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2011

WorldMathematicsTeamChampionship

AdvancedLevel

Team Round

·

Problems

1.SupposeA={x|x2_-_x_<0}and_B₌_{x x}₊1

3

é
ë

ù

û+ x+

2
3

é
ë

ù

û=1,x∈A

{

}

.

FindA∩BC _where_BC_{isthecomplementof}_B_{and [}_a_{]representsthelargestintegernot}

greaterthana.

2.SupposethelengthsofthethreesidesthatareoppositetothethreeinterioranglesA,B,and

Cof△ABCarea,b,andc,respectively,andsatisfy

a2₊_b2₊_c2_{-2 7}_a_-4_b_-6_c₊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- 2

holdsforanyrealnumberx,

findtherangeofvaluesfort.

6.Findthesolutionsetforinequality 2x_{-8+2 4-2}x-2_{≤ 2+π}_.

7.LetSnandTnbethesumofthefirstntermsofarithmeticsequences{an}and{bn

},respec-tively.Suppose_b 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+ 3}_b2_{+1+ 3}_c2₊₁_.

9.AsinFig.1,thereare12pointsAi(i

=1,2,3,…,12)ontheel-lipseE:x2
6 +

y2

3 =1.IfO
istheoriginandtheincludedanglesbe-tweenOA→iandOAi+1→(i<12)areallequaltoπ₆,find

∑12

i=1

1
|OA→|i 2

.

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totalprofitinmillionfromthesetwoinvestments?

11.LetObethecenterofthecircumscribedsphereofrectangularboxABCD-A1B1C1D1with

volume32π_{3 .Label}AB=a,BC=b,andCC1=c.

If9_{4isthesmallestvaluefor}_a12+_b42,findtheminimumdistancebetweenAandCalongthe

surfaceofthesphere.

12.Supposeθ_{∈ 0,π}

2

ỉ
è

ư

øandequationx

2_- 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)2011_and(_M_{)representsthedecimalportionof}_{M .}

FindthevalueofM·(M).

16.Suppose∠xOy=2α,0°<α<90°,Ozbisects ∠xOy,pointAisonOzandOA=a,and

MANisalinethatpassesthroughAandintersectsOxandOyatMandN,respectively.

Findthevaluefor_OM1 +_ON1 intermsofαanda.

17.Supposetheareaof△ABCis6 3,AB=6,Aisanobtuseangle,DisapointonBCsothat

BD=2DC.IfAB→·AD→=4,find_cossinB_C.

Fig.3
18.SupposeF1andF2arethetwofociofellipsex

2
a2+

y2

b2=1(a>b>0).

Also,asintheFig.3,supposetherightdirectrixoftheellipsetangentto
thecirclewithcenteratIandpassesthroughbothF1andF2andthat

IF1→·IF→2= a
4

2a2_-2_b2.

Findtheeccentricityoftheellipse.

19.Letf0(x)= 1_1-_xandf1(x)=x_{x .}-1

Definefn+2(x)=fn+1(fn(x)),n=0,1,2,….

Expressf2011(2011)infraction_qpofmostreducedterm.Findp+q.

20.Ifpositivenumbersx,y,andzsatisfyx+y+z=1and

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Team RoundAnswers

1.

{

x0<x<1_{3 or}2_3≤x<1

}

.

2.3_{2 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_.

16.2cos_{a .}α

17.2 3₇ .

18.₂2.

19.4021.

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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₎₌₃₀_x3_.

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_{+1wheretheline}_ax₊_by

-2=0passesthroughB.

3B.LetT=TNYWR.

Ifthefunctionf(x)= T+2tx-x_{takesonlargestvalue}M wheretandM_arepositive

naturalnumbers,findM .

RelayRoundAnswers

FirstRound
1A.₂₈_.
1B.3994.

SecondRound
2A.₃_.

2B.10

3,4

ỉ
è

ư
ø.

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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_=4andlet_AB_and_CD_{betwochordsofthecircle}

passingthroughP.IfthetangentstoAandBintersectatMandthetangentstoCandD
in-tersectatN,findtheequationofthelineMN .

SecondRound

5.Supposearectangularboxhasintegeredgelengthsanditsmaindiagonalis25.Whatisthe
smallestfaceintermsofareaamongallsuchpossibleboxes?

6.Ifrealnumbersx,y,andzsatisfytheequation

x2₊₂_y2₊₅_z2₊₂_xy₊₄_yz_-2_x₊₂_y₊₂_z_+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,andc_{betheoppositesidesof∠}A,∠B,and∠C,respectively.
If∠C=3∠B,comparethesizerelationshipbetweencand3b.

11.Supposethethreerealrootsfortheequationx3_-(4+_d)_x2₊₅_dx_-_d2_=0,where_d_isa

naturalnumber,representthesquareofsomerighttriangle'sthreesides.Findd.

12.SupposeF1andF2arebothfocifortheellipsex
2
m +

y2

n =1andfociforthehyperbola
x2

p
-y2

q=1(m,n,p,q∈R+).IfMisanintersectionoftheellipseandthehyperbolaand

|MF1→|·|MF2→|=1.

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FourthRound

13.Ifx=sin35°cos65°-cos65°cos5°-cos55°cos5°,whatisthevalueforx?

14.A-BCDisaregulartetrahedronwithedgelengthof24.O
isasphereinscribedinsidethetet-rahedron.O1_{isasmallspherethatistangenttotheupperthreesidesofthetetrahedronand}

thelargesphere.Whatisthevolumeofthatsmallsphere? (Expressyouranswerinterms
ofπ)

FifthRound

15.Findtheintervalsforxwherethefunctiony=

|

log2|x+1|

|

ismonotonicallydecreasing.

16.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].

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