2024年AIME2数学竞赛（英文试题中文答案）.pdf

2024骞碅IME2鏁板绔炶禌锛堣嫳鏂囷級

1銆丄mongthe900residentsofAimeville,thereare195whoownadiamondring,367whoownasetof

golfclubs,and562whoownagardenspade.Inaddition,eachofthe900residentsownsabagofcandy

hearts.Thereare437residentswhoownexactlytwoofthesethings,and234residentswhoownexactly

threeofthesethings.FindthenumberofresidentsofAimevillewhoownallfourofthesethings.

2銆丄listofpositiveintegershasthefollowingproperties:

路Thesumoftheitemsinthelistis30.

路Theuniquemodeofthelistis9.

路Themedianofthelistisapositiveintegerthatdoesnotappearinthelistitself.

Findthesumofthesquaresofalltheitemsinthelist.

3銆丗indthenumberofwaystoplaceadigitineachcellofa2脳3gridsothatthesumofthetwo

numbersformedbyreadinglefttorightis999,andthesumofthethreenumbersformedbyreadingtopto

bottomis99.Thegridbelowisanexampleofsuchanarangementbecause8+991=999and9+9+

81=99.

4銆丩et,,andbepositiverealnumbersthatsatisfythefollowingsystemofequations:

饊仴饊仴饊仸饊仸饊仹饊仹

饊仴饊仴1

log2()=

饊仸饊仸饊仸饊仸2

饊仸饊仸1

log2()=

饊仴饊仴饊仸饊仸3

饊仸饊仸1

log2()=

饊仴饊仴饊仸饊仸4

饊仛饊仛

Thenthevalueof432is,whereandarerelativelyprimepositiveintegers.Find

|log2(饊仴饊仴饊仸饊仸饊仹饊仹)|饊仛饊仛饊仜饊仜饊仛饊仛+

饊仜饊仜

饊仜饊仜.