2015_AMC_12A_美国数学竞赛

Problem1WhatisthevalueofProblem2Twoofthethreesidesofatriangleare20and15.Whichofthefollowingnumbersisnotapossibleperimeterofthetriangle?Problem3Mr.Patrickteachesmathto15students.Hewasgradingtestsandfoundthatwhenhegradedeveryone'stestexceptPayton's,theaveragegradefortheclasswas80.afterhegradedPayton'stest,theclassaveragebecame81.WhatwasPayton'sscoreonthetest?Problem4Thesumoftwopositivenumbersis5timestheirdifference.Whatistheratioofthelargernumbertothesmaller?Problem5Amelianeedstoestimatethequantity,whereandarelargepositiveintegers.Sheroundseachoftheintegerssothatthecalculationwillbeeasiertodomentally.Inwhichofthesesituationswillheranswernecessarilybegreaterthantheexactvalueof?Problem6TwoyearsagoPetewasthreetimesasoldashiscousinClaire.Twoyearsbeforethat,PetewasfourtimesasoldasClaire.Inhowmanyyearswilltheratiooftheiragesbe?Problem7Tworightcircularcylindershavethesamevolume.Theradiusofthesecondcylinderismorethantheradiusofthefirst.Whatistherelationshipbetweentheheightsofthetwocylinders?Problem8Theratioofthelengthtothewidthofarectangleis:.Iftherectanglehasdiagonaloflength,thentheareamaybeexpressedasforsomeconstant.Whatis?Problem9Aboxcontains2redmarbles,2greenmarbles,and2yellowmarbles.Caroltakes2marblesfromtheboxatrandom;thenClaudiatakes2oftheremainingmarblesatrandom;andthenCheryltakesthelast2marbles.WhatistheprobabilitythatCherylgets2marblesofthesamecolor?Problem10Integersandwithsatisfy.Whatis?Problem11Onasheetofpaper,Isabelladrawsacircleofradius,acircleofradius,andallpossiblelinessimultaneouslytangenttobothcircles.Isabellanoticesthatshehasdrawnexactlylines.Howmanydifferentvaluesofarepossible?Problem12Theparabolasandintersectthecoordinateaxesinexactlyfourpoints,andthesefourpointsaretheverticesofakiteofarea.Whatis?Problem13Aleaguewith12teamsholdsaround-robintournament,witheachteamplayingeveryotherteamexactlyonce.Gameseitherendwithoneteamvictoriousorelseendinadraw.Ateamscores2pointsforeverygameitwinsand1pointforeverygameitdraws.WhichofthefollowingisNOTatruestatementaboutthelistof12scores?Problem14Whatisthevalueofforwhich?Problem15Whatistheminimumnumberofdigitstotherightofthedecimalpointneededtoexpressthefractionasadecimal?Problem16Tetrahedronhasand.Whatisthevolumeofthetetrahedron?Problem17Eightpeoplearesittingaroundacirculartable,eachholdingafaircoin.Alleightpeoplefliptheircoinsandthosewhoflipheadsstandwhilethosewhofliptailsremainseated.Whatistheprobabilitythatnotwoadjacentpeoplewillstand?Problem18Thezerosofthefunctionareintegers.Whatisthesumofthepossiblevaluesof?Problem19Forsomepositiveintegers,thereisaquadrilateralwithpositiveintegersidelengths,perimeter,rightanglesatand,,and.Howmanydifferentvaluesofarepossible?Problem20Isoscelestrianglesandarenotcongruentbuthavethesameareaandthesameperimeter.Thesidesofhavelengthsofand,whilethoseofhavelengthsofand.Whichofthefollowingnumbersisclosestto?Problem21Acircleofradiuspassesthroughbothfociof,andexactlyfourpointson,theellipsewithequation.Thesetofallpossiblevaluesofisaninterval.Whatis?Problem22Foreachpositiveinteger,letbethenumberofsequencesoflengthconsistingsolelyofthelettersand,withnomorethanthreesinarowandnomorethanthreesinarow.Whatistheremainderwhenisdividedby12?Problem23Letbeasquareofsidelength1.Twopointsarechosenindependentlyatrandomonthesidesof.Theprobabilitythatthestraight-linedistancebetweenthepointsisatleastis,whereandarepositiveintegersand.Whatis?Problem24Rationalnumbersandarechosenatrandomamongallrationalnumbersintheintervalthatcanbewrittenasfractionswhereandareintegerswith.Whatistheprobabilitythatisarealnumber?Problem25Acollectionofcirclesintheupperhalf-plane,alltangenttothe-axis,isconstructedinlayersasfollows.Layerconsistsoftwocirclesofradiiandthatareexternallytangent.For,thecirclesinareorderedaccordingtotheirpointsoftangencywiththe-axis.Foreverypairofconsecutivecirclesinthisorder,anewcircleisconstructedexternallytangenttoeachofthetwocirclesinthepair.Layerconsistsofthecirclesconstructedinthisway.Let,andforeverycircledenotebyitsradius.Whatis