IMO_1969_partial_longlist_problem.pdf

(144 KB) Pobierz
68558419 UNPDF
IMOLonglist1969
IMOShortList/LongListProjectGroup
June19,2004
1. (Belgium1)Inanorthogonalsystemwehavetwoparabolaswithequations:
P 1 : x 2 −2py=0
P 2 : x 2 +2py=0
withp>0.ThelinetisatangentlinetoP 2 .DeterminethelocusofM,poleoftwithrespect
toP 1 .
2. (Belgium2)Findtheequationoftheequilateralhyperboleswhichpassthroughthepoints
A(,0),B(,0),C(0,).ShowthatthehyperbolespassthroughtheorthocenterHoftriangle
ABC.Findthelocusofthecentersofthosehyperboles.Andverifythatthislocuscoincideswit
thenine-pointcircleoftriangleABC.
3. (Belgium3)ThreecirclesC 1 ,C 2 ,C 3 haveonlyonepointincommon.Constructacirclewhich
istangenttothegiventhreecircles.
4. (Belgium4)AconicispassingthroughtheoriginO.ArightangleisOintersectstheconicin
pointsAandB.ProvethatthelineABpassesthroughafixedpointwhichissituatedonthe
normalonOattheconic.
5. (Belgium5)LetGbethecenterofgravityofagiventriangleOAB.Showthattheconics
whichpassthroughthepointsO,A,B,Garehyperboles.Findthelocusofthecentersofthose
hyperboles.
6. (Belgium6)Calculate cos
4
+i·sin
4
10
intwodierentwaysandconcludethat
C 1 10 −C 3 10 + 1
3 C 5 10 =2 4 .
GlobalRemark:InthosedaysanalyticgeometrywerethehighlightsinthecurriculaofBelgium.
Allthequestions,especiallythesixthquestion,arequestionswhichappearedinhandbooks.
7. (Bulgaria1)Forallnaturali>2,leth i betheinradiusofaregularn-goninscribedintoacircle
withradiusR.Provethat(n+1)h n+1 −nh n >Rforeverynaturaln>2.
RemarkByOrlando:ThisproblemwasaddedtotheIMOLongListfromoneoftheMoro-
zova/Petrakovvolumes.Thelogicalorderhowtheproblemsarelistedinthebookmakesme
assumethattheproblemwasproposedfortheIMO1969.
1
8. (Bulgaria2)Provethattheequation
x 3 +y 3 +z 3 =1969 2
hasnosolutionsinintegernumbers.
RemarkByOrlando:ThisproblemwasaddedtotheIMOLongListfromoneoftheMoro-
zova/Petrakovvolumes.Thelogicalorderhowtheproblemsarelistedinthebookmakesme
assumethattheproblemwasproposedfortheIMO1969.
9. (Bulgaria3)Findallfunctionsf(x)definedforeveryxandsatisfyingtheequation
xf(y)+yf(x)=(x+y)f(x)f(y)
forarbitraryxandy.Showthatonlytwosuchfunctionsarecontinuous.
RemarkByOrlando:ThisproblemwasaddedtotheIMOLongListfromoneoftheMoro-
zova/Petrakovvolumes.Thelogicalorderhowtheproblemsarelistedinthebookmakesme
assumethattheproblemwasproposedfortheIMO1969.
10. (Bulgaria4)Onaplane,considerasquarewithsidelength38cm,and100convexpolygons,
assumingthattheareaofsuchapolygonisalwayscm 2 ,andtheperimeterofsuchapolygon
isalways2cm.Provethatthereexistsacirculardiskofradius1cmhavingnocommonpoint
withanyofthese100polygons,butcompletelylyingintheinteriorofthesquare.
RemarkByOrlando:ThisproblemwasaddedtotheIMOLongListfromoneoftheMoro-
zova/Petrakovvolumes.Thelogicalorderhowtheproblemsarelistedinthebookmakesme
assumethattheproblemwasproposedfortheIMO1969.
11. (France1)Letaandbtwopositiveintegers.LetH(a,b)bethesetofnumbersnsatisfying
n=pa+qbwithpandqbeingpositivenumbersincludingzero.DefineH(a)=H(a,a).Show
that,ifa6=bwithaandbhavingnocommondivisor,itissucienttoknowaparticularH(a,b)
inordertoknowallothersetsH(a,b).Showthatinthiscase,thesetHcontainsallthenumbers
greaterthanorequalto
w=(a−1)(b−1)
2 othernumbers(exceptionoccursifw=0).
12. (France2)Letnbeaninteger,whichhasonly1asdivisor,thatisasquare.Inthenumber
systemwithbasen,weconsiderthelastdigitx 1 ofanintegersmallerthannandweconsider
alsothelastdigits(denotedbyx m )oftheintegerpowersx m ofthatnumber.Forwhichnumbers
xdowehavex m =0?Showthatthesequencex m hasaperiodt.Forwhichnumbersxdowe
havex t =1?Showthatifm,nandthavenocommondivisorexcept1,thenumbers
0 m ,1 m ,2 m ,...,(n−1) m
arealldierentonefromanother.Determinetheminimalconditionvalueoftasafunctionofn.
Ifndoesnotsatisfytheaboveconditions,showthatitispossibletohavex m =06=x 1 andthat
thesequenceisperiodicbutonlystartingfromacertainvaluek,independentfromx.
13. (France3)Givenapolygon,notnecessarilyconvex,withareaS,ofwhichtheverticeshave
integercoordinates,notnecessarilypositive.IfIisthenumberofpointsintheinteriorofthe
polygonandBthenumberofpointsontheborder,determinetheinteger
T=2S−B−2I+2
cbyOrlandoD¨ohring,memberoftheIMOShortList/LongListProjectGroup,page2/12
andalso w
 
