Hypatia滑铁卢数学竞赛(Grade 11).docx

Hypatia滑铁卢数学竞赛(Grade11)2020HypatiaContest(Grade11)Wednesday,April16,20201.Fornumbersaandb,thenotationa?bmeans2a+b2+ab.Forexample,1?2=2(1)+22+(1)(2)=8.(a)Determinethevalueof3?2.(b)Ifx?(?1)=8,determinethevalueofx.(c)If4?y=20,determinethetwopossiblevaluesofy.(d)If(w?2)?w=14,determineallpossiblevaluesofw.2.(a)DeterminetheequationofthelinethroughthepointsA(7,8)andB(9,0).(b)DeterminethecoordinatesofP,thepointofintersectionoftheliney=2x?10andthelinethroughAandB.(c)IsPclosertoAortoB?Explainhowyouobtainedyouranswer.3.Inthediagram,ABCDisatrapezoidwithADparalleltoBCandBCperpendiculartoAB.Also,AD=6,AB=20,andBC=30.(a)DeterminetheareaoftrapezoidABCD.(b)ThereisapointKonABsuchthattheareaofKBCequalstheareaofquadrilateralKADC.DeterminethelengthofBK.(c)ThereisapointMonDCsuchthattheareaofMBCequalstheareaofquadrilateralMBAD.DeterminethelengthofMC.C4.Thepeizi-sumofasequencea1,a2,a3,.,anisformedbyaddingtheproductsofallofthepairsofdistincttermsinthesequence.Forexample,thepeizi-sumofthesequencea1,a2,a3,a4isa1a2+a1a3+a1a4+a2a3+a2a4+a3a4.(a)Thepeizi-sumofthesequence2,3,x,2xis?7.Determinethepossiblevaluesofx.(b)Asequencehas100terms.Oftheseterms,mareequalto1andnareequalto?1.Therestofthetermsareequalto2.Determine,intermsofmandn,thenumberofpairsofdistincttermsthathaveaproductof1.(c)Asequencehas100terms,witheachtermequaltoeither2or?1.Determine,withjusti?cation,theminimumpossiblepeizi-sumofthesequence.