AIMEI十年真題(附答案)

Problem1AIME2021Calla-digitnumbergeometricifithas distinetdigitswhich,whenreadfromlefttoright,formageometricsequenee.Findthediffereneebetweenthelargestandsmallestgeometricnumbers.Problem2Thereisacomplexnumber-withimaginarypartL64andapositiveintegernsuchthatFind血.Problem3Acointhatcomesupheadswithprobability Wandtailswithprobability- /」independentlyoneachflipisflippedeighttimes.Supposethe1probabilityofthreeheadsandfivetailsisequalto25oftheprobabilityoffiveJJ= headsandthreetails.Let n,wheremandmarerelativelyprimepositiveintegers.Find皿+就.Problem4AM_17InparallelogramABCD,pointAfison月Dsothat丄。1000andpointNAN_17isonADsothatAD2021.LetPbethepointofintersectionofACandAC.Find?T.

Problem5HowmanypositiveintegersZVlessthanlQOOaretheresuchthattheequation—.Vhasasolutionfor■?(Thenotationl/:'.ldenotesthegreatestintegerthatislessthanorequalto站.)Problem75(5十.7胡_1=_!_Thesequenee(d^)satisfies =land 嘰+；JforU>1.Letbetheleastintegergreaterthan]1forwhich/：$isaninteger.Find其.Let-Let-J-二二.Considerallpossiblepositivedifferencesofpairsofelementsof'.Let.?bethesumofallofthesedifferences.Findtheremainderwhenv;isdividedby1J；丿.Problem9AgameshowoffersacontestantthreeprizesA,BandC,eachofwhichisworthawholenumberofdollarsfrom$ 1to$瞅^inclusive.ThecontestantwinstheprizesbycorrectlyguessingthepriceofeachprizeintheorderA,B,C.Asahint,thedigitsofthethreepricesaregiven.Onaparticularday,thedigitsgivenwere1JJ,K孔亠彳.Findthetotalnumberofpossibleguessesforallthreeprizesconsistentwiththehint.Problem10TheAnnualInterplanetaryMathematicsExamination(AIME)iswrittenbyacommitteeoffiveMartians,fiveVenusians,andfiveEarthlings.Atmeetings,committeememberssitataroundtablewithchairsnumberedfromIto15inclockwiseorder.CommitteerulesstatethataMartianmustoccupychairLandanEarthlingmustoccupychair15,Furthermore,noEarthlingcansitimmediatelytotheleftofaMartian,noMartiancansitimmediatelytotheleftofaVenusian,andnoVenusiancansitimmediatelytotheleftofanEarthling.ThenumberofpossibleseatingarrangementsforthecommitteeisFindProblem11Considerthesetofalltriangles JJwhereJistheoriginandPandQaresuchdistinetpointsintheplanewithnonnegativeintegercoordinates

that41工+y=200?Findthenumberofsuchdistincttriangleswhoseareaisapositiveinteger.suchProblem12InrightiXABCwithhypotenuseAB,AC=12,BC=35,andCDisthealtitudetoAU-LetwbethecirclehavingCDasadiameter.LetIbeapointoutside△蟲BCsuchthatA!andBZarebothtangenttocircle4.Theratioofmtheperimeterof△ADJtothelengthABcanbeexpressedintheform施,where筲』and arerelativelyprimepositiveintegers.Find二'十二.Problem13%+2021Thetermsofthesequeneedefinedby 1for■'areThetermsofthesequeneepositiveintegers.Findtheminimumpossiblevalueof ..Problem14350爭=刀云、丨I■',findtheFor■'—-7I,define i,where1 ' .If :and、丨I■',findtheProblem15IntriangleA￡?C,山啟=10,BC=14,andUA=16.LetDbeapointintheinteriorofDC.LetJkand^CdenotetheincentersoftrianglesABDandlr,respectively.Thecircumcirclesoftriangles壬;"?。縜nd■-汀匚meetatdistinctpointsPandDThemaximumpossibleareaof canbeexpressedintheform金_'血,wherea,b,andCarepositiveintegersandcisnotdivisiblebythesquareofanyprime.Find也+b+匕AIME2021Ofthestudentsattendingaschoolparty, 匣?