2015-AMC10A美国数学竞赛试题

Problem1WhatisthevalueofProblem2Aboxcontainsacollectionoftriangularandsquaretiles.Therearetilesinthebox,containingedgestotal.Howmanysquaretilesarethereinthebox?Problem3Annmadea3-stepstaircaseusing18toothpicks.Howmanytoothpicksdoessheneedtoaddtocompletea5-stepstaircase?Problem4Pablo,Sofia,andMiagotsomecandyeggsataparty.PablohadthreetimesasmanyeggsasSofia,andSofiahadtwiceasmanyeggsasMia.PablodecidestogivesomeofhiseggstoSofiaandMiasothatallthreewillhavethesamenumberofeggs.WhatfractionofhiseggsshouldPablogivetoSofia?Problem5Mr.Patrickteachesmathtostudents.Hewasgradingtestsandfoundthatwhenhegradedeveryone'stestexceptPayton's,theaveragegradefortheclasswas.AfterhegradedPayton'stest,thetestaveragebecame.WhatwasPayton'sscoreonthetest?Problem6Thesumoftwopositivenumbersistimestheirdifference.Whatistheratioofthelargernumbertothesmallernumber?Problem7Howmanytermsarethereinthearithmeticsequence,,,...,,?Problem8TwoyearsagoPetewasthreetimesasoldashiscousinClaire.Twoyearsbeforethat,PetewasfourtimesasoldasClaire.Inhowmanyyearswilltheratiooftheiragesbe:?Problem9Tworightcircularcylindershavethesamevolume.Theradiusofthesecondcylinderismorethantheradiusofthefirst.Whatistherelationshipbetweentheheightsofthetwocylinders?Problem10Howmanyrearrangementsofarethereinwhichnotwoadjacentlettersarealsoadjacentlettersinthealphabet?Forexample,nosuchrearrangementscouldincludeeitheror.Problem11Theratioofthelengthtothewidthofarectangleis:.Iftherectanglehasdiagonaloflength,thentheareamaybeexpressedasforsomeconstant.Whatis?Problem12Pointsandaredistinctpointsonthegraphof.Whatis?Problem13Claudiahas12coins,eachofwhichisa5-centcoinora10-centcoin.Thereareexactly17differentvaluesthatcanbeobtainedascombinationsofoneormoreofhercoins.Howmany10-centcoinsdoesClaudiahave?Problem14Problem15Considerthesetofallfractionswhereandarerelativelyprimepositiveintegers.Howmanyofthesefractionshavethepropertythatifbothnumeratoranddenominatorareincreasedby,thevalueofthefractionisincreasedby?Problem16If,and,whatisthevalueof?Problem17Alinethatpassesthroughtheoriginintersectsboththelineandtheline.Thethreelinescreateanequilateraltriangle.Whatistheperimeterofthetriangle?Problem18Hexadecimal(base-16)numbersarewrittenusingnumericdigitsthroughaswellasthelettersthroughtorepresentthrough.Amongthefirstpositiveintegers,therearewhosehexadecimalrepresentationcontainsonlynumericdigits.Whatisthesumofthedigitsof?Problem19Theisoscelesrighttrianglehasrightangleatandarea.Theraystrisectingintersectatand.Whatistheareaof?Problem20Arectanglehasareaandperimeter,whereandarepositiveintegers.Whichofthefollowingnumberscannotequal?Problem21Tetrahedronhas,,,,,and.Whatisthevolumeofthetetrahedron?Problem22Eightpeoplearesittingaroundacirculartable,eachholdingafaircoin.Alleightpeoplefliptheircoinsandthosewhoflipheadsstandwhilethosewhofliptailsremainseated.Whatistheprobabilitythatnotwoadjacentpeoplewillstand?Problem23Thezerosofthefunctionareintegers.Whatisthesumofthepossiblevaluesof?Problem24Forsomepositiveintegers,thereisaquadrilateralwithpositiveintegersidelengths,perimeter,rightanglesatand,,and.Howmanydifferentvaluesofarepossible?Problem25Letbeasquareofsidelength.Twopointsarechosenindependentlyatrandomonthesidesof.Theprobabilitythatthestraight-linedistancebetweenthepointsisatleastis,where,,andarepositiveintegerswith.Whatis?