101 algebra problems from the training of the USA IMO team.pdf

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1、ONEHUNDREDANDONEALGEBRAPROBLEMSFROMTHETRAININGOFTHEUSAIMOTEAMTituAndreescuandZumingFengContentsAcknowledgmentsiiiAbbreviationsandNotationsvPrefaceviiIntroductionix1IntroductoryProblems12AdvancedProblems93SolutionstoIntroductoryProblems194SolutionstoAdvancedProblems55Glossary11

2、3FurtherReading119iiAcknowledgmentsThankstoTiankaiLiuwhohelpedinproofreadingandpreparingsolu-tions.Manyproblemsareeitherinspiredbyorfixedfrommathematicalcontestsindifferentcountriesandfromthefollowingjournals:High-SchoolMathematics,ChinaRevistaMatematicˇaTimi¸soara,RomaniaKvant,

3、RussiaWedidourbesttocitealltheoriginalsourcesoftheproblemsintheso-lutionpart.Weexpressourdeepestappreciationtotheoriginalproposersoftheproblems.iiiAbbreviationsandNotationsAbbreviationsAHSMEAmericanHighSchoolMathematicsExaminationAIMEAmericanInvitationalMathematicsExaminationA

4、MC10AmericanMathematicsContest10AMC12AmericanMathematicsContest12,whichreplacesAHSMEARMLAmericanRegionalMathematicsLeagueIMOInternationalMathematicalOlympiadUSAMOUnitedStatesofAmericaMathematicalOlympiadMOSPMathematicalOlympiadSummerProgramPutnamTheWilliamLowellPutnamMathemati

5、calCompetitionSt.PetersburgSt.Petersburg(Leningrad)MathematicalOlympiadvviABBREVIATIONSandNOTATIONSNotationsforNumericalSetsandFieldsZthesetofintegersZnthesetofintegersmodulonNthesetofpositiveintegersN0thesetofnonnegativeintegersQthesetofrationalnumbersQ+thesetofpositiveration

6、alnumbersQ0thesetofnonnegativerationalnumbersQnthesetofn-tuplesofrationalnumbersRthesetofrealnumbersR+thesetofpositiverealnumbersR0thesetofnonnegativerealnumbersRnthesetofn-tuplesofrealnumbersCthesetofcomplexnumbersNotationsforSets,Logic,andGeometry()ifandonlyif=)impliesjAjthe

7、numberofelementsinsetAA½BAisapropersubsetofBAµBAisasubsetofBAnBAwithoutBABtheintersectionofsetsAandBA[BtheunionofsetsAandBa2AtheelementabelongstothesetAPrefaceThisbookcontainsonehundredhighlyselectedproblemsusedinthetrain-ingandtestingoftheUSAInternationalMathematicalOlympiad

8、(IMO)team.Itisnotacollectionofonehundredverydifficult,impenetra