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1、ProblemsusingCauchy-Schwarzinequality.Problem1.(JackGarfunkel)Fornon-negativenumbersa,b,c,provetheinequalitySolution.UsingtheCauchy-Schwarzinequality,wehaveThus,oneonlyneedstoproveThisisobviouslytruebecauseTheproofiscomplete.EqualityoccursifandonlyifProblem2.(VoQuocBaCan)Fornon-nega
2、tivenumbersa,b,cwithsumequalto1,provethatSolution.UsingtheCauchy-Schwarzinequality,wehaveThus,oneonlyneedstoprovethatWehaveWeproveWehaveThereforetheinequalityisequivalenttoLastinequalityisobviouslytruesotheresultfollows.Equalitydoesnothold.Problem3.(VoQuocBaCan)Fornon-negativenumber
3、sa,b,c,provethatSolution.UsingtheCauchySchwarzinequality,wehaveThus,oneonlyneedstoprovethatWithoutlossofgenerality,assume,theinequalityisequivalenttowhereUsingtheassumption,itiseasytoprovethat.Theresultfollows,equalityholdsifandonlyifProblem4.Forpositivenumbersa,b,c,provethatSolutio
4、n.Lemma.:UsingtheCauchySchwarzinequality,WehavetoproveWehaveThus,oneonlyneedstoproveWithoutlossofgenerality,normalizetheinequalitybyletThenwehaveTheinequalityisequivalenttoItisclearthatisanincreasingfunctioninr,weshouldhaveInequalityiscompletelyproved.EqualityoccursifandonlyifProble
5、m5.(PhanThanhNam)Forthenon-negativenumbersa,b,cwithsumequalto1.LetprovethatSolution.UsingtheCauchySchwarzinequality,Thus,oneonlyneedstoproveLet,thenwehaveTheinequalityisequivalenttoSo,wehaveEqualityoccursifandonlyiforanditspermutationsProblem6.(VoQuocBaCan)Forpositivenumbersa,b,c,pr
6、ovethatSolution.UsingtheCauchySchwarzinequality,wehaveSo,wehavetoproveWithoutlossofgenerality,assumethenwehaveSo,Theresultfollows.Equalityholdsifandonlyif
