trường thcs hoàng xuân hãn

Let V

2015WMTCAdvancedLevel Team RoundProblems

1.Considerasequence{an}inwhicha1=3andan+1=1+a1-an

n(nispositivenaturalnumbers)

Finda2016

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

3.Letxbeanon􀆼zeronaturalnumberandletx2-4,2x,andx +1betheedgelengthsofa

triangle.Findallthepossibleperimetersofthistriangle

4.LetMbeanon􀆼emptysetofintegers.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

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)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 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.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

12.-4

13.[0,3]

14.3

15.100

17.108

18.3-

19.4

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

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.SupposeTisa9􀆼digitnaturalnumber,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.Suppose4raysweredrawnfromapointin3􀆼spacesothatalltheanglesbetweenanytwo raysarethesame.Findcos(α)whereαisthatcommonanglemeasurement

Fig.2

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

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

P􀆼ABC

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