AMC8试题及答案.pdf

AMC8试题及答案.pdf

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Copyright2016ArtofProblemSolvingHowmanysquareyardsofcarpetarerequiredtocoverarectangularfloorthatisfeetlongandfeetwide?

（Thereare3feetinayard.）First,wemultiplytogetthatyouneedsquarefeetofcarpetyouneedtocover.Sincetherearesquarefeetinasquareyard,youdividebytogetsquareyards,soouransweris.Sincetherearefeetinayard,wedividebytoget,andbytoget.Tofindtheareaofthecarpet,wethenmultiplythesetwovaluestogethertoget.2015AMC8（ProblemsAnswerKeyResources（http:

/FirstProblemFollowedbyProblem212345678910111213141516171819202122232425AllAJHSME/AMC8ProblemsandSolutionsTheproblemsonthispagearecopyrightedbytheMathematicalAssociationofAmerica（http:

/www.maa.org）sAmericanMathematicsCompetitions（http:

/amc.maa.org）.Placement:

EasyGeometryRetrievedfromhttp:

/AMC8Problems/Problem1SolutionSolution2SeeAlsoPointisthecenteroftheregularoctagon,andisthemidpointofthesideWhatfractionoftheareaoftheoctagonisshaded?

1Solution12Solution23Solution34SeeAlsoSinceoctagonisaregularoctagon,itissplitintoequalparts,suchastriangles,etc.Theseparts,sincetheyareallequal,areoftheoctagoneach.Theshadedregionconsistsofoftheseequalpartsplushalfofanother,sothefractionoftheoctagonthatisshadedis2015AMC8Problems/Problem2ContentsSolution1Solution2Theoctagonhasbeendividedupintoidenticaltriangles（andthustheyeachhaveequalarea）.Sincetheshadedregionoccupiesoutofthetotaltriangles,theansweris.ForstarterswhatIfindhelpfulistodividethewholeoctagonupintotrianglesasshownhere:

Nowitisjustamatterofcountingthelargertrianglesrememberthatandarenotfulltrianglesandareonlyhalfforthesepurposes.Wecountitupandwegetatotalofoftheshapeshaded.Wethensimplifyittogetouranswerof.2015AMC8（ProblemsAnswerKeyResources（http:

/Problem1FollowedbyProblem312345678910111213141516171819202122232425AllAJHSME/AMC8ProblemsandSolutionsSolution3SeeAlsoCopyright2016ArtofProblemSolvingJackandJillaregoingswimmingatapoolthatisonemilefromtheirhouse.Theyleavehomesimultaneously.Jillridesherbicycletothepoolataconstantspeedofmilesperhour.Jackwalkstothepoolataconstantspeedofmilesperhour.HowmanyminutesbeforeJackdoesJillarrive?

Using,wecansetupanequationforwhenJillarrivesattheswimmingpool:

Solvingfor,wegetthatJillgetstothepoolinofanhour,whichisminutes.DoingthesameforJack,wegetthatJackarrivesatthepoolinofanhour,whichinturnisminutes.Thus,JillhastowaitminutesforJacktoarriveatthepool.2015AMC8（ProblemsAnswerKeyResources（http:

/Problem2FollowedbyProblem412345678910111213141516171819202122232425AllAJHSME/AMC8ProblemsandSolutionsTheproblemsonthispagearecopyrightedbytheMathematicalAssociationofAmerica（http:

/www.maa.org）sAmericanMathematicsCompetitions（http:

/amc.maa.org）.Retrievedfromhttp:

/AMC8Problems/Problem3SolutionSeeAlsoCopyright2016ArtofProblemSolvingTheCentervilleMiddleSchoolchessteamconsistsoftwoboysandthreegirls.Aphotographerwantstotakeapictureoftheteamtoappearinthelocalnewspaper.Shedecidestohavethemsitinarowwithaboyateachendandthethreegirlsinthemiddle.Howmanysucharrangementsarepossible?

Therearewaystoordertheboysontheend,andtherearewaystoorderthegirlsinthemiddle.Wegettheanswertobe.2015AMC8（ProblemsAnswerKeyResources（http:

/Problem3FollowedbyProblem512345678910111213141516171819202122232425AllAJHSME/AMC8ProblemsandSolutionsTheproblemsonthispagearecopyrightedbytheMathematicalAssociationofAmerica（http:

/www.maa.org）sAmericanMathematicsCompetitions（http:

/amc.maa.org）.Retrievedfromhttp:

/AMC8Problems/Problem4SolutionSeeAlsoCopyright2016ArtofProblemSolvingBillysbasketballteamscoredthefollowingpointsoverthecourseofthefirst11gamesoftheseason:

Ifhisteamscores40inthe12thgame,whichofthefollowingstatisticswillshowanincrease?

Whentheyscoreaonthenextgame,therangeincreasesfromto.Thismeanstheincreased.Becauseislessthanthescoreofeverygametheyveplayedsofar,themeasuresofcenterwillneverrise.Onlymeasuresofspread,suchasthe,mayincrease.2015AMC8（ProblemsAnswerKeyResources（http:

/Problem4FollowedbyProblem612345678910111213141516171819202122232425AllAJHSME/AMC8ProblemsandSolutionsTheproblemsonthispagearecopyrightedbytheMathematicalAssociationofAmerica（http:

/www.maa.org）sAmericanMathematicsCompetitions（http:

/amc.maa.org）.Retrievedfromhttp:

/AMC8Problems/Problem5SolutionSolution2Copyright2016ArtofProblemSolvingIn,and.Whatistheareaof?

