Let V
(1)2015WMTCAdvancedLevel Team RoundProblems
1.Considerasequence{an}inwhicha1=3andan+1=1+a1-an
n(nispositivenaturalnumbers)
Finda2016
2.Wecalla3digitnumberabcan"arithmeticsequencenumber"ifa +c=2b.How many "arithmeticsequencenumbers"arethere?
3.Letxbeanonzeronaturalnumberandletx2-4,2x,andx +1betheedgelengthsofa
triangle.Findallthepossibleperimetersofthistriangle
4.LetMbeanonemptysetofintegers.k∈Miscalledan"isolated"elementinMifbothk -1∉Mandk+1∉M SupposeM ={1,2,3,4,5}.How manysubsetsofMcontainonly one"isolated"element?
5.Findthemaximumvalueforfunctionf(x)=2 3sin2x +4sinx +8 3cosx
6.Supposeasetcontainsnintegerswithhalfevennumbersandtheotherhalfoddnumbers.If Pistheprobabilitywhenthesumofanytworandomlyselectednumbersfromthissetis even,findthelargestpositiveintegernsothatP≤ 99200
7.Findtheminimumvalueforfunctiony=3x2-7x +2-2 2x2-3x -1.
8.Supposeαisarealrootofx3+x-4=0andβisarealrootofx+3x -4=0.Findthevalue
of(α+β)
9.Let{an}and{bn}betwoarithmetic(equaldifference)sequencesandletSnandTnbethe
correspondingsumsoftheirfirstnterms,respectively.If2Sn
3Tn=
4n+19
2n+2,findthenumberof
Fig.1
primenumbersthatareintheformof83abn
n
(2)11.Letaandbberealnumbersandb≠1.Ifthethreetermsa2b,absin2x,andaform bothan
arithmetic(equaldifference)sequenceandageometric (equalproportion)sequence,find thevalueofcos(4x -π3)
12If.a>3b>0,findthemaximumvaluefor3a+88ba22-4b-16abab2
13.Supposethestraightlinex=m (1≤m≤3)intersectsfunctionsf(x)=log2(x+1)+1and
g(x)=4-2x-1atpointsMandN,respectively.Findtherangeofvaluesof|MN |.
14.Givenfunctiong(x)=||x -1|-1|(x≥0).Iffunctionf(x)=-sinπ2x(x≤0)has exactlynpointsthatarethesymmetricimagesofpointsfromg(x)reflectovertheyaxis, findthevalueofn
15.Ifasequence{an}satisfiesan+1+(-1)nan=n(nispositivenaturalnumber)andthesum
ofitsfirstntermsisSn=2550,findn
16.Givenatriangle△ABC.SupposeAB =2,BC =4,andBD,theanglebisectorof∠ABC hasthelengthof4 23 Findtheradiusoftheinscribedcircleof△ABC
17.Giventhata,b,andcarepositivenaturalnumberswhereaisprime,3b-1iscomposite, bothband2abareperfectsquares,and8a +2b+3c≤ 45.Findthesum ofallpossible valuesofabc
18.Supposein △ABC,tanB =4tanCandb2-c2=12.Findtheareaofthistriangle'sinscribed
circlewhenitscircumscribedcircleisthesmallest
Fig.2
19.Asshowninfigure2,BisthemidpointofthetangentPAtocircleO, Aisthepointoftangency,CisapointonPA,linesegmentsPDE, BFE,andCGEareallsecantlinesofcircleO,and∠FAG =∠FPB IfPA =EC =6andED =5,findthelengthofEB
20.LetVABC beatriangularpyramidwithE,F,G,andH bethe midpointsofedgesAB,AC,VB,andVC,respectively.Ifαisthe anglebetweenplanesAGHandVEF,findsinα
Team RoundAnswers
1.12
2.45
3.25,15
4.13
7.-2
8.4
9.3
10.32
12.-4
13.[0,3]
14.3
15.100
17.108
18.3-
19.4
(3)RelayRoundProblems
1A.Find 2016! +2013!
