imo shortlist problems 1959-2009 (new)

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1、INTHENAMEOFALLAHIMOShortListProblems1959–2009Collectedby:AmirHosseinParvardi(amparvardi)Problemsfrom:http://www.artofproblemsolving.com/Forum/resources.phpPublished:2010-10Email:amir_ahp_2009@yahoo.comContentsYearPageNumberofProblems19595∗6196077196196196211719631361964

2、15619651761966196319672759196838251969417119704812197150171972531219735517197458121975601519766212197764161978671719796926YearPageNumberofProblems1980722119817519198278201983812519848420198587221986902119879323198897311989993219901042819911123019921172119931202619941282

3、4199513328199613930199714626199815028199915627200016327200116928200217527200318027YearPageNumberofProblems200418830200519527200620130200720730200821426200922130∗ShortListedProblemsoftheyears1959to1966werethesame,soIjustaddedthoseproblemstotheyear1966andusedIMOproblemsfo

4、rtheyears1959–1965.ThanksOrlando(orl)forthissuggestion.IMO1959BrasovandBucharest,RomaniaDay121n+41Provethatthefractionisirreducibleforeverynaturalnumbern.14n+32Forwhatrealvaluesofxisqq√√x+2x−1+x−2x−1=Agiven√a)A=2;b)A=1;c)A=2,whereonlynon-negativerealnumbersareadmittedfo

5、rsquareroots?3Leta,b,cberealnumbers.Considerthequadraticequationincosxacosx2+bcosx+c=0.Usingthenumbersa,b,cformaquadraticequationincos2xwhoserootsarethesameasthoseoftheoriginalequation.Comparetheequationincosxandcos2xfora=4,b=2,c=−1.ThisfilewasdownloadedfromtheAoPSMathOl

6、ympiadResourcesPagePage1http://www.artofproblemsolving.com/IMO1959BrasovandBucharest,RomaniaDay24Constructarighttrianglewithgivenhypotenusecsuchthatthemediandrawntothehy-potenuseisthegeometricmeanofthetwolegsofthetriangle.5AnarbitrarypointMisselectedintheinteriorofthese

7、gmentAB.ThesquareAMCDandMBEFareconstructedonthesamesideofAB,withsegmentsAMandMBastheirrespectivebases.Thecirclescircumscribedaboutthesesquares,withcentersPandQ,intersectatMandalsoatanotherpointN.LetN0denotethepointofintersectionofthestraightlinesAFandBC.a)ProvethatNandN

8、0coincide;b)ProvethatthestraightlinesMNpassthroughafixedpointSindependentofthechoiceofM;c)Findthelocusofthemidp