鬲ofthestudentsaregirls,and=Uofthestudentsliketodance.Afterthesestudentsarejoinedby moreboystudents,allofwhomliketodance,thepartyisnow毗%girls.Howmanystudentsnowatthepartyliketodance?SolutionProblem2SquareW：#,■危hassidesoflength1>units.Isoscelestriangle門茫A.FhasbaseEM,andtheareacommontotriangleGEMandsquareAIAIEis80squareunits.FindthelengthofthealtitudetoBMinSolutionProblem3EdandSuebikeatequalandconstantrates.Similarly,theyjogatequalandconstantrates,andtheyswimatequalandconstantrates.Edcovers74kilometersafterbikingfor2hours,joggingfor3hours,andswimmingfor4hours,whileSuecovers91kilometersafterjoggingfor2hours,swimmingfor3hours,andbikingfor4hours.Theirbiking,jogging,andswimmingratesareallwholenumbersofkilometersperhour.FindthesumofthesquaresofEd'sbiking,jogging,andswimmingrates.SolutionProblem4Thereexistuniquepositiveintegers 3Tand期thatsatisfytheequationLc2+84r+2021=護.Find?+^.SolutionProblem5Arightcircularconehasbaseradiusandheight.Theconeliesonitssideonaflattable.Astheconerollsonthesurfaceofthetablewithoutslipping,thepointwherethecone'sbasemeetsthetabletracesacirculararccenteredatthepointwherethevertextouchesthetable.Theconefirstreturnstoitsoriginalpositiononthetableaftermaking Incompleterotations.Thevalueofh/Tcanbewrittenintheform,whereandarepositiveintegersandoriginalpositiononthetableaftermaking Incompleterotations.Thevalueofh/Tcanbewrittenintheform,whereandarepositiveintegersandisnotdivisiblebythesquareofanyprime.FindAtriangulararrayofnumbershasafirstrowconsistingoftheoddintegersIc?.J：：inincreasingorder.Eachrowbelowthefirsthasonefewerentrythantherowaboveit,andthebottomrowhasasingleentry.Eachentryinanyrowafterthetoprowequalsthesumofthetwoentriesdiagonallyaboveitintherowimmediatelyaboveit.Howmanyentriesinthearrayaremultiplesof?Problem7LetSbethesetofallintegers^suchthat<n<lOOfE+L).Forexample,良istheset曲二匚門 .丿孫.Howmanyofthesets:命“上- 左wdonotcontainaperfectsquare?Problem8Findthepositiveintegernsuchthat1 1 1 1arctan- +arrtan - + -十a(chǎn)rrtf-m-=—.3 4 & n 4Problem9Tenidenticalcrateseachofdimensions<ftIft/Lft.Thefirstcrateisplacedflatonthefloor.Eachoftheremainingninecratesisplaced,inturn,flatontopofthepreviouscrate,andtheorientationofeachcrateischosenatrandom.Letnbetheprobabilitythatthestackofcratesisexactly41fttall,wheremandnarerelativelyprimepositiveintegers.Find m.Problem10Let心=；beanisoscelestrapezoidwith山。HwhoseangleattheflIongerbase ；亠is.ThediagonalshavelengthJ-J,andpoint.isatdistances 'and'''fromvertices--and-;,respectively.Let」betheinthefootofthealtitudefromf'toiThedistaneeintheform臨為迢where:?and■.arepositiveintegersandisnotdivisiblebythesquareofanyprime.Find價+他Problem11Considersequencesthatconsistentirelyof .,'sand二'sandthathavethepropertythateveryrunofconsecutive..'shasevenlength,andeveryrunofconsecutive;-'：'shasoddlength.Examplesofsuchsequencesare <■..