2023-2024年数学竞赛AIME I II真题含答案（共4套）.pdf

2023-2024年数学竞赛AIMEIII真题含答案

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2023AIMEIIProblems

Problem1

Thenumbersofapplesgrowingoneachofsixappletreesformanarithmetic

sequencewherethegreatestnumberofapplesgrowingonanyofthesixtreesis

doubletheleastnumberofapplesgrowingonanyofthesixtrees.Thetotal

numberofapplesgrowingonallsixtreesisFindthegreatestnumberof

applesgrowingonanyofthesixtrees.

Problem2

Recallthatapalindromeisanumberthatreadsthesameforwardandbackward.

Findthegreatestintegerlessthanthatisapalindromebothwhenwrittenin

basetenandwhenwritteninbaseeight,suchas

Problem3

Let△beanisoscelestrianglewith∠=90°Thereexistsapoint

inside△suchthat∠=∠=∠and=10.Find

theareaof△

Problem4

Letandberealnumberssatisfyingthesystemofequations

LetbethesetofpossiblevaluesofFindthesumofthesquaresofthe

elementsof

Problem5

Letbethesetofallpositiverationalnumberssuchthatwhenthetwo

numbersandarewrittenasfractionsinlowestterms,thesumofthe

numeratoranddenominatorofonefractionisthesameasthesumofthe

numeratoranddenominatoroftheotherfraction.Thesumofalltheelements

ofcanbeexpressedintheformwhereandarerelativelyprimepositive

integers.Find+.

Problem6

ConsidertheL-shapedregionformedbythreeunitsquaresjoinedattheirsides,

asshownbelow.Twopointsandarechosenindependentlyanduniformlyat

randomfrominsidetheregion.Theprobabilitythatthemidpointofalsolies

insidethisL-shapedregioncanbeexpressedaswhereandare