Weknowthesemi-perimeterofis.Next,weuseHeronsFormulatofindthattheareaofthetriangleisjust.Splittingtheisoscelestriangleinhalf,wegetarighttrianglewithhypotenuseandleg.UsingthePythagoreanTheorem,weknowtheheightis.Nowthatweknowtheheight,theareais.2015AMC8（ProblemsAnswerKeyResources（http:

/Problem5FollowedbyProblem712345678910111213141516171819202122232425AllAJHSME/AMC8ProblemsandSolutionsTheproblemsonthispagearecopyrightedbytheMathematicalAssociationofAmerica（http:

/www.maa.org）sAmericanMathematicsCompetitions（http:

/amc.maa.org）.Retrievedfromhttp:

/AMC8Problems/Problem6Solution1Solution2SeeAlsoEachoftwoboxescontainsthreechipsnumbered,.Achipisdrawnrandomlyfromeachboxandthenumbersonthetwochipsaremultiplied.Whatistheprobabilitythattheirproductiseven?

Wecaninsteadfindtheprobabilitythattheirproductisodd,andsubtractthisfrom.Inordertogetanoddproduct,wehavetodrawanoddnumberfromeachbox.Wehaveaprobabilityofdrawinganoddnumberfromonebox,sothereisaprobabilityofhavinganoddproduct.Thus,thereisaprobabilityofhavinganevenproduct.Youcanalsomakethisproblemintoaspinnerproblem.Youhavethefirstspinnerwithequallydividedsections,andYoumakeasecondspinnerthatisidenticaltothefirst,withequalsectionsof,and.Ifthefirstspinnerlandson,tobeeven,itmustlandontwo.Youwritedownthefirstcombinationofnumbers.Next,ifthespinnerlandson,itcanlandonanynumberonthesecondspinner.Wenowhavethecombinationsof.Finally,ifthefirstspinnerendson,wehaveSincetherearepossiblecombinations,andwehaveevens,thefinalansweris.2015AMC8（ProblemsAnswerKeyResources（http:

/Problem6FollowedbyProblem812345678910111213141516171819202122232425AllAJHSME/AMC8ProblemsandSolutionsTheproblemsonthispagearecopyrightedbytheMathematicalAssociationofAmerica（http:

/www.maa.org）sAmericanMathematicsCompetitions（http:

/amc.maa.org）.Retrievedfromhttp:

/AMC8Problems/Problem7SolutionSolution2SeeAlsoCopyright2016ArtofProblemSolvingWhatisthesmallestwholenumberlargerthantheperimeterofanytrianglewithasideoflengthandasideoflength?

Weknowfromthetriangleinequalitythatthelastside,fulfills.Addingtobothsidesoftheinequality,weget,andbecauseistheperimeterofourtriangle,isouranswer.2015AMC8（ProblemsAnswerKeyResources（http:

/Problem7FollowedbyProblem912345678910111213141516171819202122232425AllAJHSME/AMC8ProblemsandSolutionsTheproblemsonthispagearecopyrightedbytheMathematicalAssociationofAmerica（http:

/www.maa.org）sAmericanMathematicsCompetitions（http:

/amc.maa.org）.Retrievedfromhttp:

/AMC8Problems/Problem8SolutionSeeAlsoCopyright2016ArtofProblemSolvingOnherfirstdayofwork,Janabelsoldonewidget.Ondaytwo,shesoldthreewidgets.Ondaythree,shesoldfivewidgets,andoneachsucceedingday,shesoldtwomorewidgetsthanshehadsoldonthepreviousday.HowmanywidgetsintotalhadJanabelsoldafterworkingdays?

ThesumofisThesumisjustthesumofthefirstoddintegers,whichis2015AMC8（ProblemsAnswerKeyResources（http:

/Problem8FollowedbyProblem1012345678910111213141516171819202122232425AllAJHSME/AMC8ProblemsandSolutionsTheproblemsonthispagearecopyrightedbytheMathematicalAssociationofAmerica（http:

/www.maa.org）sAmericanMathematicsCompetitions（http:

/amc.maa.org）.Retrievedfromhttp:

/AMC8Problems/Problem9Solution1Solution2SeeAlsoCopyright2016ArtofProblemSolvingHowmanyintegersbetweenandhavefourdistinctdigits?

ThequestioncanberephrasedtoHowmanyfour-digitpositiveintegershavefourdistinctdigits?

sincenumbersbetweenandarefour-digitintegers.Therearechoicesforthefirstnumber,sinceitcannotbe,thereareonlychoicesleftforthesecondnumbersinceitmustdifferfromthefirst,choicesforthethirdnumber,sinceitmustdifferfromthefirsttwo,andchoicesforthefourthnumber,sinceitmustdifferfromallthree.Thismeansthereareintegersbetweenandwithfourdistinctdigits.2015AMC8（ProblemsAnswerKeyResources（http:

/Problem9FollowedbyProblem1112345678910111213141516171819202122232425AllAJHSME/AMC8ProblemsandSolutionsTheproblemsonthispagearecopyrightedbytheMathematicalAssociationofAmerica（http:

/www.maa.org）sAmericanMathematicsCompetitions（http:

/amc.maa.org）.Retrievedfromhttp:

/AMC8Problems/Problem10Solution1SeeAlsoCopyright2016ArtofProblemSolvingInthesmallcountryofMathland,allautomobilelicenseplateshavefoursymbols.Thefirstmustbeavowel（A,E,I,O,orU）,thesecondandthirdmustbetwodifferentlettersamongthe21non-vowels,andthefourthmustbeadigit（0through9）.Ifthesymbolsarechosenatrandomsubjecttotheseconditions,whatistheprobabilitythattheplatewillreadAMC8?

Therei