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1、Cayley滑铁卢数学竞赛(Grade10)CharteredAccountantsSybaseInc.(Waterloo)IBMCanadaLtd.CanadianInstituteofActuariesDonotopenthecontestbookletuntilyouaretoldtodoso.Youmayuserulers,compassesandpaperforroughwork.Calculatorsarepermitted,providingtheyarenon-programmableandwithoutgraphicdisplays.PartA:Eachquestionisw
2、orth5credits.1.Thevalueof03012.()+is(A)0.7(B)1(C)0.1(D)0.19(E)0.1092.Thepiechartshowsapercentagebreakdownof1000votesinastudentelection.HowmanyvotesdidSuereceive?(A)550(B)350(C)330(D)450(E)9353.Theexpressionaaa9153isequalto(A)a45(B)a8(C)a18(D)a14(E)a214.Theproductoftwopositiveintegerspandqis100.Whati
3、sthelargestpossiblevalueofpq+?(A)52(B)101(C)20(D)29(E)255.Inthediagram,ABCDisarectanglewithDC=12.IftheareaoftriangleBDCis30,whatistheperimeterofrectangleABCD?(A)34(B)44(C)30(D)29(E)606.Ifx=2isasolutionoftheequationqx311=,thevalueofqis(A)4(B)7(C)14(D)7(E)47.Inthediagram,ABisparalleltoCD.Whatisthevalu
4、eofy?(A)75(B)40(C)35(D)55(E)508.Theverticesofatrianglehavecoordinates11,(),71,()and53,().Whatistheareaofthistriangle?(A)12(B)8(C)6(D)7(E)99.Thenumberinanunshadedsquareisobtainedbyaddingthenumbersconnectedtoitfromtherowabove.(The11isonesuchnumber.)Thevalueofxmustbe(A)4(B)6(C)9(D)15(E)10Scoring:Therei
5、snopenaltyforanincorrectanswer.Eachunansweredquestionisworth2credits,toamaximumof20credits.ABCDDACB10.Thesumofthedigitsofafive-digitpositiveintegeris2.(Afive-digitintegercannotstartwithzero.)Thenumberofsuchintegersis(A)1(B)2(C)3(D)4(E)5PartB:Eachquestionisworth6credits.11.Ifxyz+=25,xy+=19andyz+=18,t
6、henyequals(A)13(B)17(C)12(D)6(E)612.AregularpentagonwithcentreCisshown.Thevalueofxis (A)144(B)150(C)120(D)108(E)7213.Ifthesurfaceareaofacubeis54,whatisitsvolume? (A)36(B)9(C)8138(D)27(E)162614.Thenumberofsolutionsxy,()oftheequation3100xy+=,wherexandyarepositiveintegers,is(A)33(B)35(C)100(D)101(E)971
7、5.Ify55=and28x=,thenxy+equals(A)13(B)28(C)3316.RectangleABCDhaslength9andwidth5.Diagonalisdividedinto5equalpartsatW,X,Y,andZareaoftheshadedregion.(A)36(B)365(C)18(D)41065(E)2106517.IfNpq=()()()+75243isaperfectcube,wherepandqarepositiveintegers,thesmallestpossiblevalueofpq+is(A)5(B)2(C)8(D)6(E)1218.Q
8、isthepointofintersectionofthediagonalsofonefaceofacubewhoseedgeshavelength2units.ThelengthofQRis(A)2(B)8(C)5(D)12(E)619.Mr.Andersonhasmorethan25studentsinhisclass.Hehasmorethan2butfewerthan10boysandmorethan14butfewerthan23girlsinhisclass.Howmanydifferentclasssizeswouldsatisfytheseconditions?(A)5(B)6
9、(C)7(D)3(E)420.EachsideofsquareABCDis8.AcircleisdrawnthroughAandDsothatitistangenttoBC.Whatistheradiusofthiscircle?(A)4(B)5(C)6(D)42(E)5.25PartC:Eachquestionisworth8credits.21.WhenBettysubstitutesx=1intotheexpressionaxxc32+itsvalueis5.Whenshesubstitutesx=4theexpressionhasvalue52.Onevalueofxthatmakes
10、theexpressionequaltozerois(A)2(B)52(C)3(D)72(E)422.Awheelofradius8rollsalongthediameterofasemicircleofradius25untilitbumpsintothissemicircle.Whatisthelengthoftheportionofthediameterthatcannotbetouchedbythewheel?(A)8(B)12(C)15(D)17(E)2023.Therearefourunequal,positiveintegersa,b,c,andNsuchthatNabc=+53
11、5.ItisalsotruethatNabc=+454andNisbetween131and150.Whatisthevalueofabc+?(A)13(B)17(C)22(D)33(E)3624.Threerugshaveacombinedareaof2002m.Byoverlappingtherugstocoverafloorareaof1402m,theareawhichiscoveredbyexactlytwolayersofrugis242m.Whatareaofflooriscoveredbythreelayersofrug?(A)122m(B)182m(C)242m(D)362m(E)422m25.Onewaytopacka100by100squarewith10000circles,eachofdiameter1,istoputthemin100rowswith100circlesineachrow.Ifthecirclesarerepackedsothatthecentresofanythreetangentcirclesformanequilateraltriangle,whatisthemaximumnumberofadditionalcirclesthatcanbepacked?(A)647(B)1442(C)1343(D)1443(E)1344