2018-AMC-10B-試題及答案

1、Problem1Katebakesa20-inchby18-inchpanofcornbread.Thecornbreadiscutintopiecesthatmeasure2inchesby2inches.Howmanypiecesofcornbreaddoesthepancontain?(A)90(B)100(C)180(D)200(E)360SoluticmProblem2Samdrove96milesin90minutes.Hisaveragespeedduringthefirst30minuteswas60mph(milesperhour),andhisaveragespeeddur

2、ingth。second30minuteswas65mph.Whatwashisaveragespeed,inmph,duringthelast30minutes?(A)64(B)65(C)66(D)67(E)68Problem3Intheexpression(X)+(X)eachblankistobefilledinwithoneofthedigits1,2,3,or4,v/itheachdigitbeingusedonce.Howmanydifferentvaluescanbeobtained?(A)2(B)3(C)4(D)6(E)24Problem4Athree-dimensionalr

3、ectangularboxwithdimensionsX,匕andZhasfaceswhosesurfaceareasare24,24,48,48,72,and72squareunits-WhatisX+Y+Z?(A)18(B)22(C)24(D)30(E)36Problem5Howmanysubsetsof(2,3,4,5,6,7.8,9containatleastoneprimenumber?(A)128(B)192(C)224(D)240(E)256Problem6Aboxcontains5chips,numbered1,2.3,4,and5.Chipsaredrawnrandomlyo

4、neatatimewithoutreplacementuntilthesumofthevaluesdrawnexceeds4.Whatistheprobabilitythat3drawsarerequired?(a)A(D)土(c):(D)!(E)Problem7Inthefigurebelow,Ncongruentsemicirclesaredrawnalongadiameterofalargesemicircle,withtheirdiameterscoveringthediameterofthelargesemicirclewithnooverlap.LetAbethecombineda

5、reaofthesmallsemicixlesandBbetheareaoftheregionInsidethelargesemicirclebutoutsidethesmallsemicircles.TheratioAtBls1:18.WhatisNR(C)18(D)19(E)36(A)16(B)17貿(mào)泌nProblem8Saramakesastaircaseoutoftoothpicksasshown:Thisisa3-stepstaircaseanduses18toothpicks.Howmanystepswouldbeinastaircasethatused180toothpicks?

6、(A)10(B)M(C)12QJ)24(E)30Problem9Thefacesofeachof7standarddicearelabeledwiththeintegersfrom1to6.Letpbetheprobabilitythatwhenall7dicearerolled,thesumofthenumbersonthetopfacesis10.Whatothersumoccurswithth©sameprobabilityp?(A)13(B)26(C)32(D)39(E)42SectionProblem10Intherectangularparallelepipedshown

7、,AI3=3tBC-1,andCG=2.PointMisthemidpointofFGWhatisthevolumeoftherectangularpyramidwithbaseBCHEandapexM?(A)1(B)|(D)|(E)2:040Problem11Whichofth©followingexpressionsisnovoraprim©numberwhenpisaprim©number?(A)/+的(B)/+24(C)/+26(D)/+43(E)護(hù)+96Problem12LinesegmentABisadiameterofacirclewithAB=24

8、.PointC.notequalioAorB,liesonthecircle.AspointCmovesaroundthecircle,thecentroid(centerofmass)ofAABCtracesoutaclosedcurvemissingtwopoints.Tothenearestpositiveinteger,whatistheareaoftheregionboundedbythiscurve?(A)25(B)38(C)50(D)63(E)75SlutianProblem13Howmanyofthefirst2018numbersInthesequence101,1001,1

9、0001,100001,.aredivisibleby101?(A)253(B)504(C)505(D)506(E)1009SolutionProblem14Alistof2018positiveIntegershasauniquemode,whichoccursexactly10times.WhatIstheleastnumberofdistinctvaluesthatcanoccurinthelist?(A)202(B)223(C)224(D)225(E)234SolutionProblem15Aclosedboxwithasquarebaseistobewrappedwithasquar

