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1、Hypatia滑铁卢数学竞赛(Grade11)2005HypatiaContest(Grade11)Wednesday,April20,20051.Fornumbersaandb,thenotationabmeansa2?4b.Forexample,53=52?4(3)=13.(a)Evaluate23.(b)Findallvaluesofksuchthatk2=2k.(c)Thenumbersxandyaresuchthat3x=yand2y=8x.Determinethevaluesofxandy.2.GwenandChrisareplayingagame.Theybeginwithapi
2、leoftoothpicks,andusethefollowingrules:?Thetwoplayersalternateturns?Onanyturn,theplayercanremove1,2,3,4,or5toothpicksfromthepile?Thesamenumberoftoothpickscannotberemovedontwodi?erentturns?Thelastpersonwhoisabletoplaywins,regardlessofwhetherthereareanytoothpicksremaininginthepileForexample,ifthegameb
3、eginswith8toothpicks,thefollowingmovescouldoccur:Gwenremoves1toothpick,leaving7inthepileChrisremoves4toothpicks,leaving3inthepileGwenremoves2toothpicks,leaving1inthepileGwenisnowthewinner,sinceChriscannotremove1toothpick.(Gwenalreadyremoved1toothpickononeofherturns,andthethirdrulesaysthat1toothpickc
4、annotberemovedonanotherturn.)(a)Supposethegamebeginswith11toothpicks.Gwenbeginsbyremoving3toothpicks.Chrisfollowsandremoves1.ThenGwenremoves4toothpicks.ExplainhowChriscanwinthegame.(b)Supposethegamebeginswith10toothpicks.Gwenbeginsbyremoving5toothpicks.ExplainwhyGwencanalwayswin,regardlessofwhatChri
5、sremovesonhisturn.(c)Supposethegamebeginswith9toothpicks.Gwenbeginsbyremoving2toothpicks.ExplainhowGwencanalwayswin,regardlessofhowChrisplays.2005HypatiaContestPage23.Inthediagram,ABCisequilateralwithsidelength4.PointsP,QandRarechosenonsidesAB,BCandCA,respectively,suchthatAP=BQ=CR=1.BQRCAP(a)Determi
6、netheexactareaofABC.Explainhowyougotyouranswer.(b)DeterminetheexactareasofPBQandPQR.Explainhowyougotyouranswers.4.Anarrangementofasetisanorderingofallofthenumbersintheset,inwhicheachnumberappearsexactlyonce.Forexample,312and231aretwoofthepossiblearrangementsof1,2,3.(a)Determinethenumberoftriples(a,b
7、,c)wherea,bandcarethreedi?erentnumberschosenfrom1,2,3,4,5withac.Explainhowyougotyouranswer.(b)Howmanyarrangementsof1,2,3,4,5,6containthedigits254consecutivelyinthatorder?Explainhowyougotyouranswer.(c)Alocalpeakinanarrangementoccurswherethereisasequenceof3numbersinthearrangementforwhichthemiddlenumberisgreaterthanbothofitsneighbours.Forexample,thearrangement35241of1,2,3,4,5contains2localpeaks.Determine,withjusti?cation,theaveragenumberoflocalpeaksinall40320possiblearrangementsof1,2,3,4,5,6,7,8.