AMC 10B ProblemsWord格式文档下载.docx

AMC 10B ProblemsWord格式文档下载.docx

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ofthepizza.Dangottheleftovers.Whatisthesequenceofthesiblingsindecreasingorderofthepartofpizzatheyconsumed?

（A）Alex,Beth,Cyril,Dan（B）Beth,Cyril,Alex,Dan（C）Beth,Cyril,Dan,Alex

（D）Beth,Dan,Cyril,Alex（E）Dan,Beth,Cyril,Alex

Problem5

David,Hikmet,Jack,Marta,Rand,andToddwereina 

12-personracewith 

6 

otherpeople.Randfinished 

placesaheadofHikmet.Martafinished 

1 

placebehindJack.Davidfinished 

2 

placesbehindHikmet.Jackfinished 

placesbehindTodd.Toddfinished 

placebehindRand.Martafinishedin 

6thplace.Whofinishedin 

8thplace?

（A）David（B）Hikmet（C）Jack（D）Rand（E）Todd

Problem6

Marleypracticesexactlyonesporteachdayoftheweek.Sherunsthreedaysaweekbutneverontwoconsecutivedays.OnMondaysheplaysbasketballandtwodayslatergolf.Sheswimsandplaystennis,butsheneverplaystennisthedayafterrunningorswimming.WhichdayoftheweekdoesMarleyswim?

（A）Sunday（B）Tuesday（C）Thursday（D）Friday（E）Saturday

Problem7

Considertheoperation"

minusthereciprocalof,"

definedby 

.Whatis 

（A）

（B）

（C）0（D）

（E）

Problem8

TheletterFshownbelowisrotated90º

clockwisearoundtheorigin,thenreflectedinthe 

-axis,andthenrotatedahalfturnaroundtheorigin.Whatisthefinalimage?

Problem9

Theshadedregionbelowiscalledashark'

sfinfalcata,afigurestudiedbyLeonardodaVinci.Itisboundedbytheportionofthecircleofradius 

3 

andcenter（0,0） 

thatliesinthefirstquadrant,theportionofthecirclewithradius3/2 

andcenter（0,3/2） 

thatliesinthefirstquadrant,andthelinesegmentfrom（0,0） 

to（3,0）.Whatistheareaoftheshark'

sfinfalcata?

Problem10

Whatarethesignandunitsdigitoftheproductofalltheoddnegativeintegersstrictlygreaterthan-2015?

（A）Itisanegativenumberendingwitha1（B）Itisapositivenumberendingwitha1.

（C）Itisanegativenumberendingwitha5.（D）Itisapositivenumberendingwitha5.

（E）Itisanegativenumberendingwitha0.

Problem11

Amongthepositiveintegerslessthan 

100,eachofwhosedigitsisaprimenumber,oneisselectedatrandom.Whatistheprobablilitythattheselectednumberisprime?

Problem12

Forhowmanyintegers 

isthepoint

insideoronthecircleofradius 

10 

centeredat（5,5）?

（A）11（B）12（C）13（D）14（E）15

Problem13

Theline

formsatrianglewiththecoordinateaxes.Whatisthesumofthelengthsofthealtitudesofthistriangle?

（A）20（B）

Problem14

Let 

and 

bethreedistinctone-digitnumbers.Whatisthemaximumvalueofthesumoftherootsoftheequation

（A）15（B）15.5（C）16（D）16.5（E）17

Problem15

ThetownofHamlethas 

peopleforeachhorse, 

4 

sheepforeachcow,and 

ducksforeachperson.Whichofthefollowingcouldnotpossiblybethetotalnumberofpeople,horses,sheep,cows,andducksinHamlet?

（A）41（B）47（C）59（D）61（E）66

Problem16

Al,Bill,andCalwilleachrandomlybeassignedawholenumberfrom 

to 

10,inclusive,withnotwoofthemgettingthesamenumber.WhatistheprobabilitythatAl'

snumberwillbeawholenumbermultipleofBill'

sandBill'

snumberwillbeawholenumbermultipleofCal'

s?

Problem17

Thecentersofthefacesoftherightrectangularprismshownbelowarejoinedtocreateanoctahedron,Whatisthevolumeoftheoctahedron?

Problem18

Johannhas 

64 

faircoins.Heflipsallthecoins.Anycointhatlandsontailsistossedagain.Coinsthatlandontailsonthesecondtossaretossedathirdtime.Whatistheexpectednumberofcoinsthatarenowheads?

（A）32（B）40（C）48（D）56（E）64

Problem19

In△ABC,∠C=90º

andAB=12.SquaresABXY 

andACWZ 

areconstructedoutsideofthetriangle.ThepointsX,Y,Z,andW 

lieonacircle.Whatistheperimeterofthetriangle?

（C）

（D）30（E）32

Problem20

Erintheantstartsatagivencornerofacubeandcrawlsalongexactly 

7 

edgesinsuchawaythatshevisitseverycornerexactlyonceandthenfindsthatsheisunabletoreturnalonganedgetoherstartingpoint.Howmanypathsaretheremeetingtheseconditions?

