2024年AIME1数学竞赛（英文）

第页2024年AIME1数学竞赛（英文）1、EverymorningAyagoesfora-kilometer-longwalkandstopsatacoffeeshopafterwards.Whenshewalksataconstantspeedofskilometersperhour,thewalktakesherhours,includingminutesspentinthecoffeeshop.Whenshewalksatkilometersperhour,thewalktakesherhoursandminutes,includingminutesspentinthecoffeeshop.SupposeAyawalksatkilometersperhour.Findthenumberofminutesthewalktakesher,includingtheminutesspentinthecoffeeshop.2、Thereexistrealnumbersand,bothgreaterthan,suchthat.Find.3、AliceandBobplaythefollowinggame.Astackoftokensliesbeforethem.TheplayerstaketurnswithAlicegoingfirst.Oneachturn,theplayerremoveseithertokenortokensfromthestack.Whoeverremovesthelasttokenwins.FindthenumberofpositiveintegerslessthanorequaltoforwhichthereexistsastrategyforBobthatguaranteesthatBobwillwinthegameregardlessofAlice'splay.4、Jenentersalotterybyselectingfourdistinctelementsof.Thenfourdistinctelementsofaredrawnatrandom.Jenwinsaprizeifatleasttwoofhernumbersaredrawn,andshewinsthegrandprizeifallfourofhernumbersaredrawn.TheprobabilitythatJenwinsthegrandprizegiventhatJenwinsaprizeis,whereandarerelativelyprimepositiveintegers.Find.5、Rectanglehasdimensionsand,andrectanglehasdimensionsand.Points,,,andlieonlineinthatorder,andandlieonoppositesidesofline,asshown.Points,,,andlieonacommoncircle.Find.6、Considerthepathsoflengththatfollowthelinesfromthelowerleftcornertotheupperrightcorneronangrid.Findthenumberofsuchpathsthatchangedirectionexactlyfourtimes,asintheexamplesshownbelow.7、Findthegreatestpossiblerealpartof,whereisacomplexnumberwith.Here.8、Eightcirclesofradiuscanbeplacedtangenttosideofsothatthecirclesaresequentiallytangenttoeachother,withthefirstcirclebeingtangenttoandthelastcirclebeingtangentto,asshown.Similarly,circlesofradiuscanbeplacedtangenttointhesamemanner.Theinradiusofcanbeexpressedas,whereandarerelativelyprimepositiveintegers.Find.9、Let,,,andbepointsonthehyperbolasuchthatABCDisarhombuswhosediagonalsintersectattheorigin.Findthegreatestrealnumberthatislessthanforallsuchrhombi.10、Lethavesidelengths,,and.Thetangentstothecircumcircleofatandintersectatpoint,andintersectsthecircumcircleat.Thelengthofisequalto,whereandarerelativelyprimepositiveintegers.Find.11、Eachvertexofaregularoctagonisindependentlycoloredeitherredorbluewithequalprobability.Theprobabilitythattheoctagoncanthenberotatedsothatalloftheblueverticesmovetopositionswheretherehadbeenredverticesis,whereandarerelativelyprimepositiveintegers.Find.12、Define.Findthenumberofintersectionsofthegraphsof.13、Letbetheleastprimenumberforwhichthereexistsanintegernsuchthatisdivisibleby.Findtheleastpositiveintegersuchthatisdivisibleby.14、Letbeatetrahedronsuchthat,,and.ThereexistsapointIinsidethetetrahedronsuchthatthedistancesfromItoeachofthefacesofthetetrahedronareallequal.Thisdistancecanbewrittenintheform,where,,andarepositiveintegers,andarerelativelyprime,andisnotdivisiblebythesquareofanyprime.Find.15、Letbethesetofrectangularboxeswithsurfaceareaandvolume.Letbetheradiusofthesmallestspherethatcancontaineachoftherectangularboxesthatareelementsof.Thevalueofcanbewrittenas,whereandarerelativ