<，圧：,andAABAA,whileBUABisnotsuchasequenee.Howmanysuchsequenceshavelength14?Problem12Onalongstraightstretchofone-waysingle-lanehighway,carsalltravelatthesamespeedandallobeythesafetyrule:thedistanceformthebackofthecaraheadtothefrontofthecarbehindisexactlyonecarlengthforeach15kilometersperhourofspeedorfractionthereof(Thusthefrontofacartraveling52kilometersperhourwillbefourcarlengthsbehindthebackofthecarinfrontofit.)Aphotoelectriceyebythesideoftheroadcountsthenumberofcarsthatpassinonehour.Assumingthateachcaris4meterslongandthatthecarscantravelatanyspeed,letMbethemaximumwholenumberofcarsthatcanpassthephotoelectriceyeinonehour.Findthequotientwhen Misdividedby10.Problem13Lety)= +nix+ +g刊+砒/+<3占護+ +心迥/+agy3Supposethatpfo.o)=^(i5o)=p(—iso)=r(Q>i)=i)=i)= —i)=P(久2)=0

Thereisapointforallsuchpolynomials,whereThereisapointforallsuchpolynomials,where，芒,and'arepositiveintegers, andarerelativelyprime,and「-1.Find視+*+毗SolutionProblem14LetA￡?beadiameterofcirclea.ExtendABthroughAtoC.Pointflieson-sothatlineJistangentto忖.Point,isthefootoftheperpendicularfrom.Itoline：.,〔Suppose—It,andlet『denotethemaximumpossiblelengthofsegmentBP.Find皿.SolutionProblem15AsquarepieceofpaperhassidesoflengthIm.Fromeachcornerawedgeiscutinthefollowingmanner:ateachcorner,thetwocutsforthewedgeeachstartatdistanee、、、fromthecorner,andtheymeetonthediagonalatanangleof60°(seethefigurebelow).Thepaperisthenfoldedupalongthelinesjoiningtheverticesofadjacentcuts.Whenthetwoedgesofacutmeet,theyaretapedtogether.Theresultisapapertraywhosesidesarenotatrightanglestothebase.Theheightofthetray,thatis,theperpendiculardistaneebetweentheplaneofthebaseandtheplaneformedbytheupperedges,canbewrittenintheformv?n,wheremand池arepositiveintegers,m.<lOU",andisnotdivisiblebythe■'thpowerofanyprime.Find比宀-:―.Problem1AIME2007Problem1AIME2007Howmanypositiveperfectsquareslessthan 1I1aremultiplesof24?Problem2A100footlongmovingwalkwaymovesataconstantrateof6feetpersecond.Alstepsontothestartofthewalkwayandstands.Bobstepsontothestartofthewalkwaytwosecondslaterandstrollsforwardalongthewalkwayataconstantrateof4feetpersecond.Twosecondsafterthat,Cyreachesthestartofthewalkwayandwalksbrisklyforwardbesidethewalkwayataconstantrateof8feetpersecond.Atacertaintime,oneofthesethreepersonsisexactlyhalfwaybetweentheothertwo.Atthattime,findthedistaneeinfeetbetweenthestartofthewalkwayandthemiddleperson.Problem3ThecomplexnumberRisequalto9+加，where6isapositiverealnumberand：嚴(yán)一一i.Giventhattheimaginarypartsof一and一arethesame,whatisequalto?Problem4Threeplanetsorbitastarcircularlyinthesameplane.Eachmovesinthesamedirectionandmovesatconstantspeed.Theirperiodsare60,84,and1-'l:.Thethreeplanetsandthestararecurrentlycollinear.Whatisthefewestnumberofyearsfromnowthattheywillallbecollinearagain?Problem5TheformulaforconvertingaFahrenheittemperature Ftothecorresponding廠匸=_BiCelsiustemperatureis田 AnintegerFahrenheittemperatureisconvertedtoCelsius,roundedtothenearestinteger,convertedbacktoFahrenheit,andagainroundedtothenearestinteger.ForhowmanyintegerFahrenheittemperaturesbetween32and1000inclusivedoestheoriginaltemperatureequalthefinaltemperature?

