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Givenapyramid P ABCD wherebase ABCD isarightangletrapezoid,∠ A =90°,.. AB ∥ CD ,and PD ⊥plane ABCD..[r]

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2014WorldMathematicsTeamChampionship AdvancedLevel

Team Round·Problems

1.Definefunction

F(x,y)=|x-1|+|x-2|+|x-3|+|x-4|+|y-1|+|y-2|+|y-3|

FindtheminimumvalueforF(x,y)

2.SupposeA={(x,y)|y≥2x2}andB={(x,y)|x2+(y-a)2≤5}.IfA∩B=B,findtherange

ofvaluesfora

3.Supposepositiveintegersaandbarerelativelyprimesandwhenbisdividedbya,4and7are theirremainderandquotient,respectively.Leta1,a2,a3,a4,… beallthenumbersa(in

ascendingorder)thatsatisfytheaboveconditions,finda2014

4.Iftherangeofvaluesforthefunctionf(x)=log10

2x2-(a+2)x+a2+4

é ë

ù

ûisallreal

numbers,findthedomainoff(x)

5.Considerafunctionf(x)onrealnumbersRandsatisfiesthefollowingconditions: (a)f(2+x)=f(2-x),

(b)f(4-x)=-f(4+x),and (c)f(x)=x2when0≤x≤2.

Findthevalueforf(2015)

6.Solvetheequation273x2+2y+273y2+2z+273z2+2x=1 (x,y,z∈R)for(x,y,z).

7.Supposef(x)=|x3-x|-|x3+x|.Iftheequationf2(x)+2|f(x)|+n-1=0(n∈R)has

exactly3distinctrealroots,findthevalueforn

8.Findallpossiblepositiveintegersolutionsxandyfor2 x+y- x- y=3

9.Let{an}beageometric(equalproportion)sequencewithan>0.Supposea4a2n-4=4n(n≥3)

andletSnbethesumofthefirstntermsofthesequence{log2a2n-1}

Findthelargestposi-tiveintegernthatsatisfiesS2n-1≤2015

10.LetP ABCbeatetrahedronthatisinscribedinsidesphereO.IfAC=BC=6,∠ACB= 90°,andPB=12isthediameterofsphereO,findthevolumeofP ABC

11.Suppose,foranypositiveintegersmandn,functionfsatisfiesthefollowingconditions: (a)f(1,1)=2,

(b)f(m,n+1)=f(m,n)+(-1)n·2,

(c)f(m+1,1)=(-1)m·2f(m,1).

Findthevalueforf(2015,2016)

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13.AsshownintheFig.1,allthecentersofsemi circlesO1,O2,O3,…,andOnareonACand

Fig.1

eachsemi circletangentstoitsneighborsatpointsB1,B2,B3,

… ,andBn.SupposeCM1isalsotangenttothesesemi circles

withpointsoftangencyatM1,M2,M3,…,andMn.IfAB1=2

and∠M1O1C=θ,findthevalueofO1B1+O2B2+O3B3+…+ OnBn(intermsofθ)whenn→+∞

14.If2≤x2y≤4and -2≤3y2

x ≤-1wherex,y∈R

,findthesumofthemaximumandmini-mumvaluesofyx47

15.Let △ABC1and △ABC2beisoscelesrighttrianglesboth withequalsidelengthof1.If

foldingalongABtomakethetwohalfplanesinformingadihedralangleof60°,whatisthe maximumpossiblelengthforC1C2?

16.Supposethethreedifferentedgelengthsofarectangularboxarem,n,and1.Ifmandn

satisfy3m+2n+6mn=9m2+4n2

+1,findthelengthofthisrectangularbox'smaindiago-nal

17.Supposef(x)=2cos2x-2 3sinxcosx.Iftherangeofvaluesoff(x)is [0,1]when

x∈[12π,t],findthevaluefort

18.Supposethesequence{an}satisfiesa1=0andan+1=an+(n+1)·2n-1.Findthemaximum

valueamongallCn=a n

3n-1

19.GivenapyramidP ABCD wherebaseABCDisarightangletrapezoid,∠A=90°,

AB∥CD,andPD⊥planeABCD.IfPD=CD=2andAB=AD=1,findthevolumeofthe spherethatisdeterminedbythe4pointsP,B,C,andD

20.Givenpositiveintegersa,b,andcsuchthata<canda+c=2b.Findthenumberof digitnumbersabcthatsatisfytheseconditions

Team RoundAnswers

1.6

2.a≥1018

3.28221

4.{x|x≠4}

5.-1

6.(x,y,z)= -1æè 3,-13,-13öø

7.1

8.(x,y)=(16,9)or(9,16)

9.22

10.36

11.-22015-2.

