2016年袋鼠数学竞赛理工学院JC

 Singapore Math Kangaroo Contest 2017 Junior College Contest Paper Name: School: INSTRUCTIONS: 1. Please DO NOT OPEN the contest booklet until the Proctor has given permission to start. 2. TIME : 1 hour and 30 minutes 3. There are 30 questions in this paper. 3 points, 4 points and 5 points will be awarded for each correct question in Section A, B and C respectively. Points are not deducted for unanswered questions. 1 point will be deducted for every wrong answer. 4. Shade your answers neatly in the answer entry sheet. 5. PROCTORING : No one may help any student in any way during the contest. 6. Calculators are not allowed. 7. Students are required to shade in your Name, Index number, Level and School in the Answer sheet provided. 8. MINIMUM TIME: Students must stay in the exam hall for at least 1 hour and 15 minutes. 9. Students must show detailed working and transfer their final answers to the answer entry sheet. 10. Any additional papers is not allowed for this contest. Sufficient space will be provided in the contest paper. 11. You must return this contest paper to the proctor. Rough Working

SingaporeMathKangarooContest2017–JC,Grade11/12

SectionA(Correct–3points|Unanswered–0points|Wrong–deduct1point)Question120×17

=

2+0+1+7(A)3.4

(B)17

(C)34

(D)201.7

(E)340

Question2

Tombuiltafigurineofhisbrotherusingtheratioof1:87.Iftheheightofthefigurineis2cm,whatisactualtheheightofhisbrother?

(A)1.74m(B)1.62m(C)1.86m(D)1.94m(E)1.70m

Question3

Inthefigurebelow,wecansee10islandsthatareconnectedby15bridges.Ifallbridgesareopen,whatisthesmallestnumberofbridgesthatmustbeclosedinordertostopthetrafficbetweenAandB?

AB(A)1(B)2(C)3(D)4(E)5

1

SingaporeMathKangarooContest2017–JC,Grade11/12

Question4

Itisgiventhat75%ofaisequalto40%ofb.Whichoptionbelowisalwaystrue?

(A)15a=8b(B)7a=8b(C)3a=2b(D)5a=12b(E)8a=15b

Question5

Fourofthefollowingfiveclippingsarepartofthesamegraphofthequadraticfunction.Whichclippingisnotpartofthisgraph?

(A)(B)

(C)(D)

(E)

Question6

GivenacirclewithcenterOwithdiametersABandCXsuchthatOB=BC.Whatfractionoftheareaofthecircleistheshaded?

(A)

25

(B)

13

(C)

27

(D)

38

(E)

411

2

SingaporeMathKangarooContest2017–JC,Grade11/12

Question7

Abarconsistsof2whiteand2greycubesgluedtogetherasshowninthepicturebelow.Whichfigurecanbebuiltfrom4suchbars?

(A)(B)(C)(D)(E)

Question8

Whichquadrantcontainsnopointsonthegraphofthelinearfunctionf(x)=−3.5x+7?

(A)I(D)IV

(B)II

(E)Allquadrantscontainpoints.

(C)III

3

SingaporeMathKangarooContest2017–JC,Grade11/12

Question9

Eachofthefollowingfiveboxesisfilledwithredandblueballsaslabeled.Benwantstotakeoneballoutoftheboxeswithoutlooking.Fromwhichboxshouldhetaketheballtohavethehighestprobabilityofgettingablueball?

(A)(D)

(B)(E)

(C)

Question10

Findthegraphwhichhasthemostnumberofcommonpointswiththegraphofthefunctionf(x)=x?

(A)g1(x)=x2(B)g2(x)=x3(C)g3(x)=x4(D)g4(x)=−x4

(E)g5(x)=−x

SectionB(Correct–4points|Unanswered–0points|Wrong–deduct1point)

Question11

ThreemutuallytangentcircleswithcentresA,B,Chavetheradii3,2and1,respectively.WhatistheareaofthetriangleABC?

(A)6

√(B)43√(C)32(D)9

√(E)264

SingaporeMathKangarooContest2017–JC,Grade11/12

Question12

Thepositivenumberpislessthan1,andthenumberqisgreaterthan1.Whichoneofthefollowingnumbersisthelargest?

(A)p·q(B)p+q(C)

pq

(D)p(E)q

Question13

TworightcylindersAandBhavethesamevolume.IftheradiusofthebaseofBis10%largerthanthebaseofA.HowmuchlargeristheheightofAthantheheightofB?

(A)5%(B)10%(C)11%(D)20%(E)21%

Question14

Eachfaceofthepolyhedronshownbelowiseithertriangleorasquare.Eachsquareissurroundedby4triangles,whileeachtriangleissurroundedby3squares.Ifthereare6squaresintotal,howmanytrianglesarethere?

(A)5(B)6(C)7(D)8(E)9

5

SingaporeMathKangarooContest2017–JC,Grade11/12

Question15

Wehavefourtetrahedraldice,perfectlybalanced,withtheirfacesnumbered2,0,1and7.Ifwerollallfourofthesedice,whatistheprobabilitythatwecanformthenumber2017usingexactlyoneofthethreevisiblenumbersfromeachdie?

(A)

1256

(B)

6364

(C)

81256

(D)

332

(E)

2932

Question16

Thepolynomial5x3+ax2+bx+24hasintegercoefficientsaandb.Whichofthefollowingiscertainlynotarootofthepolynomial?