Problem6Afrogisplacedattheoriginonthenumberline,andmovesaccordingtothefollowingrule:inagivenmove,thefrogadvancestoeithertheclosestpointwithagreaterintegercoordinatethatisamultipleof3,ortotheclosestpointwithagreaterintegercoordinatethatisamultipleof13.Amovesequeneeisasequeneeofcoordinateswhichcorrespondtovalidmoves,beginningwith0,andendingwith39.Forexample,:' 「 『°…上二-'isamovesequenee.Howmanymovesequencesarepossibleforthefrog?Problem71OQ0Let 冷=丄FindtheremainderwhenNisdividedby1000.(〔*」isthegreatestintegerlessthanorequaltok,and貳】istheleastintegergreaterthanorequalto金.)Problem8Thepolynomial'(旳iscubic.Whatisthelargestvalueoffrforwhichthepolynomials⑦何=X2+(fc-?)?一冷andQ誰)=+(2h-4S)a?+karebothfactorsofarebothfactorsofProblem9InrighttriangleABCwithrightangleG,C蟲=30andCB—1&.ItslegsCAandCEareextendedbeyond>!andB.Points01andOzlieintheexteriorofthetriangleandarethecentersoftwocircleswithequalradii.ThecirclewithcenterHistangenttothehypotenuseandtotheextensionofleg 心◎,thecirclewithcenter^istangenttothehypotenuseandtotheextensionofleg,andthecirclesareexternallytangenttoeachother.Thelengthoftheradiusofeithercirclecanbeexpressedas ,where卩and°arerelativelyprimepositiveintegers.Find P+工

Problem10Ina6x4grid(6rows,4columns),12ofthe24squaresaretobeshadedsothattherearetwoshadedsquaresineachrowandthreeshadedsquaresineachcolumn.LetNbethenumberofshadingswiththisproperty.FindtheremainderwhenJVisdividedby1000.Problem11

Foreachpositiveinteger-,let如耗Foreachpositiveinteger-,let如耗denotetheuniquepositiveintegersuch2097that_!.Forexample,'1and匯期I八I'findthat_!.Forexample,theremainderwhen-isdividedby1000.Problem12InisoscelestriangleABC,AislocatedattheoriginandBislocatedat(20,0).Point匚;isinthefirstquadrantwith-I；L；andangle “ 1.Iftriangle型朋rotatedcounterclockwiseaboutpoint.iuntiltheimageofliesonthepositive-axis,theareaoftheregioncommontotheoriginalandtherotatedtriangleisintheform+Qv3+rv6+巴where卩g,真序aretherotatedtriangleisintheformintegers.Find審沐、''m.Problem13Asquarepyramidwithbase止口andvertex/.haseightedgesoflength4.Aplanepassesthroughthemidpointsof.l..,」?；，and①龍.Theplane'sintersectionwiththepyramidhasanareathatcanbeexpressedas亞.FindP.\ E =/"I\//\I /p*l;l.3i%cProblem14

Asequeneeisdefinedovernon-negativeintegralindexesinthefollowingway:伽=血1=3, =厲怛+2007減06+a2O07Findthegreatestintegerthatdoesnotexceed&20辭衛(wèi)〔KUProblem15LetABCbeanequilateraltriangle,andletDandFbepointsonsidesUCandAB,respectively,withFJ1=bandCD=2.Pointi?liesonsideC且suchthatangleDEF=60°.TheareaoftriangleDEFis14\/3.Thetworational,and■isanintegernotdivisiblebythesquareofaprime.Findpossiblevaluesofthelengthofside../.arewhere關(guān)possiblevaluesofthelengthofside../.arewhere關(guān)and玄areProblem1AIME2006Problem1AIME2006Inquadrilateral /LB\sarightangle,diagonalACisperpendicularto—-J一二］and'''IFindtheperimeterof-U；」LetJVbethenumberofconsecutive0'sattherightendofthedecimalrepresentationoftheproduct 1 ；■■■'■■!J1-1!1'.findtheremainderwhen，:『isdividedby1000.Problem5Thenumber/皿蟲+唄“Q十皿+2Wcanbewrittenas'■■1 ‘where'■■1 ‘where:亠遲andarepositiveintegers.Find?Dp.Problem6LetSbethesetofrealnumbersthatcanberepresentedasrepeatingdecimalsoftheform where也』，嗆aredistinctdigits.FindthesumoftheelementsofSProblem7Anangleisdrawnonasetofequallyspacedparallellinesasshown.Theratiooftheareaofshadedregiontotheareaofshadedregion總is11/5.FindtheratioofshadedregionptotheareaofshadedregionA-Problem8HexagonABCDEFisdividedintofiverhombuses, 