（A）6（B）9（C）12（D）18（E）24

Problem21

CozytheCatandDashtheDogaregoingupastaircasewithacertainnumberofsteps.However,insteadofwalkingupthestepsoneatatime,bothCozyandDashjump.Cozygoestwostepsupwitheachjump（thoughifnecessary,hewilljustjumpthelaststep）.Dashgoesfivestepsupwitheachjump（thoughifnecessary,hewilljustjumpthelaststepsiftherearefewerthan5stepsleft）.SupposetheDashtakes19fewerjumpsthanCozytoreachthetopofthestaircase.Let

denotethesumofallpossiblenumbersofstepsthisstaircasecanhave.Whatisthesumofthedigitsof 

（A）9（B）11（C）12（D）13（E）15

Problem22

Inthefigureshownbelow,ABCDE 

isaregularpentagonandAG=1.WhatisFG+JH+CD?

Problem23

beapositiveintegergreaterthan4suchthatthedecimalrepresentationof 

n!

endsin 

zerosandthedecimalrepresentationof（2n）!

endsin3k 

zeros.Let 

denotethesumofthefourleastpossiblevaluesof 

.Whatisthesumofthedigitsof 

（A）7（B）8（C）9（D）10（E）11

Problem24

Aarontheantwalksonthecoordinateplaneaccordingtothefollowingrules.Hestartsattheorigin

facingtotheeastandwalksoneunit,arrivingat

.Forn=1,2,3,…,rightafterarrivingatthepoint 

ifAaroncanturn90º

leftandwalkoneunittoanunvisitedpoint

hedoesthat.Otherwise,hewalksoneunitstraightaheadtoreach

.Thusthesequenceofpointscontinues

andsooninacounterclockwisespiralpattern.Whatis

（A）（-22,-13）（B）（-13,-22）（C）（-13,22）（D）（13,-22）（E）（22,-13）

Problem25

Arectangularboxmeasures

where

and 

areintegersand

.Thevolumeandsurfaceareaoftheboxarenumericallyequal.Howmanyorderedtriples

arepossible?

（A）4（B）10（C）12（D）21（E）26

2015AMC10BSolutions

C

Mariedoesherworktwicein 

hourand 

40 

minutes.Therefore,onetaskshouldtake 

50 

minutestofinish. 

minutesafter2:

PMis3:

30 

PM,soouranswerisB 

Letthefirstnumberbe 

andthesecondbe 

.Wehave

.Wearegivenoneofthenumbersis 

28.If 

weretobe 

28, 

wouldnotbeaninteger,thus

whichgives

A

Letthepizzahave 

60 

slices,sincetheleastcommonmultipleof（5,3,4）=60.Therefore,Alexate（1/5）*60=12 

slices,Bethate（1/3）*60=20 

slices,andCyrilate（1/4）*60=15 

slices.Danmusthaveeaten60-（12+20+15）=13 

slices.Indecreasingorder,weseetheansweris（C）Beth,Cyril,Dan,Alex.

BecauseMartawas 

6th,Jackwas 

5th,soToddwas 

3rd.Thus,Randwas 

2ndandthe8thplacefinisherwas（B）Hikmet.

MarleydoesbasketballonMondayandgolfonWednesday.SincetherearefourconsecutivedaysbetweenWednesdayandMondayexclusive,shemustrunonTuesday.Thus,shemustalsorunonThursdayandSaturdayorFridayandSunday.Shecan'

trunonThursdayandSaturdaybecauseshewouldhavetodotennisafterrunning,whichisnotallowed.Thus,sherunsonFridayandSunday,soshemustdotennisonThursday,sosheswimson（E）Saturday

so 

.Also,

.

Thus, 

Thefirstrotationmovesthebaseofthe 

F 

tothenegative 

-axis,andthestemtothepositive 

-axis.Thereflectionthenmovesthestemtothenegative 

-axis,withthebaseunchanged.Thenthehalfturnmovesthestemtothepositive 

axisandthebasetothepositive 

-axis,choice 

E.

Theareaoftheshark'

sfinfalcataisjusttheareaofthequarter-circleminustheareaofthesemicircle.Thequarter-circlehasradius 

soithasarea

.Thesemicirclehasradius3/2 

soithasarea

.Thus,theshadedareais

B

Since-5>

-2015,theproductmustendwitha 

5.

Themultiplicandsaretheoddnegativeintegersfrom 

-1 

to-2013.Thereare（2013+1）/2=1007 

ofthesenumbers.Since

theproductisnegative.

Therefore,theanswermustbe（C）Itisanegativenumberendingwitha5.

Theonedigitprimenumbersare2,3,5,and 

7.Sothereareatotalof4*4=16 

waystochooseatwodigitnumberwithbothdigitsasprimesand4waystochooseaonedigitprime,foratotalof4+16=20 

ways.Outofthese2,3,5,7,23,37,53,and 

73 

areprime.Thustheprobabilityis8/20=2/5.B.

Theequationofthecircleis

.Plugginginthegivenconditionswehave

whichsimplifiesto

andtherefore

or

.So 

rangesfrom-5 

5,foratotalof 

11 

values.A

Wefindthex-interceptsandthey-interceptstofindtheintersectionsoftheaxesandtheline.If

then

.If 

is 

0,then

.Ourthreeverticesare（0,0）,（5,0）,and（0,12）.Twoofouraltitudesare 

5 

12.Sincetheareaofthetriangleis 

30,letting 

beourhypotenuse,ourfinalaltitudehastobe30

（2）/h 

.BythePythagoreanTheorem,

sothesumofouraltitudesis281