2024年AIME1数学竞赛（英文）.docVIP

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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.FindthenumberofpositiveintegerslessthanorequaltoforwhichthereexistsastrategyforBobthatguaranteesthatBobwillwinthegameregardlessofAlicesplay.

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、Eightcirclesofradiuscanbeplacedtangenttosideofsothatthec