.
14. (France4)WehavearighttriangleOAB( ˆ B=90 )andacirclewithcenterlocatedon[OB],
tangentonOA.FromAtwotangentscanbedrawnatthegivencircle.LetTbethecontact
withthecircle,dierentfromthatwithOA.Showthatthemedian,lineofgravity,fromBinthe
triangleOABintersectATinapointMsuchthat
|MB|=|MT|.
,andanotherrelationamong(n),(n−3)and(n).
Finallyrepresent(n)asaresultofthevaluesoftherestofthedivisionofnby6.Whatcanbe
saidabout(n)andthenumber1+ n(n+6)
12 ?Determine
thenumberoftriplets(x,y,z)inafunctiondivisibleby6.Whatcanbesaidaboutthisnumber
regarding (n+6)(2n 2 +9n+12)
72
12 ,and(n)andthenumber (n+3) 2
?
GlobalRemark:AllthesequestionsaremadebyMr.Warusfel,whowasafanaticsupporterof
thenewmath.SecondlyhefollowstheFrenchtraditionofthequestionsfortheexam”Compo-
sitiondesMathem´ematiques”andthismeansthateveryquestionisacomplexinvestigationofa
problemandnotaproblemonitsown.
16. (GreatBritain1)Considerthepolynomial
P(x)=a 0 x k +a 1 x k−1 +...+a k
witha 0 ,a 1 ,...,a k beingintegers.SuchapolynomialissaidtobedivisiblebymifP(x)isa
multipleforeveryintegerx.Showthata 0 k!isamultipleofmifP(k)isdivisiblebym.Prove
alsothefollowingproperty:Ifa 0 ,kandmarepositiveintegerswitha 0 k!amultipleofm,thenit
ispossibletofindapolynomialdivisiblebymofwhicha 0 x k hasthehighestdegree.
(DarijGrinberg’stranslationfromMorozova/Petrakov:)Apolynomial
P(x)=a 0 x k +a 1 x k−1 +...+a k ,
wherea 0 ,a 1 ,...,a k areintegers,issaidtobedivisiblebym,ifmdividesthenumberP(x)for
everyintegerx.ShowthatifthepolynomialP(x)isdivisiblebym,thenthenumberk!a 0 is
divisiblebym.Alsoshowthatifthenumbersa 0 ,kandmarepositiveintegerssuchthatk!a 0 is
divisiblebym,thenyoucanalwaysfindapolynomial
P(x)=a 0 x k +a 1 x k−1 +...+a k
divisiblebym.
17. (GreatBritain2)Givenarethenegativeintegersa,b,x,ywiththeassumptionthata,bhave
nocommonfactor.Provethatab−a−brepresentsthegreatestnumberwhichcannotbewritten
intheformax+by.
cbyOrlandoD¨ohring,memberoftheIMOShortList/LongListProjectGroup,page3/12
simplerelationbetween(n)and n+2
2
15. (France5)Let(n)bethenumberofintegerpairs(x,y)suchthatx+y=n,0yx.Let
(n)bethenumberintegertriplets(x,y,z)suchthatx+y+z=n,0zyx.Finda
 
