AMC 10A 试题及答案解析.docx
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AMC10A试题及答案解析
2007AMC10A
1、Onetickettoashowcostsatfullprice.Susanbuys4ticketsusingacouponthatgiveshera25%discount.Pambuys5ticketsusingacouponthatgiveshera30%discount.HowmanymoredollarsdoesPampaythanSusan?
Solution
=theamountPamspent=theamountSusanspent
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Pampays10moredollarsthanSusan
2、Defineand.Whatis?
Solution
3、Anaquariumhasarectangularbasethatmeasures100cmby40cmandhasaheightof50cm.Itisfilledwithwatertoaheightof40cm.Abrickwitharectangularbasethatmeasures40cmby20cmandaheightof10cmisplacedintheaquarium.Byhowmanycentimetersdoesthewaterrise?
Solution
Thevolumeofthebrickis.Thusthewatervolumerose.
4、Thelargeroftwoconsecutiveoddintegersisthreetimesthesmaller.Whatistheirsum?
Solution
Letthetwoconsecutiveoddintegersbe,.Then,soandtheirsumis.
5、Aschoolstoresells7pencilsand8notebooksfor.Italsosells5pencilsand3notebooksfor.Howmuchdo16pencilsand10notebookscost?
Solution
Weletcostofpencilsincents,numberofnotebooksincents.Then
Subtractingtheseequationsyields.Backwardssolvinggives.Thustheansweris.
6、AtEuclidHighSchool,thenumberofstudentstakingtheAMC10was60in2002,66in2003,70in2004,76in2005,78and2006,andis85in2007.Betweenwhattwoconsecutiveyearswastherethelargestpercentageincrease?
Solution
Wecomputethepercentageincreases:
1.
2.
3.
4.
5.
Theansweris.
Infact,theanswerfollowsdirectlyfromexaminingthedifferencesbetweeneachyear.Thelargestdifferencesare,anditiseasytoseethatduetothedecreasedstartingnumberofstudentsin2002thatthatwillbeouranswer.
7、LastyearMr.JohnQ.Publicreceivedaninheritance.Hepaid20%infederaltaxesontheinheritance,andpaid10%ofwhathehasleftinstatetaxes.Hepaidatotalofforbothtaxes.Howmanydollarswastheinheritance?
Solution
Afterpayinghistaxes,hehasoftheinheritanceleft.Sinceisoftheinheritance,thewholeinheritanceis.
8、Trianglesandareisosceleswithand.PointDisinside.,and.Whatisthedegreemeasureof?
Solution
Weanglechase,andfindoutthat:
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9、Realnumbersandsatisfytheequationsand.Whatis?
Solution
And
Substitutiongives,andsolvingforyields.Thus.
10、TheDunbarfamilyconsistsofamother,afather,andsomechildren.Theaverageageofthemembersofthefamilyis,thefatherisyearsold,andtheaverageageofthemotherandchildrenis.Howmanychildrenareinthefamily?
Solution
Letbethenumberofchildren.Thenthetotalagesofthefamilyis,andthetotalnumberofpeopleinthefamilyis.So
11、Thenumbersfromtoareplacedattheverticesofacubeinsuchamannerthatthesumofthefournumbersoneachfaceisthesame.Whatisthiscommonsum?
Solution
Thesumofthenumbersonthetopfaceofacubeisequaltothesumofthenumbersonthebottomfaceofthecube;thesenumbersrepresentalloftheverticesofthecube.Thustheansweris.
12、Twotourguidesareleadingsixtourists.Theguidesdecidetosplitup.Eachtouristmustchooseoneoftheguides,butwiththestipulationthateachguidemusttakeatleastonetourist.Howmanydifferentgroupingsofguidesandtouristsarepossible?
Solution
Eachtouristhastopickinbetweentheguides,sofortouriststherearepossiblegroupings.However,sinceeachguidemusttakeatleastonetourist,wesubtractthecaseswhereaguidehasnotourist.Thustheansweris.
13、Yanissomewherebetweenhishomeandthestadium.Togettothestadiumhecanwalkdirectlytothestadium,orelsehecanwalkhomeandthenridehisbicycletothestadium.Heridestimesasfastashewalks,andbothchoicesrequirethesameamountoftime.WhatistheratioofYan'sdistancefromhishometohisdistancefromthestadium?
Solution
LetthedistancefromYan'sinitialpositiontothestadiumbeandthedistancefromYan'sinitialpositiontohomebe.Wearetryingtofind,andwehavethefollowingidentitygivenbytheproblem:
Thusandtheansweris
14、Atrianglewithsidelengthsintheratioisinscribedinacircleofradius.Whatistheareaofthetriangle?
Solution
Since3-4-5isaPythagoreantriple,thetriangleisarighttriangle.Sincethehypotenuseisadiameterofthecircumcircle,thehypotenuseis.Thentheotherlegsareand.Theareais
15、Fourcirclesofradiusareeachtangenttotwosidesofasquareandexternallytangenttoacircleofradius,asshown.Whatistheareaofthesquare?
Solution
Thediagonalhaslength.Thereforethesideshavelength,andtheareais
AlternateSolution
Exten