凡場and兀asshown.Rhombuses只Q,R?andSarecongruent,andeachhasarea#2006.LetJfbetheareaofrhombusT.GiventhatKisapositiveinteger,findthenumberofpossiblevaluesforK.Problem9Thesequenee厲1，厲厶，…isgeometricwith =aandcommonratio幾whereand■'arepositiveintegers.Giventhatfindthenumberofpossibleorderedpairs〔血Problem10Eightcirclesofdiameter1arepackedinthefirstquadrantofthecoordinteplaneasshown.Letregion^betheunionoftheeightcircularregions.Line'withslope3,dividesintotworegionsofequalarea.Line''sequationcanbe

expressedintheform:洱—■搶==LwhereM；￡and arepositiveintegerswhosegreatestcommondivisoris1.FindProblem11Acollectionof8cubesconsistsofonecubewithedge-length ■-foreachinteger—違初匚3atoweristobebuiltusingall8cubesaccordingtotherules:Anycubemaybethebottomcubeinthetower.Thecubeimmediatelyontopofacubewithedge-lengthkmusthaveedge-lengthatmostfe+2.Let/bethenumberofdifferenttowersthancanbeconstructed.Whatistheremainderwhen?isdividedby1000?Problem12Findthesumofthevaluesof'suchthat:⑴丄：丨-川一汽宀*whereismeasuredindegreesand||-|：'1■■-■-Problem13Foreachevenpositiveinteger 逸let即曲Senotethegreatestpowerof2thatdividesX'-Forexample,9〔卻〕=&and刃序〕=皿Foreachpositiveinteger

Se=Xg(2k\let Jfe=i Findthegreatestintegernlessthan1000suchthatSn.\saperfectsquare.Problem14Atripodhasthreelegseachoflength5feet.Whenthetripodissetup,theanglebetweenanypairoflegsisequaltotheanglebetweenanyotherpair,andthetopofthetripodis4feetfromthegroundInsettingupthetripod,thelower1footofonelegbreaksoff.Letftbetheheightinfeetofthetopofthetripodfromthegroundwhenthebrokentripodissetup.ThenHcanbewrittenmintheform嚴(yán)"wheremand打arepositiveintegersandnisnotdivisiblebythesquareofanyprime.Find陰十魯那么thesquareofanyprime.Find陰十魯那么「-(ThenotationIdenotesthegreatestintegerthatislessthanorequalto )Problem15Giventhatasequeneesatisfies—and Jr■-'sforallintegersGiventhatasequeneesatisfiesnfindtheminimumpossiblevalueofAIME2005Sixcirclesformaringwithwitheachcircleexternallytangenttotwocirclesadjacenttoit.Allcirclesareinternallytangenttoacircle<withradius30.Let.、betheareaoftheregioninsidecircle匸andoutsideofthesixcirclesinthering.FindProblem2Foreachpositiveinteger^let*Sfrdenotetheincreasingarithmeticsequeneeofintegerswhosefirsttermis1andwhosecommondiffereneeis Forexample,''isthesequenee Forhowmanyvaluesofdoes--containtheterm2005?Problem3Howmanypositiveintegershaveexactlythreeproperdivisors,eachofwhichislessthan50?Problem4Thedirectorofamarchingbandwishestoplacethemembersintoaformationthatincludesallofthemandhasnounfilledpositions.Iftheyarearrangedinasquareformation,thereare5membersleftover.Thedirectorrealizesthatifhearrangesthegroupinaformationwith7morerowsthancolumns,therearenomembersleftover.Findthemaximumnumberofmembersthisbandcanhave.Problem5Roberthas4indistinguishablegoldcoinsand4indistinguishablesilvercoins.Eachcoinhasanengravingofonefaceononeside,butnotontheother.Hewantstostacktheeightcoinsonatableintoasinglestacksothatnotwoadjacentcoinsarefacetoface.Findthenumberofpossibledistunguishablearrangementsofthe8coins.Problem8Theequation嚴(yán)九_2+2111r+2=護透和+Lhasthreerealroots.Giventhatmtheirsumisnwhere^andnarerelativelyprimepositiveintegers,findm+n.Problem9Twentysevenunitcubesarepaintedorangeonasetoffourfacessothattwonon-paintedfacesshareanedge.The27cubesarerandomlyarrangedtoforma直過丫汎/cube.Giventheprobabilityoftheentiresurfaceareaofthelargercubeisorangeis1'where護孑忌andaredistinctprimesand甌兔andarepositiveintegers,find心?"