(A)1(B)-1(C)3(D)5(E)6

6

SingaporeMathKangarooContest2017–JC,Grade11/12

Question17

Juliahas2017chips.1009ofthemareblackandtherestarewhite.Sheplacedtheminasquarepatternasshowninthefigurebelow.Howmanychipsofeachcolourareleftaftershehasformedthelargestpossiblesquareusingherchips?

(A)None

(C)40blackonesand41whiteones(E)40whiteonesand41blackones

(B)40ofeach(D)41ofeach

Question18

Twoconsecutivenumbersaresuchthatthesumsoftheirdigitsineachofthemaremultiplesof7.Howmanydigitsdoesthesmallernumberhave?

(A)3(B)4(C)5(D)6(E)7

7

SingaporeMathKangarooContest2017–JC,Grade11/12

Question19

InaconvexquadrilateralABCD,thediagonalsareperpendiculartoeachother.Thesideshavelengths|AB|=2017,|BC|=2018and|CD|=2019.WhatisthelengthofAD?

(A)2016(B)2018(C)

√

20202−4(D)

√

20182+2(E)2020

Question20

TaylorattemptstobeagoodlittleKangaroo,butlyingistoomuchfun.Therefore,everythirdstatementshesaysisaliewhiletherestaretrue.(Sometimesshestartswithalieandsometimeswithoneortwotruestatements.)

Taylorthinksofa2-digitnumberandtellsherfriendaboutit:”Oneofitsdigitsisa2.””Itislargerthan50.””Itisanevennumber.””Itislessthan30.”

”Itisdivisiblebythree.””Oneofitsdigitsisa7.”

WhatisthesumofthedigitsinTaylor’snumber?

(A)9(B)12(C)13(D)15(E)17

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SingaporeMathKangarooContest2017–JC,Grade11/12

SectionC(Correct–5points|Unanswered–0points|Wrong–deduct1point)

Question21

Whenyouremovethelastdigitinapositiveinteger,itisequalto1/14oftheoriginalnumber.Howmanysuchpositiveintegersarethere?

(A)0(B)1(C)2(D)3(E)4

Question22

Thepictureshowsaregularhexagonwitheachsidelengthsequalto1.Theflowerwasconstructedwithsectorsofcirclesofradius1andcentersontheverticesofthehexagon.Whatistheareaoftheflower?

(A)

π2

(B)

2π3

√

(C)23−π

(D)

π√+32

√

(E)2π−339

SingaporeMathKangarooContest2017–JC,Grade11/12

Question23

Considerthesequenceanwitha1=2017andan+1=

an−1

.Whatisthevalueofa2017?an

(A)−2017(B)

−12016

(C)

20162017

(D)1(E)2017

Question24

Consideraregulartetrahedron.Itsfourcornersarecutoffbyfourplanes,eachpassingthroughthemidpointsofthreeadjacentedgesasshowninthefigurebelow.Whatistheratioofthevolumeoftheresultingsolidtothevolumeoftheoriginaltetrahedron?

(A)

45

(B)

34

(C)

23

(D)

12

(E)

13

Question25

Thelengthsofthethreesidesofaright-angledtriangleaddupto18.Thesquaresofthelengthsofthesidesaddupto128.Whatistheareaofthetriangle?(A)18

(B)16

(C)12

(D)10

(E)9

10

SingaporeMathKangarooContest2017–JC,Grade11/12

Question26

Youaregiven5boxes,5blackand5whiteballs.Youchoosehowtoputtheballsintheboxes(eachboxhastocontainatleastoneball).Youropponentcomesanddrawsoneballfromoneboxofhischoiceandhewinsifhedrawsawhiteball.Otherwise,youwin.Howshouldyouarrangetheballsintheboxestohavethebestchancetowin?

(A)Youputonewhiteandoneblackballineachbox.

(B)Youarrangealltheblackballsinthreeboxes,andallthewhiteballsintwoboxes.(C)Youarrangealltheblackballsinfourboxes,andallthewhiteballsinonebox.(D)Youputoneblackballineverybox,andaddallthewhiteballsinonebox.(E)Youputonewhiteballineverybox,andaddalltheblackballsinonebox.

Question27

Nineintegersarewritteninthecellsofa3×3table.Thesumoftheninenumbersisequalto500.Itisknownthatthenumbersinanytwoneighboringcells(withacommonside)differby1.Whatisthenumberinthecentralcell?

(A)50(B)54(C)55(D)56(E)57

Question28

If|x|+x+y=5andx+|y|−y=10whatisthevalueofx+y?(A)1(E)5

(B)2

(C)3

(D)4

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SingaporeMathKangarooContest2017–JC,Grade11/12

Question29

Howmanythree-digitpositiveintegersABCaretheresuchthat(A+B)Cisathree-digitintegerandanintegerpowerof2?

(A)15(B)16(C)18(D)20(E)21

Question30

Eachofthe2017peoplelivingonanislandiseitheraliar(andalwayslies)oratruth-teller(andalwaystellsthetruth).Morethanonethousandofthemtakepartinabanquet,allsittingtogetherataroundtable.Eachofthemsays:”Ofthetwopeoplebesideme,oneisaliarandtheotheroneatruth-teller.”Howmanytruth-tellersarethereontheislandatmost?

(A)1683(B)668(C)670(D)1344(E)1343

12

Rough Working

Rough Working