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2015

WMTCAdvancedLevel

Team RoundProblems

1.Considerasequence{an}inwhicha1=3andan+1=1+a_1-an

n(nispositivenaturalnumbers).

Finda2016.

2.Wecalla3􀆼digitnumberabcan"arithmeticsequencenumber"ifa +c=2b.How many
"arithmeticsequencenumbers"arethere?

3.Letxbeanon􀆼zeronaturalnumberandletx2_-_4,2_x,and_{x +1betheedgelengthsofa}

triangle.Findallthepossibleperimetersofthistriangle.

4.LetMbeanon􀆼emptysetofintegers.k∈Miscalledan"isolated"elementinMifbothk
-1∉Mandk+1∉M ._SupposeM ={1,2,3,4,5}.How manysubsetsofM_containonly
one"isolated"element?

5.Findthemaximumvalueforfunctionf(x)=2 3sin2x +4sinx +8 3cosx.

6.Supposeasetcontainsnintegerswithhalfevennumbersandtheotherhalfoddnumbers.If
Pistheprobabilitywhenthesumofanytworandomlyselectednumbersfromthissetis
even,findthelargestpositiveintegernsothatP≤ 99₂₀₀.

7.Findtheminimumvalueforfunctiony=3x2_-7_{x +2}_{-2 2}_x2_-3_{x -1.}

8.Supposeαisarealrootofx3_+x-₄₌_0and_β_{isarealrootof}_x+3_{x -}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

primenumbersthatareintheformof8₃a_bn

n.

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11.Letaandbberealnumbersandb≠1.Ifthethreetermsa2_b_,_absin2x_,and_{aform bothan}

arithmetic(equaldifference)sequenceandageometric (equalproportion)sequence,find
thevalueofcos

(

4x -π₃

)

.

12If.a>3b>0,findthemaximumvaluefor3a+8₈_ba22_-4b-16_abab2.

13.Supposethestraightlinex=m (1≤m≤3)intersectsfunctionsf(x)=log2(x+1)+1and

g(x)=4-2x-1_atpoints_M_andN_{,respectively.Findtherangeofvaluesof}_{|MN |.}

14.Givenfunctiong(x)=||x -1|-1|(x≥0).Iffunctionf(x)=-sinπ₂x(x≤0)has
exactlynpointsthatarethesymmetricimagesofpointsfromg(x)reflectoverthey􀆼axis,

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 2_{3 .Findtheradiusoftheinscribedcircleof△}ABC.

17.Giventhata,b,andcarepositivenaturalnumberswhereaisprime,3b-1iscomposite,
bothband2ab_{areperfectsquares,and8}a +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.LetV􀆼ABC 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.
1

12.-4 3.

13.[0,3].

14.3.

15.100.

17.108.

18.3- 5.

19.4 3.

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RelayRoundProblems

1A._{Find 2016}! +2013!

2015! +2014!
é

ë

êê ù_ûúú 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._{Findthesumofrealrootsofequation2}_x3_-10_x2₊₇_{x +10=}₀_.

3B.LetT =TR.GivenarectanglewithintegervaluelengthsasitsdimensionsandareaofS =

T2_-₁_._{Findthelengthoftheshortestpossiblediagonalofthisrectangle}_.

RelayRoundAnswers

1A.₂₀₁₅_.
1B.6041.

2A.₃_.
2B.48.

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IndividualRoundProblems

Fig.1

1.Useacertainnumberofidenticalcubesofedgelength1tocomposeageometric
solid.Amongallthesolidswithfrontandleftsideviewslooklikethefigure1,

whatisthevolumeofthesolidthathasthesmallestvolume?

2.Computelog2(2log73×22log79)2log37.

3.Findtheminimumvalueofthefunctiony=4cos2_{x -4cos}_{x -6}_.

4.SupposeMisanaturalnumberand0<M <a<M +1where2a_+a3₌₄₇_._{FindM .}

5.Iftanα= 2,findsin2α-cos2α.

6.SupposeTisa9􀆼digitnaturalnumber,nisapositivenaturalnumber,and3T +1=10n_.

FindthesumofdigitsofT.

7.If 3sin20°=cos40°+sinxand0°≤x<360°,findx.

8.Suppose,in △ABC,cosA =1_{3 and}BC = 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.Suppose4raysweredrawnfromapointin3􀆼spacesothatalltheanglesbetweenanytwo
raysarethesame.Findcos(α)whereαisthatcommonanglemeasurement.

Fig.2

13.Thefigure2consistsofnine1×1squares.Themiddlesquareisdividedbya
lineasindicatedinthefigure.IfapointMisrandomlyplacedinsidethat
middlesquare,whatistheprobabilityofthispointMislandedintheshaded
region?

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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_._Let_f-1₍_{x)betheinversefunctionof}_f₍_x)_._{Findthevalueof}

f-1(3)_.

19.GivenasphereOthathasdiameterPC=2.LetAandBbetwopointsonthisspheresuch
thatAB = 2 and ∠APC =∠BPC =45°.Findthevolumeofthetriangularpyramid

P􀆼ABC.

20.Giventhatan= 5×6+ 7×8+…+ (2n+3)(2n+4)andbn=an

n
é
ë

êê ù_ûúú 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°.

8.3_{2 3}.

9.14_{3 ≤}a10≤15.

10.15.

11.1.

12.-1₃.

13.11₁₂.

14.9.

15.1₃.

16.2.

17.2-|x -4|.

18ln2. .

19.1₃.

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