2023年AMC10美國數學競賽A卷附中文翻譯和答案

2023AMC10美國數學競賽A卷

1.Acellphoneplancosts$20eachmonth,plus50pertextmessagesent,plus100for

eachminuteusedover30hours.InJanuaryMichellesent100textmessagesand

talkedfor30.5hours.Howmuchdidshehavetopay?

(A)$24.00(B)$24.50(C)$25,50(D)$28.00(E)$30.00

2.Asmallbottleofshampoocanhold35millilitersofshampoo,Whereasalarge

bottlecanhold500millilitersofshampoo.Jasminewantstobuytheminimumnumber

ofsmallbottlesnecessarytocompletelyfillalargebottle.Howmanybottlesmustshe

buy?

(A)11(B)12(C)13(D)14(E)15

3.Suppose[ab]denotestheaverageofaandb,and{abc}denotestheaverageofa,b,

andc.Whatis{{110}[01]0}?

57

(A)(B)(D)(E)

1818

4.LetXandYbethefollowingsumsofarithmeticsequences:

X=10+12+14+...+100.

Y=12+14+16+...+102.

Whatisthevalueof|r-x|?

(A)92(B)98(C)100(D)102(E)112

5.Atanelementaryschool,thestudentsinthirdgrade,fourthgrade,andfifthgrade

runanaverageof12,15,and10minutesperday,respectively.Therearetwiceas

manythirdgradersasfourthgraders,andtwiceasmanyfourthgradersasfifthgraders.

Whatistheaveragenumberofminutesrunperdaybythesestudents?

88

(A)12(B)(C)(D)13(E)14

7

6.SetAhas20elements,andsetBhas15elements.Whatisthesmallestpossible

numberofelementsinAUB,theunionofAandB?

(A)5(B)15(C)20(D)35(E)300

7.WhichofthefollowingequationsdoesNOThaveasolution?

(A)(x+7)2=0(C)C-2=0

(D)4-8=0(E)'3x|-4=0

8.Lastsummer30%ofthebirdslivingonTownLakeweregeese,25%wereswans,

10%wereherons,and35%wereducks.Whatpercentofthebirdsthatwerenotswans

weregeese?

(A)20(B)30(C)40(D)50(E)60

9.Arectangularregionisboundedbythegraphsoftheequationsy=a,y=-b,x=-c,and

x=d,wherea,b,c,anddareallpositivenumbers.Whichofthefollowingrepresents

theareaofthisregion?

(A)ac+ad+be+bd(B)ac-ad+be-bd(C)ac+ad-be-bd

(D)-ac—ad+be+bd(E)ac—ad—be+bd

10.Amajorityofthe20studentsinMs.Deameanor'sclassboughtpencilsatthe

schoolbookstore.Eachofthesestudentsboughtthesamenumberofpencils,andthis

numberwasgreaterthan1.Thecostofapencilincentswasgreaterthanthenumber

ofpencilseachstudentbought,andthetotalcostofallthepencilswas$17.71.What

wasthecostofapencilincents?

(A)7(B)11(C)17(D)23(E)77

11.SquareEFGHhasonevertexoneachsideofsquareABCD.PointEisonABwith

AE=7,EB.WhatistheratiooftheareaofEFGHtotheareaofABCD?

4925

(A)(B)

6432

12.Theplayersonabasketballteammadesomethree-pointshots,sometwo-point

shots,someone-pointfreethrows.Theyscoredasmanypointswithtwo-pointshotsas

withthree-pointshots.Theirnumberofsuccessfulfreethrowswasonemorethantheir

numberofsuccessfultwo-pointshots.Theteam'stotalscorewas61points.How

manyfreethrowsdidtheymake?

(A)13(B)14(C)15(D)16(E)17

13.Howmanyevenintegersaretherebetween200and700whosedigitsareall

differentandcomefromtheset{1,2,5,7,8,9}?

(A)12(B)20(C)72(D)120(E)200

14.Apairofstandard6-sidedfairdiceisrolledonce.Thesumofthenumbersrolled

determinesthediameterofacircle.Whatistheprobabilitythatthenumericalvalueof

theareaofthecircleislessthanthenumericalvalueofthecircle'scircumference?

11

(A)(B)

3612

15.Royboughtanewbattery-gasolinehybridcar.Onatripthecarranexclusivelyon

itsbatteryforthefirst40miles,thenranexclusivelyongasolinefortherestofthe

trip,usinggasolineatarateof0.02gallonspermile.Onthewholetripheaveraged

55milespergallon.Howlongwasthetripinmiles?