2015! +2014! é
ë
êê ùûúú where[n]representsthelargestintegernotgreaterthann
1B.LetT=TR (ThenumberyouwillReceive).Leta,b,T,c,anddbefivedifferentpositive
integersthatarelistedinascendingorderfromsmallesttolargest.IftheiraverageisT,find thelargestpossiblevalueford
2A.Givenasequence{an}thatsatisfiesa1=7,a2=29,an+2=7an+1-10anwheren=0,1,2,
….Findtheunits(last)digitfortheterma2015
2B.LetT =TR.Supposethevolumeoftheexternalsphereofarectangularsolidis8π 2T
Findthemaximumpossiblesurfaceareaofthisrectangularsolid
3A.Findthesumofrealrootsofequation2x3-10x2+7x +10=0.
3B.LetT =TR.GivenarectanglewithintegervaluelengthsasitsdimensionsandareaofS =
T2-1.Findthelengthoftheshortestpossiblediagonalofthisrectangle. RelayRoundAnswers
1A.2015. 1B.6041
2A.3. 2B.48
(4)IndividualRoundProblems
Fig.1
1.Useacertainnumberofidenticalcubesofedgelength1tocomposeageometric solid.Amongallthesolidswithfrontandleftsideviewslooklikethefigure1,
whatisthevolumeofthesolidthathasthesmallestvolume?
2.Computelog2(2log73×22log79)2log37
3.Findtheminimumvalueofthefunctiony=4cos2x -4cosx -6.
4.SupposeMisanaturalnumberand0<M <a<M +1where2a+a3=47.FindM
5.Iftanα= 2,findsin2α-cos2α
6.SupposeTisa9digitnaturalnumber,nisapositivenaturalnumber,and3T +1=10n.
FindthesumofdigitsofT
7.If 3sin20°=cos40°+sinxand0°≤x<360°,findx
8.Suppose,in △ABC,cosA =13 andBC = 6.Findthelengthofthelongestchordofthis triangle'scircumscribedcircle
9.Givenanarithmetic (EqualDifference)sequence {an}withfirstterma1andcommon
differenced.If-1≤a2≤2and3≤a5≤5,findtherangeofvaluesofa10
10.Supposethelengthsofboththeedgesandmainbodydiagonalsofarectangularsolidare naturalnumbersandthisrectangularsolidhasavolumeof220.Findthelengthofitsmain bodydiagonal
11.How manyrealrootsdoestheequationx3-x2-1=0have?
12.Suppose4raysweredrawnfromapointin3spacesothatalltheanglesbetweenanytwo raysarethesame.Findcos(α)whereαisthatcommonanglemeasurement
Fig.2
13.Thefigure2consistsofnine1×1squares.Themiddlesquareisdividedbya lineasindicatedinthefigure.IfapointMisrandomlyplacedinsidethat middlesquare,whatistheprobabilityofthispointMislandedintheshaded region?
(5)15.Ifthesolutionsetforinequality 5-x2 >ax +5
3 is(-2,1),findthevalueofreal numbera
16.Giventhatf(x)=22x+x2+2+2-x1-x.Iff(t)=4,findthevalueoff(-t)
17.Giventhaty=f(x)isarealevenfunctionandf(x -2)=f(x +2).Also,f(x)=x +2 whenx∈ [-2,0].Findtheanalyticformulaforf(x)whenx∈ [2,6]
18.Giventhatf(lnx)=x2-1.Letf-1(x)betheinversefunctionoff(x).Findthevalueof
f-1(3).
19.GivenasphereOthathasdiameterPC=2.LetAandBbetwopointsonthisspheresuch thatAB = and ∠APC =∠BPC =45°.Findthevolumeofthetriangularpyramid
PABC
20.Giventhatan= 5×6+ 7×8+…+ (2n+3)(2n+4)andbn=an
n é ë
êê ùûúú wheren=1,2, 3,… ([x]representsthelargestintegerthatisnotlargerthanx).LetSn=b1+b2+b3+
…+bn.FindthesumofallnsuchthatSn≤5455 IndividualRoundAnswers
1.5
2.20
3.-7
4.3
5.2 23+1
6.27
7.190°or350°
8.32 3
9.143 ≤a10≤15
10.15
11.1
12.-13
13.1112
14.9
15.13
16.2
17.2-|x -4|
18ln2
19.13