再+總"巴Problem10Triangle:口liesintheCartesianPlaneandhasanareaof70.ThecoordinatesofBandCare爲(wèi)Wnd⑺孤現(xiàn)^respectively,andthecoordinatesofAare °>ThelinecontainingthemediantosideBChasslope■氐FindthelargestpossiblevalueofP十殳Problem11AsemicirclewithdiameteriscontainedinasquarewhosesideshavelengthGiventhemaximumvalueofAsemicirclewithdiameteriscontainedinasquarewhosesideshavelengthGiventhemaximumvalueof^ism'*^find +乩精選Forpositiveintegerslet ：〔denotethenumberofpositiveintegerdivisorsForpositiveintegersof冷including1andn-Forexample,廠(D=】and?、?4“DefineS(n)byH〒仁門珂H〒仁門珂2；；珂匚Let「denotethenumberofpositiveintegersS<■'1.'|k-with幻學(xué)屯S<■'1.'|k-with幻學(xué)屯odd,andlet1.Fromanylatticepoint''theparticlemayonlymovetoi1 1 or:denotethenumberofpositiveintegers?V』〕with妙他even.Find -「IProblem13AparticlemovesintheCartesianPlaneaccordingtothefollowingrules:(d+1』+1).2.Therearenorightangleturnsintheparticle'sHowmanydifferentpathscantheparticletakefromProblem14Considerthepoints月?調(diào)*占(1仏 。)乜nd 3ThereisauniquesquareSsuchthateachofthefourpointsisonadifferentsideofS-LetKbetheareaofS.FindtheremainderwhenLOKisdividedby1000.Problem15Triangle匚上；has匕(："Theincircleofthetriangleevenlytrisectsthemedian.Ulftheareaofthetriangleis巴從￡爲(wèi)where'and'■areintegersandisnotdivisiblebythesquareofaprime,find民」諄AIME2004Thedigitsofapositiveintegernarefourconsecutiveintegersindecreasingorderwhenreadfromlefttoright.Whatisthesumofthepossibleremainderswhenisdividedby37?Problem2SetNeonsistsofmconsecutiveintegerswhosesumis2m*andsetUconsistsof2tticonsecutiveintegerswhosesumis血Theabsolutevalueofthediffereneebetweenthegreatestelementof AandthegreatestelementofCis99.Findm.Problem3Aconvexpolyhedron丿has26vertices,60edges,and36faces,24ofwhicharetriangular,and12ofwhicharequadrilaterals.Aspacediagonalisalinesegmentconnectingtwonon-adjacentverticesthatdonotbelongtothesameface.Howmanyspacediagonalsdoes .have?Problem4Asquarehassidesoflength2.Set"isthesetofalllinesegmentsthathavelength2andwhoseendpointsareonadjacentsidesofthesquare.Themidpointsofthelinesegmentsinset$enclosearegionwhoseareatothenearesthundredthish.Find100民Problem5AlphaandBetabothtookpartinatwo-dayproblem-solvingcompetition.Attheendofthesecondday,eachhadattemptedquestionsworthatotalof500points.Alphascored160pointsoutof300pointsattemptedonthefirstday,andscored140pointsoutof200pointsattemptedonthesecondday.Betawhodidnotattempt300pointsonthefirstday,hadapositiveintegerscoreoneachofthetwodays,andBeta'sdailysuccessrate(pointsscoreddividedbypointsattempted)oneachdaywaslessthanAlpha'sonthatday.Alpha'stwo-daysuccessratiowas300/500=3/5.Thelargestpossibletwo-dayprimepositiveintegers.WhatisProblem6Anintegeriscalledsnakelikeifitsdecimalrepresentationsatisfies仮 偽4丄ifsisoddand論A心44if￡iseven.Howmanysnakelikeintegersbetween1000and9999havefourdistinetdigits?Problem7LetCbethecoefficientof ntheexpansionoftheproduct〔1-0f〕〔l+對〕〔1-錨〕…〔1+14b〕〔1-lfeJ-FindPI-Problem8Definearegular'-pointedstartobetheunionoflinesegments'二「 ：suchthatthepointsepe%■??