12.[32,36]

13.1+cos2cosθ θ

14.-14411

15.2

16.76

17.π6

18.89

19.8 23π

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RelayRound·Problems

FirstRound

1A.Findthenumberof3 digitnumbersabc(a,b,andcaredistinct)suchthat|a-c|=5 1B.LetT=TNYWR (TheNumberYou WillReceive)andS=T

3.Ifx+y+z=S,findthe maximumvalueforxyz+3(xy+yz+zx)

SecondRound

2A.Supposeabcdefabcisa9 digitnumberwithdef=2abc.Ifabcdefabcisaproductofthe

squaresof4distinctprimenumbers,findthesumofallpossible3 digitnumbersabc

2B.LetT=TNYWR (TheNumberYou WillReceive).SupposecistheunitsdigitforT.How

many5 digitnumbersintheformof12abccanhave3asremainderwhenitisdividedby7? ThirdRound

3A.Givenasequence{an}wherean=n(n!).LetSnbethesumofthefirstntermsof{an}.Find

thesmallestnumbernsothatSn≥2014

(Note:n! =n×(n-1)×(n-2)×…×2×1)

3B.LetT= TNYWR (The Number You WillReceive).Findthenumberofnegativevalue

solutionstotheequation -x5+x4-3x3+5x2-2x+T=0.

RelayRoundAnswers

FirstRound

1A.72 1B.1088

SecondRound

2A.650 2B.14

ThirdRound

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IndividualRound·Problems

FirstRound

1.IfABCD A′B′C′D′isarectangularsolid,how manytetrahedronscanbeformedusingthe

centerofthisrectangularsolidasvertexandthreepointsfromverticesA,B,D,B′,C′,andD′

toformthebase?

2.Ifx5+5x4+10x3+10x2-5x+1=0(x≠-1),findthevaluefor(x+1)4.

3.Supposex∈Rand[x]representsthelargestintegernotlargerthanx.Ifthefunctionf(x)= [x]

x -a(x>0)hasexactlythreezeros,findtherangeofpossiblevaluesfora

4.LetA(2,2)beapointonthexy coordinatesystem,Bapointontheliney=x+1andCa pointonthex axis.Findtheminimumperimeterforallsuchpossibletriangles△ABC SecondRound

5.Suppose,exceptforPA,thelengthofalledgesintetrahedronP ABChasalengthof1.Find therangeofpossiblevaluesforthelengthofPA?

6.Giventhatf(log10(log310))=5forfunctionf(x)=ax

3+bsinx

x2+c (a,b,c∈R).Findthevalue

forf(log10(log103))

7.SupposesetsAandBaredefinedtobeA={x,xy,log10(xy)}andB={0,|x|,y}.IfA=B,

findỉèx+y1ưø+x2+1 y2 æ è

ö ø+x

3+1 y3 æ è

ư

ø+…+x

2014+ y2014 ỉ

è

ö ø

8.Randomlypicktwodiagonals(includingbothfacediagonalsandbodydiagonals)fromacube Whatistheprobabilitythatthesetwodiagonalsareperpendicular?

ThirdRound

9.Givenx5+5sinx+2m=0and16y5+5sinycosy-m=0wherem∈Randx,y∈ -π

6,

π

6

æ è

ö ø

Findthevalueforcos(x+2y)

10.GivenarectangularboxABCD A1B1C1D1withthebaseABCDbeingasquareandthatE

andFarepointsonedgesBB1andDD1,respectively,sothatAB=DF=B1E=2andBE=

1.FindthevolumeofthepyramidD1 AEC1F

11.LetA,B,andCbetheinterioranglesoftriangle△ABCthatareoppositetothesidesa,b,

andc,respectively.Iftheareaof△ABCisS=12bccosA=2 2anda=2 5-2 2,findthe valueforb+c

12.Supposexi(i=1,2,3,4,5)arenon negativerealnumbersandx1+x2+x3+x4+x5=1

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FourthRound

Fig.1 13.CircleC,x axis,y axis,andthecurvey=x3(x>0)aretangentto

eachotherasshownintheFig.1.FindtheradiusofcircleC

14.Findtheareaoftheregionthatisboundedbythecurve |x-2|-|y+1|=|2x-7|

FifthRound

15.AsshownintheFig.2,A1,A2,A3,andA4arepointsonthex axisand B1,B2,B3,andB4arepointsonthecurvey2=kx(k>0).Suppose

Fig.2

pointsC1,C2,andC3arepointsonA2B2,A3B3,andA4B4,

respectively,so that A1B1C1A2, A2B2C2A3, and A3B3C3A4areallsquareswithareas,respectively,S1,S2,

andS3.IfOA1=1andS2=2S1,findS3

(Theresultcan-notcontaink)

16.Considerthesequence{an}witha1=14.Denotethesumof

itsfirstntermsasSn.Ifanisthearithmeticaverageof Snand Sn-1foralln≥2,findthe

valuefora2014

IndividualRoundAnswers

FirstRound

1.8

2.10

3.æ34,45

è ù û

4.26 SecondRound

5.(0,3)

6.-5

7.0

8.103 ThirdRound

9.1

10.43

11.6

12.13 FourthRound

13.2 3-

14.3 FifthRound

15.2+

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