2023年AMC10美国数学竞赛A卷附中文翻译和答案

2023AMC10美国数学竞赛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)国(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美国数学竞赛A卷

1.某通讯公司手机每月基本费为20美元，每传送一则简讯收5美分（一美元

=100美分）。若通话超过30小时，超过的时间每分钟加收10美分。已知小美

一月份共传送了100条简讯及通话30.5小时，则她需要付多少美元？

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

2.小瓶装有35毫升的洗发液，大瓶可装500毫升的洗发液。小华至少要买多少瓶小

瓶的洗发液才干装满一个大瓶的洗发液？

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

3.若以[ab]表达a,b两数的平均数，以｛abc｝表达a,b,c三数的平均

数，则Hl10｝[01]0｝之值为什么？

4.设X和丫为下列等差级数之和：

X=10+12+14+…+100.

丫=12+14+16+…+102.

则叵三用之值为什么？

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

5.在某小学三年级，四年级及五年级的学生，天天分别平均跑12,15,及10分钟,

已知三年级的学生人数是四年级人数的两倍，四年级的学生人数是五年级学生人数

的两倍。试问所有这些学生天天平均跑几分钟？

(A)12(B)四(C)园(D)13(E)14

37

6.已知集合A中有20个元素，集合B中有15个元素.AUB是集合A和集合B

的联集,它是由集合A与集合B中所有元素所形成的集合，则集合AUB中至少有多

少个元素？

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

7.下列哪个方程式没有解？

(A)|(X+7)2=O[(B)卜3^+5=0(C)|Q-2=0

(D)|«-8=o|(E)卜3%|-4=0

8.去年夏季保护区里有30%是鹅，25%是鸳鸯，10%是苍鹰，35%是鸭子.试问不是

鸳鸯的鸟类中鹅占多少比例？

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

9.某个矩形是由y=a,y=-b,x=-c,与x=d,的圆形所围成的，其中a,b,c,,d

均为正数。试问下列何者可以表达这个矩形的面积？

(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.戴老师班上30位学生中超过半数的学生买了同一种铅笔，这些学生所买的铅笔

都多于1支，且每个人所买的数目相同。以美分计，每支铅笔的价格都是整数，且

比每位同学所买的支数多。若买铅笔共花了17.71美元(1美元=100美分)，则每支

铅笔的价格为多少美分？

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

11.已知正方形EFGH的顶点分别在正方形ABCD的四边上.若E点在AB上且

AE=7・EB.试问EFGH的面积与ABCD的面积比值多少？

12.某篮球队投进一些三分球、两分球及一分的罚球。他们三分球所得的分数与两

分球所得的分数相同，且罚球投进的球数比两分球投进的球数多一球。若此球队总

共得到61分，则此球队罚球共投进了多少球？

(A)13(B)14(C)15(D)16(E)17。

13.在200至700中有多少个偶数，其各位数字都不相同，且各位数字是取自{1,2,

5,7,8,9}?

(A)12(B)20(C)72(D)120(E)200。

14.投掷两个有六面的公正骰子一次，用两个骰子出现点数的和作为一个圆的直径。

试问圆面、积的数值小于圆周长的数值之机率为多少？

15.罗先生买了一部新型的油电车。在某路程中，这部车子在前40公里只使用电池;

在之后的、旅途中只使用汽油，而此时每公里需用0.02加仑的汽油。若用同量的这

些汽油恰可跑完所有路程，则平均每加仑须跑55公里。试问此路程是多少公里？

(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)图(D)3画(E)6

17.在八项的数列4、B、aD、E、F、G、〃中，。的值是5,且任何连续三项的和

都是30。试问力+〃的值是多少？

(A)17(B)18(C)25(D)26(E)43。

18.如图所示，4B,。三圆的半径均为1,圆力与圆6外切，圆。与逅相切于灌]

的中点。试问在圆。的内部且在圆力与圆月外部阴影区域的面积是多少？

(A)3-1(B)](C)2(D)子(E)1+(。

19.某城乡1991年的人口数是一个完全平方数；十年后，增长了150人，当时人口

数比一个完全平方数多9；到了2023年，人口又再增长了150人，这时人口数又是

一个完全平方数。试问这二十年间此城乡人口成长率的比例最接近下列何者？

(A)42(B)47(C)52(D)57(E)62。

20.在一个半径为r的圆周上随意的取两点，从这两点依顺时针方向各画一条长度

为r的弦，则这两弦会相交的机率是多少？

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