(A)140(B)240(C)440(D)640(E)840

16.Whichofthefollowinginequalto,9-6拒+也+6拒?

(D)國(E)向

17.Intheeight-termsequenceA,B,C,D,E,F,G,H,thevalueofCis5andthesum

ofanythreeconsecutivetermsis30.WhatisA+H?

(A)17(B)18(C)25(D)26(E)43

18.CirclesA,B,andCeachhaveradius1.CirclesAandBshareonepointof

tangency.CircleChasapointoftangencywiththemidpointofAB.Whatisthearea

insideCircleCbutoutsideCircleAandCircleB?

(A)3-j(B)j(C)②(D),(E)1+|

19.In1991thepopulationofatownwasaperfectsquare.Tenyearslater,afteran

increaseof150people,thepopulationwas9morethanaperfectsquare.Now,in2023,

withanincreaseofanother150people,thepopulationisonceagainaperfectsquare.

Whichofthefollowingisclosesttothepercentgrowthofthetown'spopulation

duringthistwenty-yearperiod?

(A)42(B)47(C)52(D)57(E)62

20.Twopointsonthecircumferenceofacircleofradiusrareselectedindependently

andatrandom.Fromeachpointachordoflengthrisdrawninaclockwisedirection.

Whatistheprobabilitythatthetwochordsintersect?

(A)(B)(C)(D)(E)

21.Twocounterfeitcoinsofequalweightaremixedwith8identicalgenuinecoins.

Theweightofeachofthecounterfeitcoinsisdifferentfromtheweightofeachofthe

genuinecoins.Apairofcoinsisselectedatrandomwithoutreplacementfromthe10

coins.Asecondpairisselectedatrandomwithoutreplacementfromtheremaining8

coins.Thecombinedweightofthefirstpairisequaltothecombinedweightofthe

secondpair.Whatistheprobabilitythatall4selectedcoinsaregenuine?

~T\mimnyinj

(A)—(B)—(C)—(D)—(E)—

1113151916

22.EachvertexofconvexpentagonABCDEistobeassignedacolor.Thereare6

colorstochoosefrom,andtheendsofeachdiagonalmusthavedifferentcolors.How

manydifferentcoloringsarepossible?

(A)2500(B)2880(C)3120(D)3250(E)3750

23.Sevenstudentscountfrom1to1000asfollows:

,Alicesaysallthenumbers,exceptsheskipsthemiddlenumberineachconsecutive

groupofthreenumbers.ThatisAlicesays1,3,4,6,7,9,997,999,1000.

,BarbarasaysallofthenumbersthatAlicedoesn'tsay,exceptshealsoskipsthe

middlenumberineachconsecutivegropeofthreenumbers.

,CandicesaysallofthenumbersthatneitherAlicenorBarbarasays,exceptshealso

skipsthemiddlenumberineachconsecutivegroupofthreenumbers.

,Debbie,Eliza,andFatimasayallofthenumbersthatnoneofthestudentswiththe

firstnamesbeginningbeforetheirsinthealphabetsay,excepteachalsoskipsthe

middlenumberineachofherconsecutivegroupsofthreenumbers.

,Finally,Georgesaystheonlynumberthatnooneelsesays.

WhatnumberdoesGeorgesay?

(A)37(B)242(C)365(D)728(E)998

24.Twodistinctregulartetrahedrahavealltheirverticesamongtheverticesofthe

sameunitcube.Whatisthevolumeoftheregionformedbytheintersectionofthe

tetrahedra?

1叵走旦

(A)(B)(C)(D)(E)

1212126

25.LetRbeasquareregionandaninteger.ApointXintheinteriorofRis

calledn-raypartitionaliftherearenraysemanatingfromXthatdivideRintoN

trianglesofequalarea.Howmanypointsare100-raypartitionalbutnot60-ray

partitional?

(A)1500(B)1560(C)2320(D)2480(E)2500

2023AMC10美國數學競賽A卷

1.某通訊公司手機每月基本費為20美元，每傳送一則簡訊收5美分（一美元

=100美分）。若通話超過30小時，超過的時間每分鐘加收10美分。已知小美

一月份共傳送了100條簡訊及通話30.5小時，則她需要付多少美元？

(A)$24.00(B)$24.50(C)$25.50(D)$28.00(E)$30.00

2.小瓶裝有35毫升的洗發(fā)液，大瓶可裝500毫升的洗發(fā)液。小華至少要買多少瓶小

瓶的洗發(fā)液才干裝滿一個大瓶的洗發(fā)液？

(A)11(B)12(C)13(D)14(E)15

3.若以[ab]表達a,b兩數的平均數，以｛abc｝表達a,b,c三數的平均

數，則Hl10｝[01]0｝之值為什么？

4.設X和丫為下列等差級數之和：

X=10+12+14+…+100.

