AMC10美国数学竞赛A卷附中文翻译和答案

实用文档之"2011AMC10美国数学竞赛A卷"Acellphoneplancosts$20eachmonth,plus50pertextmessagesent,plus100foreachminuteusedover30hours.InJanuaryMichellesent100textmessagesandtalkedfor30.5hours.Howmuchdidshehavetopay?(A)$24.00 (B)$24.50 (C)$25.50 (D)$28.00 (E)$30.00Asmallbottleofshampoocanhold35millilitersofshampoo,Whereasalargebottlecanhold500millilitersofshampoo.Jasminewantstobuytheminimumnumberofsmallbottlesnecessarytocompletelyfillalargebottle.Howmanybottlesmustshebuy?(A)11 (B)12 (C)13 (D)14 (E)15Suppose[abjdenotestheaverageofaandb,and{abc}denotestheaverageofa,b,andc.Whatis{{110}[01]0}?(A)£ (B)A(C)1 (D)1 (E)£9 18 3 18 3LetXandYbethefollowingsumsofarithmeticsequences:X=10+12+14+...4-100.Y=12+14+16+...+102.Whatisthevalueofy_x?(A)92 (B)98 (C)100 (D)102 (E)112Atanelementaryschool,thestudentsinthirdgrade,fourthgrade,andfifthgraderunanaverageof12,15,and10minutesperday,respectively.Therearetwiceasmanythirdgradersasfourthgraders,andtwiceasmanyfourthgradersasfifthgraders.Whatistheaveragenumberofminutesrunperdaybythesestudents?(A)12 (B)卫(C)色(D)13 (E)143 7SetAhas20elements,andsetBhas15elements.WhatisthesmallestpossiblenumberofelementsinAljB,theunionofAandB?(A)5 (B)15 (C)20 (D)35 (E)300WhichofthefollowingequationsdoesNOThaveasolution?(A)(x+7)2=0 (B)网+5=0 (C)Q-2=0(D)77-8=0 (E)卜3止4=0Lastsummer30%ofthebirdslivingonTownLakeweregeese,25%wereswans,10%wereherons,and35%wereducks.Whatpercentofthebirdsthatwerenotswansweregeese?(A)20 (B)30 (C)40 (D)50 (E)60Arectangularregionisboundedbythegraphsoftheequationsy=a,y=-b,x=-c,andx=d,wherea,b,c,anddareallpositivenumbers.Whichofthefollowingrepresentstheareaofthisregion?(A)ac+ad+be+bd(B)ac-ad+be-bd(C)ac+ad-be-bd(D)-ac-ad+be+bd(E)ac-ad-be+bdAmajorityofthe20studentsinMs.Deameanor'sclassboughtpencilsattheschoolbookstore.Eachofthesestudentsboughtthesamenumberofpencils,andthisnumberwasgreaterthan1.Thecostofapencilincentswasgreaterthanthenumberofpencilseachstudentbought,andthetotalcostofallthepencilswas$17.71.Whatwasthecostofapencilincents?(A)7 (B)11 (C)17 (D)23 (E)77SquareEFGHhasonevertexoneachsideofsquareABCD.PointEisonABwithAE=7•EB.WhatistheratiooftheareaofEFGHtotheareaofABCD?(A)竺(B)今(C)I(D)些(E)64 32 8 8~T~Theplayersonabasketballteammadesomethree-pointshots,sometwo-pointshots,someone-pointfreethrows.Theyscoredasmanypointswithtwo-pointshotsaswiththree-pointshots.Theirnumberofsuccessfulfreethrowswasonemorethantheirnumberofsuccessfultwo-pointshots.Theteam'stotalscorewas61points.Howmanyfreethrowsdidtheymake?(A)13 (B)14 (C)15 (D)16 (E)17Howmanyevenintegersaretherebetween200and700whosedigitsarealldifferentandcomefromtheset{1,2,5,7,8,9}?(A)12 (B)20 (C)72 (D)120(E)200Apairofstandard6-sidedfairdiceisrolledonce.Thesumofthenumbersrolleddeterminesthediameterofacircle.Whatistheprobabilitythatthenumericalvalueoftheareaofthecircleislessthanthenumericalvalueofthecircle'scircumference?(A)1 (B)1 (C)1 (D)1 (E)A36 12 6 4 18Royboughtanewbattery-gasolinehybridcar.Onatripthecarranexclusivelyonitsbatteryforthefirst40miles,thenranexclusivelyongasolinefortherestofthetrip,usinggasolineatarateof0.02gallonspermile.Onthewholetripheaveraged55milespergallon.Howlongwasthetripinmiles?(B)240(C)440(B)240(C)440(D)640(E)WhichofthefollowinginequaltoJ9-6#+J9+6衣?(A)3J2 (B)2J6 (C)至(D)3/ (E)6Intheeight-termsequenceA,B,C,D,E,F,G,H,thevalueofCis5andthesumofanythreeconsecutivetermsis30.WhatisA+H?(A)17 (B)18 (C)25 (D)26 (E)43CirclesA,B,andCeachhaveradius1.CirclesAandBshareonepointoftangency.CircleChasapointoftangencywiththemidpointofAB.WhatistheareainsideCircleCbutoutsideCircleAandCircleB?(A)3-1 (B)1 (C)2 (D)主(E)i+l2 2 4 2In1991thepopulationofatownwasaperfectsquare.Tenyearslater,afteranincreaseof150people,thepopulationwas9morethanaperfectsquare.Now,in2011,withanincreaseofanother150people,thepopulationisonceagainaperfectsquare.Whichofthefollowingisclosesttothepercentgrowthofthetown,spopulationduringthisytar)p七@od?