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2012

WorldMathematicsTeamChampionship

AdvancedLevel

Team Round

·

Problems

1.GivenatetrahedronPABCinwhich ∠APB=∠BPC=∠CPA=90°and

PA=PB=PC=1.IfPABC′sinscribedspherehasaradiusofr,then1_risbetweenwhichtwo
consecutiveintegers?

2.If(5x+2y)25₊_x25₊₆_x+2_{y=0,findthevaluefor6}_x+2_y.

Fig.1

3.InFig.1,letABCDbeatetrahedronwithedgeAD=
2andalltheothered-geshavealengthof1.SupposeM andNare midpointsofABandCD,

respectively.FindtheminimumdistancemovingfromMtoN

overthesur-faceofABCD.

4.Letxbeanyrealnumberandlet[x]representsthegreatestintegerlessthan
orequaltox.Forexample,[2.3]=2and[-2.3]=-3.Findallpossiblereal
solutionsforequationé_ëx₂ù_û+éx₄

ë
ù
û+

x
6

é
ë

ù
û+

x
8
é
ë

ù
û+…+

x
2012
é
ë

ù
û=x.

5.Randomlypickfivenumbersfromtennumbers1,2,3,…,10.Arrangethese5numbersfrom
smalltolargeandlabelthema1,a2,a3,a4,anda5.Arrangetheremainingfivenumbersfrom

largetosmallandlabelthemb1,b2,b3,b4,andb5.Findthevaluefor

|a1-b1|+|a2-b2|+|a3-b3|+|a4-b4|+|a5-b5|.

6.Ifthesystemofequations

{

_xmx2₊₍2+_m-1)x+2m_x=0-1=0hasonlyonerealsolutionforx,findm.

7.How manyfunctionscanbedefinedsothatthefunction′sdomainis
A={ab|ab-a-2b-2=0,a,b∈Z}
anditsrangeisB=x|sinx-æ π₆

è
ư

ø=1,0<x<5π

{

}

?

8.Ifα,_β,γ ∈ 0,π_èỉ ₂ư_ø and sin2_α _{+ sin}2

β+ sin2γ = 1,find the maximum value for
sinα+sinβ+sinγ

cosα+cosβ+cosγ.

9.Ifintegermsatisfiestheequation 1+1ỉ_è _mư_øm+1= 1+ 1ỉ_è ₂₀₁₂ư_ø2012,findthevalueform.

10.Supposen_{isanaturalnumberlargerthan0.If}f(n)representsthenumberofrootsin[0,π]
forequationsinx=cos(nx),findf(1)+f(2)+f(3)+…+f(100).

11.Let{an}beasequencethatsatisfiesan+3-an+2+an+1-an=0.IfSnrepresentsthesumof
thefirstntermsof{an}andS2012=2012,finda1+a3.

12.Supposethestraightline2x-y-12=0andtheparabolay2₌₄_x_{intersectatpoints}_Aand_B._Let_C

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13.Ifrealnumbersxandysatisfy

x+3y-23≤0,
2x+y-11≥0,
x-2y+2≤0,
ì

ỵ

í findtherangeofvaluesofx+_xy+4

+1 .

14.Arrangethepositiverootsforequationx·cosx+sinx_{=0inascendingorder}a1,a2,…,an,
…._{Amongthefollowingconclusions,selectthecorrectstatements}.

① 0<an+1-an<π
2;
② _2<π an+1-an<π;
③2an+1>an+2+an;
④2an+1<an+2+an.

15.Findtheintervalswherethefunctiony= cos|π₂₊₂x| ismonotonicallydecreasing.

16.Supposethegraphofquadraticfunctionf(x)=-x2₊_bx+_c₍_Δ=_b2₊₄_{ac>0)intersects}

x-axisatAandB.IfpointP(x0,f(x0))wheref(x0)≠0isonthecurveandPAis

perpendiculartoPB,findf(x0).

Fig.2

17.LetA = (1,1).SupposeBandCaretwopointsontheparabolay=x2_such

thatAB_{isperpendicularto}BC._{Amongallpossible}BandC,findtheareaof
thesmallestcircumscribedcircleof△ABC.

18.Ifthesolutionsetfortheinequality x+1+2x≤bis [-1,3],findthe
solutionsetfortheinequality |x-1|-10 ≤b.

19.AsinFig.2,ABisthediameteroftheCircleO,OD⊥BCatFonBCand
intersectsthecircleatE.If∠AEC=∠ODB ,OA=5,andBC =8,findDE.

20.FindallclosedintervalsMsothatthefunctionf(x)=1₃x2₊2

3x-113 hasMasbothits
domainandrange.

Team RoundAnswers

1.4and5.

2.0.

3.1.

4.0,8,10.

5.25.

6.1_{2 or0}.

7.36.

8.₂2.

9.-2013.

10.5025.

11.2.

12.(1,2)or(4,-4).

13.é17₉,13₃
ë

ù
û.

14.② and ④.

15.kπ
2 +
π
4,
kπ
2 +
π
2
é

ë
ù

û,k∈Z.

16.1.

17.π.

18.[-17,-1]∪[3,19].

19.10₃.