10、esheetofwrappingpaper.Theboxiscenteredonthewrappingpaperwiththeverticesofthebaselyingonthemidlinesofthesquaresheetofpaper,asshowninthefigureontheleft.Thefourcornersofthewrappingpaperaretobefoldedupoverthesidesandbroughttogethertomeetatthecenterofthetopofthebox,pointAinthefigureontheright.Theboxhasba

11、selengthwandheighth.Whatistheareaofthesheetofwrappingpaper?(A)2(也+h)2(B)二”)(c)2w2+4uh,(D)2w2(E)w2hSolutionProblem16Let01,2,-52018beastrictlyincreasingsequenceofpositiveintegerssuchthat+*''+2018=2018”'氣Whatistheremainderwhena+a*+a|olgisdividedby6?(A)0(B)1(C)2(D)3(E)4Problem17InrectanglePQ

12、RS,PQ=8andQR=6.PointsAandBlieonPQ,pointsCandDlieonQR,pointsEandFlieon反。，andpointsGandHlieonSPsothatAP=BQ<4andtheconvexoctagonABCDEFGHisequilateral.Thelengthofasideofthisoctagoncanbeexpressedintheformk+my/n9wherek,andnareintegersandnisnotdivisiblebythesquareofanyprimaWhatisAr+m+n?(A)1(B)7(C)21(D)9

13、2(E)106Problem18Threeyoungbrother-sisterpairs-fromdifferentfamiliesneedtotakeatripinavan.Thesesixchildrenwilloccupythesecondandthirdrowsinthevan,eachofwhichhasthreeseats.Toavoiddisruptions,siblingsmaynotsitrightneKttoeachotherinthesamerow,andnochidmaysitdirectlyinfrontofbisorhersibling.Howmanyseatin

14、garrangementsarepossibleforthistrip?(A)60(B)72(C)92(D)96(E)120SolutionProblem19JoeyandChloeandtheirdaughterZoeallhavethesamebirthday.Joeyis1yearolderthanChloe,andZoeIsexactly1yearoldtoday.Todayisthefirstofthe9birthdaysonwhichChioe'sagewillbeanintegralmultipleofZoe'sage.Whatwillbethesumofthet

15、wodigitsofJoey'sagethenexttimehisageisamultipleofZoesage?(A)7(B)8(C)9(D)10(E)11Problem20Afunctionfisdefinedrecursivelyby/(I)=/(2)=1and/(n)=/(n-1)-f(n-2)+nforallintegersn>3.Whatisf(2018)?(A)2016(B)2017(C)2018(D)2019(E)2020Problem21Marychoseaneven4-digitnumbern.Shewrotedownallthedivisorsofninin

16、creasingnorderfromlefttoright:1,2,n.AtsomemomentMarywrote323asadivisorofn.乙Whatisthesmallestpossiblevalueofthenextdivisorwrittentotherightof323?(A)324(B)330(C)340(D)361(E)646Problem22Realnumbersxandyarechosenindependentlyanduniformlyatrandomfromtheinterval0,1.Whichofthefollowingnumbersisclosesttothe

17、probabilitythatxyand1arethesidelengthsofanobtusetriangle?(A)0.21(B)0.25(C)0.29(D)0.50(E)0.79Problem23Howmanyorderedpairs(a,b)ofpositiveintegerssatisfytheequationa3+63=20lcm(a,b)+12gcd(a,b)fwheregcd(a5b)denotesthegreatestcommondivisorofaandb,andlcm(a,b'Jdenotestheirleastcommonmultiple?(A)0(B)2(C)4(D)6(E)8Problem24LetABCDEFbearegularhexagonwithsidelength1.DenotebyX,匕andZthemidpointsofsidesAB,CD,andEFfrespectively.Whatistheareaoftheconvexhexagonwhoseinterioristheintersectiono