}2 ,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= 1

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-idformedbyrotatingPA_{thatisoutsideofthesphereformedby☉}Ais .

Team RoundAnswers

1.[-2,-1]∪(5,+∞).

2.ỉ_è-∞,-3₂ù_û∪[3,+∞).
3.(2,+∞).

4.135.

5.1+ 17₂ ;-3.
6.±1or-1₂.
7.3-3 3

4 ,3+3 34
é

ë

ù
û.

8.4.
9.3· 5ỉ₃

è
ư
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.

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RelayRound

·

Problems

FirstRound

1A .Solvetheinequalityfora,2a2+(4 2-7)a+(3-2 2)<0.
1B ._Let_A_{=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

hasalengthof5andthecosineofoneofitsanglesislim_n_→+∞₂An-1.Ifweuseoneofsidesofthis

righttriangleasaxisandrotatethetrianglearoundthisaxistogetasolidofrevolution,find
thelargestvolumeofallsuchsolids.

ThirdRound

3A ._{Supposethatthedirectrixofaparabola}_y2₌₂_px₍_p_>0)_{istangenttothecircle}

x2₊_y2_-4_x₊₂_y_-4=0_._Find_p.

3B ._Let_A_{=theanswerpassedfromyourteammate.Abaghas}_x_blackand(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 .2

5[2n-(-1)n·2-n].

2B._16π.

ThirdRound

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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_|=22is_x₌ _.

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,and_x_{+ 3}_y_{≥ 6}_._{Thenthemaximum}

valueofx+yis .

7.Ifasequence{an}isdefinedasa1=2,a2=5andan+1=an+an+2,thena2010= .

8.If{x|-1≤x≤3}isthesolutionsetforinequality -x2₊₂_x_+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

pointN_{infinitely.Use}M_{asthecoordinateoriginandthestraightlinethrough}Mtoward
rightas

thex_{axistoformarectangularCartesiancoordinate,thenthecoordinateofthepoint}Nis

.

12.Suppose∠B=15°and ∠C=30°in △ABC.LetDbeapointonBCsothatADistheangle
bisectorfor∠A.IfAD→=λAB→+μAC→(λ,_μ∈R),thenthevalueofλ

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FourthRound

13.Foranyrealnumberθ,movethestraightlinel:xcosθ+ysinθ=2andformaregion.Thearea
oftheregionis .

14.GivenarectangleABCD withAB=4andBC=6andasquareAEFGoflength 13that
sharesavertexAwiththerectangleABCD.Supposethissquare,onthesameplaneas

ABCD,rotatesaroundpointA_{foronerevolution,thenthevaluerangeforthelength}CEis
.

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₊₄_y2₌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
ư
ø.

12.6- 2₂ .

FourthRound
13.4π.

14.[13,3 13].
FifthRound
15.1₃a3_.

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