20.[-4,-1]or
-4,1+3 5₂
é

ë

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RelayRound

·

Problems

FirstRound

1A.Iff(cosα)=cos(2012α),findthevaluefor_f_sin π

2012
ỉ

è

ư
ø.

1B.LetT = TNYWR (TheNumberYou WillReceive).

Letf(x)beanoddfunctiondefinedonRandf(x+2)= -f(x)forallrealx.Iff(3)=

2m-3

m+1 andf(1)>-T,findtherangeofvaluesform.

SecondRound

2A._Giventhatlogmx,lognx,andlog2xformanarithmetic(equaldifference)sequenceandx≠1.

Findthevalueforlogmn·log2(2m).

2B._{LetT= TNYWR (TheNumberYou WillReceive).}

Giventhat{an}isanarithmetic(equaldifference)sequence.IfSnisthesum ofthefirstn
termsandST=ST+11,finda8.

ThirdRound

3A.If_f(x)= 1

2a₊_x+₂a1_-_x-1andf(1)+f(-1)=2₃,finda.

3B.LetT = TNYWR (TheNumberYou WillReceive).

SupposethelengthofthesquareABCDisT.LetEbeapointthatmovesalongthecirclewith
AD_{asthediameter.Findtherangeofvaluesfor}EB+EC.

RelayRoundAnswers

FirstRound

1A._-1_.
1B.-1<m<2

3.

SecondRound

2A.₂_.
2B.₀_.

ThirdRound

3A.₁_.

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IndividualRound

·

Problems

FirstRound

1.IfM=cos2_x+_x_sin_x_where0<_x<π

2,findtheintegerportionofM .

2.Ifalltheneighboringintersectingpointsofstraightliney=1andcurvef(x)=sinωx+æ_è π₃ö_ø

haveafixeddistanceof4,findf(1)+f(2)+…+f(2013).

3.Givenageometric(equalproportion)sequence{an}withbothitsfirstterma1>0andcommon

ratioq>0.Letbn= an+ a2012-n wherenisanaturalnumberand0<n<2012.Ifbkisthe
smallesttermofthesequence{bn},findk.

4.Giventhatthestraightliney=m(m<0)andthecurvey=cosx_{intersectontherightsideof}
they-axis.Arrangethehorizontalcoordinatesoftheseinterceptionsinascendingorder
x1,x2,x3,….Ifx1,x2,andx3formageometric(equalproportion)sequence,findm.

SecondRound

5.Letf(x)=x3_-6_x2₊₁₃_x_-6and_f(_a)=7,_f(b)=1,find_a₊_b.

6.IfS=min

{

_|_x-1|3 , 1_|_x-9|

}

forrealnumbersx≠1or9,findSmax.

7.Findtherangeofvaluesofslopeksothatthestraightliney=
kx+1wouldintersecttheel-lipse3x2₊_y2_{=1attheFirstQuadrant(}_x_>0and_y>0)_.

8.Givenasequence{an}withanexplicitformulaan=(-1)n(2n-1)wheren
ispositiveinte-gers.LetSnbethesumofthissequence′sfirstnterms.WritetheexplicitformulaforS2k.

ThirdRound

9.Findtheacuteanglexsatisfyingtheequation(sin2x+cosx)(sinx-cosx)=2cos2_x.

10.Givenarighttrianglewithhypotenuseoflength4x-2.Ifoneoftheothersideshasalength
of415-3x,findthevaluerangeofthelengthofitsthirdside.

11.Givennon-negativenumbersa,b,c,x,y,andzwitha+b+c+x+y+z=1,and
abc+xyz=₃₆1,findthelargestpossiblevalueforabz+bcx+cay.

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FourthRound

13.LetObethevertexoftheparabolay2₌₂_px₍_p≠0)andlet_Aand_B_{betwonon-vertexpoints}

onthisparabola.IftheslopesofOAandOBarek1andk2,respectively,andk1andk2

areal-sotherootsofequationx2₊₄_x_{-2=0,findtheslopeofstraightline}_AB.

14.Given △ABC.Leta,b,andcbetheoppositesidestointerioranglesA,B,andC,
respectively.If_a1=_b1+_c1,findthemaximumpossiblevalueforsin(A).

FifthRound

15.Supposey=f(x)isanevenfunctionand[f(x1)-f(x2)](x1-x2)<0forany

x1,x2∈(-∞,0].Findtherangeofvaluesforxthatsatisfyf(x+1)<f(2x-1).

16.ConsideracubeABCD-A1B1C1D1ofedgelength1.LetO1bethespheretangentstoeachof

thiscube′s12edgesandletsphereO2bethespheretangentstoeachofthiscube′s6faces.

SupposeM1andM2representtheareaoftheregionsthatareintersectedbytheplaneD1AC

withspheresO1andO2respectively.FindM1-M2.

IndividualRoundAnswers

FirstRound

1.1.

2.1_{2 or-}1₂.

3.1006.

4.-1₂.

SecondRound

5.4.

6.1₂.

7.(- 3,0).

8.S2k=2k.

ThirdRound

9.π₃.

10.(0,4

3).

11.₁₀₈1.

12.- 15.

FourthRound

13.1₂.

14.₈15.

FifthRound

15.(-∞,0)∪(2,+∞).

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