(A)42 (B)47 (C)52 (D)57 (E)62Twopointsonthecircumferenceofacircleofradiusrareselectedindependentlyandatrandom.Fromeachpointachordoflengthrisdrawninaclockwisedirection.Whatistheprobabilitythatthetwochordsintersect?(A)1 (B)1 (C)1 (D)1 (E)16 5 4 3 2Twocounterfeitcoinsofequalweightaremixedwith8identicalgenuinecoins.Theweightofeachofthecounterfeitcoinsisdifferentfromtheweightofeachofthegenuinecoins.Apairofcoinsisselectedatrandomwithoutreplacementfromthe10coins.Asecondpairisselectedatrandomwithoutreplacementfromtheremaining8coins.Thecombinedweightofthefirstpairisequaltothecombinedweightofthesecondpair.Whatistheprobabilitythatall4selectedcoinsaregenuine?(A)Z (B)2 (C)11 (D)” (E)经11 13 15 19 16EachvertexofconvexpentagonABCDEistobeassignedacolor.Thereare6colorstochoosefrom,andtheendsofeachdiagonalmusthavedifferentcolors.Howmanydifferentcoloringsarepossible?(A)2500 (B)2880 (C)3120 (D)3250 (E)3750Sevenstudentscountfrom1to1000asfollows:,Alicesaysallthenumbers,exceptsheskipsthemiddlenumberineachconsecutivegroupofthreenumbers.ThatisAlicesays1,3,4,6,7,9, 997,999,1000.,BarbarasaysallofthenumbersthatAlicedoesn'tsay,exceptshealsoskipsthemiddlenumberineachconsecutivegropeofthreenumbers.,CandicesaysallofthenumbersthatneitherAlicenorBarbarasays,exceptshealsoskipsthemiddlenumberineachconsecutivegroupofthreenumbers.,Debbie,Eliza,andFatimasayallofthenumbersthatnoneofthestudentswiththefirstnamesbeginningbeforetheirsinthealphabetsay,excepteachalsoskipsthemiddlenumberineachofherconsecutivegroupsofthreenumbers.•Finally,Georgesaystheonlynumberthatnooneelsesays.WhatnumberdoesGeorgesay?(A)37 (B)242 (C)365 (D)728 (E)998Twodistinctregulartetrahedrahavealltheirverticesamongtheverticesofthesameunitcube.Whatisthevolumeoftheregionformedbytheintersectionofthetetrahedra?(A)1 (B)显(C)乖(D)1 (E)12 12 12 6显6LetRbeasquareregionandn4aninteger.ApointXintheinteriorofRiscalledn-raypartitionaliftherearenraysemanatingfromXthatdivideRintoNtrianglesofequalarea.Howmanypointsare100-raypartitionalbutnot60-raypartitional?(A)1500 (B)1560 (C)2320 (D)2480 (E)25002011AMC10美国数学竞赛A卷1.某通讯公司手机每个月基本费为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.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能装满一个大瓶的洗发液？(A)11 (B)12 (C)13 (D)14 (E)15.若以[ab]表示a,b两数的平均数，以｛abc｝表示a,b,c三数的平均数，贝I」｛｛110｝[01]0｝之值为何？(A)I(B)A(C)1 (D)2. (E)I9 18 3 18 3.设X和丫为下列等差级数之和：X=10+12+14+…+100.Y=12+14+16+•••+102.则yx之值为何？(A)92 (B)98 (C)100 (D)102 (E)112.在某小学三年级，四年级及五年级的学生，每天分别平均跑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。试问所有这些学生每天平均跑几分钟？(A)12 (B)至(0!! (D)13 (E)143 7.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.下列哪个方程式没有解?(A)(%+7)2=0 (B)卜3x|+5=0 (C)，7-2=0(D)J7-8=0 (E)卜3乂-4=0.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百分比？(A)20 (B)30 (C)40 (D)50 (E)60.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示这个矩形的面积？(A)ac+ad+be+bd(B)ac-ad+be-bdac+ad-be-bd-ac-ad+be+bd(E)ac-ad-be+bd10.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)7711.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？(A)49 (B)25 (C)1 (D)至(E)64 32 8 8714~T12.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。.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。.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？(A)±(B)1(C)1(D)1(E)Ao36 12 6 4 18.罗先生买了一部新型的油电车。在某旅程中，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全部旅程，则平均每加仑须跑55公里。试问此旅程是多少公里？(A)140(B)240(C)440(D)640(E)840。.下列何者等于庐4方+标扬？(A)3"(B)2#(C)述(D)3仃(E)62 ..在八项的数列A、B、C、D、E、F、G、H中，C的值是5,且任何连续三项的和都是30。试问A+H的值是多少？(A)17(B)18(C)25(D)26(E)43。.如图所示，A,B,C三圆的半径均为1,圆A与圆B外切，圆C与而相切于荏的中点。试问在圆C的内部且在圆A与圆B外部阴影区域的面积是多少？(A)3-工(B)三(C)2(D)主(E)1+L。2 2 4 219.某城镇1991年的人口数是一个完全平方数；十年后，增加了150人，当时人口数比一个完全平方数多9；到了2011年，人口又再增加了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镇人口成长率的百分比最接近下列何者？(A)42(B)47(C)52(D)57(E)62。.在一个半径为r的圆周上随意的取两点，从这两点依