trường thcs hoàng xuân hãn

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2014

WorldMathematicsTeamChampionship

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_}and_B_={(_x_,_y_)|_x2₊₍_y_-_a₎2_≤5}_._If_A_∩_B₌_B_,_findtherange

ofvaluesfora.

3.Supposepositiveintegersaandb_{arerelativelyprimesandwhen}b_isdividedbya,4and7are
theirremainderandquotient,respectively.Leta1,a2,a3,a4,… beallthenumbersa(in

ascendingorder)thatsatisfytheaboveconditions,finda2014.

4.Iftherangeofvaluesforthefunctionf(x)=log10 1

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)=x2_when0≤_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_|_._{Iftheequation}_f2₍_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 ABC_{beatetrahedronthatisinscribedinsidesphere}O.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_·2_f₍_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≤x2_y_{≤4and -2≤}3y2

x ≤-1wherex,y∈R

,findthesumofthemaximumandmini-mumvaluesofy_x47.

15.Let △ABC1and △ABC2beisoscelesrighttrianglesboth withequalsidelengthof1.If

foldingalongABtomakethetwohalfplanesinformingadihedralangleof60°,whatisthe
maximumpossiblelengthforC1C2?

16.Supposethethreedifferentedgelengthsofarectangularboxarem,n,and1.Ifmandn

satisfy3m+2n+6mn=9m2₊₄_n2

+1,findthelengthofthisrectangularbox'smaindiago-nal.

17.Supposef(x)=2cos2_x_{-2 3sin}_x_cos_x._{Iftherangeofvaluesof}_f₍_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
3 digitnumbersabcthatsatisfytheseconditions.

Team RoundAnswers

1.6.

2.a≥101₈.

3.28221.

4.{x|x≠4}.

5.-1.

6.(x,y,z)= -1ỉ_è ₃,-1₃,-1₃ư_ø.

7.1.

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

9.22.

10.36 2.

11.-22015_-2_.

12.[32,36].

13.1+cos_2cos_{θ .}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._Let_T_{=TNYWR (TheNumberYou WillReceive)and}_S₌T

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

SecondRound

2A._Supposeabcdefabc_{isa9 digitnumberwith}def₌₂abc._Ifabcdefabc_{isaproductofthe}

squaresof4distinctprimenumbers,findthesumofallpossible3 digitnumbersabc.

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

many5 digitnumbersintheformof12abc_{canhave3asremainderwhenitisdividedby7?}
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_-3_x3₊₅_x2_-2_x₊_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₊₅_x4₊₁₀_x3₊₁₀_x2_-5_x₊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₊_b_sin_x

x2₊_c (a,b,c∈R).Findthevalue

forf(log10(log103)).

7.SupposesetsAandBaredefinedtobeA={x,xy,_log10(xy)}andB={0,|x|,y}.IfA=B,

findỉ_èx+_y1ư_ø+x2₊1
y2
ỉ
è

ư

ø+x

3₊1
y3
ỉ
è

ư

ø+…+x

2014₊ 1
y2014
ỉ

è

ư
ø.

8.Randomlypicktwodiagonals(includingbothfacediagonalsandbodydiagonals)fromacube.
Whatistheprobabilitythatthesetwodiagonalsareperpendicular?

ThirdRound

9.Givenx5_+5sin_x₊₂_m_=0and16_y5_+5sin_y_cos_y_-_m_=0where_m_∈_R_and_x,_y_{∈ -}π

6,

π

6

ỉ
è

ư
ø.

Findthevalueforcos(x+2y).

10.GivenarectangularboxABCD A1B1C1D1withthebaseABCDbeingasquareandthatE

andF_{arepointsonedges}BB1andDD1,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- 6.

14.3.
FifthRound

15.2+ 2.

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