Givenapyramid P ABCD wherebase ABCD isarightangletrapezoid,∠ A =90°,.. AB ∥ CD ,and PD ⊥plane ABCD..[r]
(1)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}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)
(2)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]
,findthevaluefort18.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
(3)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
(4)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
(5)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+