USA-AMC_8-2010-42

USAAMC820101AtEuclidHighSchool,themathematicsteachersareMrs.Germain,Mr.Newton,andMrs.Young.Thereare11studentsinMrs.Germain’sclass,8inMr.Newton,and9inMrs.Young’sclassaretakingtheAMC8thisyear.HowmanymathematicsstudentsatEuclidHighSchoolaretakingthecontest?(A)26(B)27(C)28(D)29(E)302Ifa@b=a×ba+b,fora,bpositiveintegers,thenwhatis5@10?(A)310(B)1(C)2(D)103(E)503Thegraphshowsthepriceoffivegallonsofgasolineduringthefirsttenmonthsoftheyear.Bywhatpercentisthehighestpricemorethanthelowestprice?0$5$10$15$20Price12345678910Month(A)50(B)62(C)70(D)89(E)1004Whatisthesumofthemean,median,andmodeofthenumbers,2,3,0,3,1,4,0,3?(A)6.5(B)7(C)7.5(D)8.5(E)95Aliceneedstoreplacealightbulblocated10centimetersbelowtheceilingofherkitchen.Theceilingis2.4metersabovethefloor.Aliceis1.5meterstallandcanreach46centimetersaboveherhead.Standingonastool,shecanjustreachthelightbulb.Whatistheheightofthestool,incentimeters?http://www.artofproblemsolving.com/ThisfilewasdownloadedfromtheAoPSMathOlympiadResourcesPagePage1USAAMC82010(A)32(B)34(C)36(D)38(E)406Whichofthefollowinghasthegreatestnumberoflineofsymmetry?(A)EquilateralTriangle(B)Non-squarerhombus(C)Non-squarerectangle(D)IsoscelesTriangle(E)Square7Usingonlypennies,nickels,dimes,andquarters,whatisthesmallestnumberofcoinsFreddiewouldneedsohecouldpayanyamountofmoneylessthanonedollar?(A)6(B)10(C)15(D)25(E)998AsEmilyisridingherbikeonalongstraightroad,shespotsErmensonskatinginthesamedirection1/2mileinfrontofher.Aftershepasseshim,shecanseehiminherrearmirroruntilheis1/2milebehindher.Emilyridesataconstantrateof12milesperhour.Ermensonskatesataconstantrateof8milesperhour.ForhowmanyminutescanEmilyseeErmenson?(A)6(B)8(C)12(D)15(E)169Ryangot80%oftheproblemsona25-problemtest,90%ona40-problemtest,and70%ona10-problemtest.WhatpercentofallproblemsdidRyananswercorrectly?(A)64(B)75(C)80(D)84(E)86106pepperonicircleswillexactlyfitacrossthediameterofa12-inchpizzawhenplaced.Ifatotalof24circlesofpepperoniareplacedonthispizzawithoutoverlap,whatfractionofthepizzaiscoveredwithpepperoni?(A)12(B)23(C)34(D)56(E)7811Thetopofonetreeis16feethigherthanthetopofanothertree.Theheightofthe2treesareataratioof3:4.Infeet,howtallisthetallertree?(A)48(B)64(C)80(D)96(E)11212Ofthe500ballsinalargebag,80%areredandtherestareblue.Howmanyoftheredballsmustberemovedsothat75%oftheremainingballsarered?(A)25(B)50(C)75(D)100(E)15013Thelengthsofthesidesofatriangleininchesarethreeconsecutiveintegers.Thelengthoftheshortersideis30%oftheperimeter.Whatisthelengthofthelongestside?(A)7(B)8(C)9(D)10(E)1114Whatisthesumoftheprimefactorsof2010?(A)67(B)75(C)77(D)201(E)210http://www.artofproblemsolving.com/ThisfilewasdownloadedfromtheAoPSMathOlympiadResourcesPagePage2USAAMC8201015Ajarcontains5differentcolorsofgumdrops.30%areblue,20%arebrown,15%red,10%yellow,andtheother30gumdropsaregreen.Ifhalfofthebluegumdropsarereplacedwithbrowngumdrops,howmanygumdropswillbebrown?(A)35(B)36(C)42(D)48(E)6416Asquareandacirclehavethesamearea.Whatistheratioofthesidelengthofthesquaretotheradiusofthecircle?(A)√π2(B)√π(C)π(D)2π(E)π217Thediagramshowsanoctagonconsistingof10unitsquares.TheportionbelowPQisaunitsquareandatrianglewithbase5.IfPQbisectstheareaoftheoctagon,whatistheratioXQQY?PXYQ(A)25(B)12(C)35(D)23(E)3418Adecorativewindowismadeupofarectanglewithsemicirclesateitherend.TheratioofADtoABis3:2.AndABis30inches.Whatistheratiooftheareaoftherectangletothecombinedareaofthesemicircle.http://www.artofproblemsolving.com/ThisfilewasdownloadedfromtheAoPSMathOlympiadResourcesPagePage3USAAMC82010ABDC(A)2:3(B)3:2(C)6:π(D)9:π(E)30:π19ThetwocirclespicturedhavethesamecenterC.ChordADistangenttotheinnercircleatB,ACis10,andchordADhaslength16.Whatistheareabetweenthetwocircles?ABCD(A)36π(B)49π(C)64π(D)81π(E)100π20Inaroom,2/5ofthepeoplearewearinggloves,and3/4ofthepeoplearewearinghats.Whatistheminimumnumberofpeopleintheroomwearingbothahatandaglove?(A)3(B)5(C)8(D)15(E)20http://www.artofproblemsolving.com/ThisfilewasdownloadedfromtheAoPSMathOlympiadResourcesPagePage4USAAMC8201021Huiisanavidreader.SheboughtacopyofthebestsellerMathisBeautiful.Onthefirstday,sheread1/5ofthepagesplus12more,andontheseconddaysheread1/4oftheremainingpagesplus15more.Onthethirddaysheread1/3oftheremainingpagesplus18more.Shethenrealizesshehas62pagesleft,whichshefinishesthenextday.Howmanypagesareinthisbook?(A)120(B)180(C)240(D)300(E)36022Thehundredsdigitofathree-digitnumberis2morethantheunitsdigit.Thedigitsofthethree-digitnumberarereversed,andtheresultissubtractedfromtheoriginalthree-digitnumber.Whatistheunitsdigitoftheresult?(A)0(B)2(C)4(D)6(E)823SemicirclesPOQandROSpassthroughthecenterofcircleO.WhatistheratioofthecombinedareasofthetwosemicirclestotheareaofcircleO?P(−1,1)Q(1,1)R(−1,−1)S(1,−1)O(A)√24(B)12(C)2π(D)23(E)√2224Whatisthecorrectorderingofthethreenumbers,108,512,and224?(A)224<108<512(B)224<512<108(C)512<224<108(D)108<512<224(E)108<224<512http://www.artofproblemsolving.com/ThisfilewasdownloadedfromtheAoPSMathOlympiadResourcesPagePage5USAAMC8201025Everydayatschool,Joclimbsaflightof6stairs.Joecantakethestairs1,2,or3atatime.Forexample,Jocouldclimb3,then1,then2.InhowmanywayscanJoclimbthestairs?(A)13(B)18(C)20(D)22(E)24http://www.artofproblemsolving.com/ThisfilewasdownloadedfromtheAoPSMathOlympiadResourcesPagePage6