*carecoplanarandnothreeofthemarecollinear,eachofthewlinesegmentsintersectsatleastoneoftheotherlinesegmentsatapointotherthananendpoint,alloftheanglesatPxFg,熱arecongruent,allofthenlinesegments丹?■、RtRarecongruent,andthepathR丹.B兔…弓PnPiturnscounterclockwiseatanangleoflessthan180degreesateachvertex.Therearenoregular3-pointed,4-pointed,or6-pointedstars.Allregular5-pointedstarsaresimilar,buttherearetwonon-similarregular7-pointedstars.Howmanynon-similarregular1000-pointedstarsarethere?Problem9LetG召叮beatrianglewithsides3,4,and5,andf匚bea6-by-7rectangle.Asegmentisdrawntodividetriangle m；intoatriangleandatrapezoidlandanothersegmentisdrawntodividerectangleUmintoatrianglefandatrapezoid'i-^suchthatlUissimilarto〔3：and\issimilarto旳.TheminimumvalueoftheareaofUicanbewrittenintheform where‘a(chǎn)nd晞arerelativelyprimepositiveintegers.Find「匸+口〞Problem10Acircleofradius1israndomlyplacedina15-by-36rectangle :、msothatthecircleliescompletelywithintherectangle.Giventhattheprobabilitythatthecirclewillnottouchdiagonal磁、；is' '■wherethecirclewillnottouchdiagonal磁、；is' '■where:譏andarerelativelyprimepositiveintegers.Find:飛*匚“Problem11Asolidintheshapeofarightcircularconeis4inchestallanditsbasehasa3-inchradius.Theentiresurfaceofthecone,includingitsbase,ispainted.Aplaneparalleltothebaseoftheconedividestheconeintotwosolids,asmallercone-shapedsolidCandafrustum-shapedsolid^insuchawaythattheratiobetweentheareasofthepaintedsurfacesof CandFandtheratiobetweenthevolumesofGandFarebothequalto Giventhat電=悔仏where筲』and arerelativelyprimepositiveintegers,find'：―Problem12Let*bethesetoforderedpairs也丄；Let*bethesetoforderedpairs也丄；suchthatV、■〔 、丄and［隔〔［logsI牛j|and arebotheven.Giventhattheareaofthegraphofwheref&and■-arerelativelyprimepositiveintegers,findnotation denotesthegreatestintegerthatislessthanorequalto

ThepolynomialPM=(L+ThepolynomialPM=(L+摩+滬+…-ha:17)3—xlThas34complexrootsoftheform蠢=尸血［皿^{2喬aj=1、盂3??■、34側(cè)曲：'：<二■.<：去￡二」5、…二<■''and知“Giventhat,,;i'■■'-'：，!'"I':匕.■r-wheremandwarerelativelyprimepositiveintegers,findm+mProblem14Aunicornistetheredbya20-footsilverropetothebaseofamagician'scylindricaltowerwhoseradiusis8feet.Theropeisattachedtothetoweratgroundlevelandtotheunicornataheightof4feet.Theunicornhaspulledtheropetaut,theendoftheropeis4feetfromthenearestpointonthetower,anda—thelengthoftheropethatistouchingthetoweris ■feet,where汽列andarepositiveintegers,andisprime.Find二—亠巴Problem15rl ifx=1=<磊 ifxisdivisibleby10Forallpositiveintegers^,let^x+1otherwise anddefineasequeneeasfollows: X1=n；and陽41=/(^n-)forallpositiveintegers■'.Let知*—bethesmallestsuchthatr■■-—丄.(Forexample,'=‘‘a(chǎn)nd'厘強:!：".)Let bethenumberofpositiveintegers ?：suchthat貞對=-游.Findthesumofthedistinctprimefactorsof皿.SolutionAIME2003Giventhat即=5where；=and讓arepositiveintegersandkisaslargeaspossible,find'上口Problem2OnehundredconcentriccircleswithradiihN ?§100aredrawninaplane.Theinteriorofthecircleofradius1iscoloredred,andeachregionboundedbyconsecutivecirclesiscoloredeitherredorgreen,withnotwoadjacentregionsthesamecolor.Theratioofthetotalareaofthegreenregionstotheareaofthecircleofradius100canbeexpressedas'■where江』thecircleofradius100canbeexpressedas'■where江』and arerelativelyprimepositiveintegers.Find^<r-—"■Problem3Lettheset5=Lettheset5={8,5,1,13,H3,21,2}Susanmakesalistasfollows:foreachtwo-elementsubsetof^shewritesonherlistthegreaterofthes