丫=12+14+16+…+102.

則叵三用之值為什么？

(A)92(B)98(C)100(D)102(E)112

5.在某小學三年級，四年級及五年級的學生，天天分別平均跑12,15,及10分鐘,

已知三年級的學生人數是四年級人數的兩倍，四年級的學生人數是五年級學生人數

的兩倍。試問所有這些學生天天平均跑幾分鐘？

(A)12(B)四(C)園(D)13(E)14

37

6.已知集合A中有20個元素，集合B中有15個元素.AUB是集合A和集合B

的聯集,它是由集合A與集合B中所有元素所形成的集合，則集合AUB中至少有多

少個元素？

(A)5(B)15(C)20(D)35(E)300

7.下列哪個方程式沒有解？

(A)|(X+7)2=O[(B)卜3^+5=0(C)|Q-2=0

(D)|?-8=o|(E)卜3%|-4=0

8.去年夏季保護區(qū)里有30%是鵝，25%是鴛鴦，10%是蒼鷹，35%是鴨子.試問不是

鴛鴦的鳥類中鵝占多少比例？

(A)20(B)30(C)40(D)50(E)60

9.某個矩形是由y=a,y=-b,x=-c,與x=d,的圓形所圍成的，其中a,b,c,,d

均為正數。試問下列何者可以表達這個矩形的面積？

(A)ac+ad+be+bd(B)ac-ad+be-bd(C)ac+ad-be-

bd

(D)-ac-ad+be+bd(E)ac-ad-be+bd

10.戴老師班上30位學生中超過半數的學生買了同一種鉛筆，這些學生所買的鉛筆

都多于1支，且每個人所買的數目相同。以美分計，每支鉛筆的價格都是整數，且

比每位同學所買的支數多。若買鉛筆共花了17.71美元(1美元=100美分)，則每支

鉛筆的價格為多少美分？

(A)7(B)11(C)17(D)23(E)77

11.已知正方形EFGH的頂點分別在正方形ABCD的四邊上.若E點在AB上且

AE=7?EB.試問EFGH的面積與ABCD的面積比值多少？

12.某籃球隊投進一些三分球、兩分球及一分的罰球。他們三分球所得的分數與兩

分球所得的分數相同，且罰球投進的球數比兩分球投進的球數多一球。若此球隊總

共得到61分，則此球隊罰球共投進了多少球？

(A)13(B)14(C)15(D)16(E)17。

13.在200至700中有多少個偶數，其各位數字都不相同，且各位數字是取自{1,2,

5,7,8,9}?

(A)12(B)20(C)72(D)120(E)200。

14.投擲兩個有六面的公正骰子一次，用兩個骰子出現點數的和作為一個圓的直徑。

試問圓面、積的數值小于圓周長的數值之機率為多少？

15.羅先生買了一部新型的油電車。在某路程中，這部車子在前40公里只使用電池;

在之后的、旅途中只使用汽油，而此時每公里需用0.02加侖的汽油。若用同量的這

些汽油恰可跑完所有路程，則平均每加侖須跑55公里。試問此路程是多少公里？

(A)140(B)240(C)440(D)640(E)840。

16.下列何者等于何9—6人乩朋目?

(A)3典⑻2胸(C)圖(D)3畫(E)6

17.在八項的數列4、B、aD、E、F、G、〃中，。的值是5,且任何連續(xù)三項的和

都是30。試問力+〃的值是多少？

(A)17(B)18(C)25(D)26(E)43。

18.如圖所示，4B,。三圓的半徑均為1,圓力與圓6外切，圓。與逅相切于灌]

的中點。試問在圓。的內部且在圓力與圓月外部陰影區(qū)域的面積是多少？

(A)3-1(B)](C)2(D)子(E)1+(。

19.某城鄉(xiāng)1991年的人口數是一個完全平方數；十年后，增長了150人，當時人口

數比一個完全平方數多9；到了2023年，人口又再增長了150人，這時人口數又是

一個完全平方數。試問這二十年間此城鄉(xiāng)人口成長率的比例最接近下列何者？

(A)42(B)47(C)52(D)57(E)62。

20.在一個半徑為r的圓周上隨意的取兩點，從這兩點依順時針方向各畫一條長度

為r的弦，則這兩弦會相交的機率是多少？

(A)|(B)|(0|(D)|(E)|