18. (GreatBritain3)Considerasolidcomposedofarightcircularcylinder,withheighthand
radiusr,andthehemisphereplaceonthetopofit,radiusrandcenterO.Thatsolidisplacedon
ahorizontaltable.Awireisattachedononesideatapointofthebaseofthecylinder,stretched
verticallyandattachedatthetopatpointPofthehemisphere.LetbetheanglewhichOP
makeswiththehorizontaltable.Showthat,ifissucientlysmallandthewireisalittlebit
displaced,notanymoreintheverticalplane,thewireiscomingloose.
Remark:Seemstoberatherascienceolympiadproblemthanamathematicalolympiadproblem.
19. (GreatBritain4)LetABbeperpendiculartoCDinapointXinsuchawaythat|CX|=|XD|.
ShowthatwithsomerestrictionsthereexistsatruncatedconeonwhichA,B,C,Dareranged.
Whatarethoserestrictions?
20. (GreatBritain5)Wehaveu 0 =1,u 1 =1andu n+2 =au n+1 +bu n ,n0,withaandbbeing
naturalnumbers.Writeu n asapolynomialinaandbandproveyourresult.Ifbisaprime,show
thatbdividesa(u 9 −1).
21. (GreatBritain6)GivenapointOandthreelengthsx,y,z.Showthatanequilateraltriangle
ABCwithverticessatisfyingOA=x,OB=y,OC=zexistsifandonlyiftheinequalities
y+zx,z+xy,x+yzarefulfilled.(Hereby,thepointsO,A,B,Caresupposedtolie
ononeplane.)
RemarkByOrlando:ThisproblemwasaddedtotheIMOLongListfromoneoftheMoro-
zova/Petrakovvolumes.Thelogicalorderhowtheproblemsarelistedinthebookmakesme
assumethattheproblemwasproposedfortheIMO1969.
GlobalRemark:ThequestionproposedbyGreatBritainare-inthewhole-ratherpeculiar.The
questionsaretotallyinfluencedbythetraditionsoftheA−levelexamsinEngland.Itmust
honestlybesaidthattheproposalsofalotofwesterncountrieswereanalogoustothecurriculum
situationintheircountry.
22. (Hungary1)
126.InanisoscelestriangleABC,wehaveAB=ACand ] BAC=20 .LetDbethepointon
theedgeABsuchthatAD=CD,andletEbethepointontheedgeACsuchthatBC=CE.
Find ] CDE.
RemarkByOrlando:ThisproblemwasaddedtotheIMOLongListfromoneoftheMoro-
zova/Petrakovvolumes.Thelogicalorderhowtheproblemsarelistedinthebookmakesme
assumethattheproblemwasproposedfortheIMO1969.
23. (Hungary2)Provetheinequalityforanyn2 N
n X
k 3 < 5
4 .
k=1
RemarkByOrlando:ThisproblemwasaddedtotheIMOLongListfromoneoftheMoro-
zova/Petrakovvolumes.Thelogicalorderhowtheproblemsarelistedinthebookmakesme
assumethattheproblemwasproposedfortheIMO1969.
cbyOrlandoD¨ohring,memberoftheIMOShortList/LongListProjectGroup,page4/12
1
68558419.001.png
 
24. (Hungary3)Given4000pointsintheplanesuchthatnothreeofthesepointsarecollinear.
Showtheexistenceof1000pairwisedisjunctquadrilateralswhoseverticesarethegivenpoints.
(Twoquadrilateralsarecalleddisjunctifthereisnopointlyingintheinteriorofbothquadrilat-
erals.)
RemarkByOrlando:ThisproblemwasaddedtotheIMOLongListfromoneoftheMoro-
zova/Petrakovvolumes.Thelogicalorderhowtheproblemsarelistedinthebookmakesme
assumethattheproblemwasproposedfortheIMO1969.
25. (Mongolia1)Findthecircumradiusoftheisoscelestrianglewhosesidelengthsaretherootsof
aquadraticequationx 2 −ax+b=0.
RemarkByOrlando:ThisproblemwasaddedtotheIMOLongListfromoneoftheMoro-
zova/Petrakovvolumes.Thelogicalorderhowtheproblemsarelistedinthebookmakesme
assumethattheproblemwasproposedfortheIMO1969.
26. (Netherlands1)Theverticesofapolygonwith(n+1)sidesarechosenontheperimeterof
agivenregularpolygonwithnsidesinsuchawaythattheydividetheperimeterintwoequal
parts.Howcanthisbedonesothattheareaofthepolygonwith(n+1)sidesis(i)maximaland
(ii)minimal.
(DarijGrinberg’stranslationfromMorozova/Petrakov:)Consideran(n+1)-gonwith
verticeslyingonthesidesofafixedregularn-gonanddividingtheperimeterofthisn-goninto
n+1equalparts.[Ofcourse,twocertainverticesofthe(n+1)-gonwillhavetolieononeside
ofthen-gon.]Howshouldtheverticesofthe(n+1)-gonbeplacedonthesidesofthen-gonin
ordertohavetheareaofthe(n+1)-gon
a)maximal?
b)minimal?
27. (Netherlands2)AsemicirculararcisdrawnonABasdiameter.Cisapointonotherthan
AandB,andDisthefootoftheperpendicularfromCtoAB.Weconsiderthethreecircles
1 , 2 , 3 ,alltangenttothelineAB.Ofthese, 1 isinscribedintriangleABC,while 2 and 3
arebothtangenttoCDandto,oneoneachsideofCD.Provethat 1 , 2 , 3 haveasecond
tangentincommon.
Remark:ThisquestionwaschosenasthefourthIMOquestion.
cbyOrlandoD¨ohring,memberoftheIMOShortList/LongListProjectGroup,page5/12

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin