罗马尼亚IMO国家队选拔考试2005

April2005

RomanianIMOTST2005

DayI

1.Solvetheequation3x=2xy+1inpositiveintegers.

2.Letn≥1beanintegerandletXbeasetofn2+1positiveintegerssuchthatinanysubsetofXwithn+1elementsthereexisttwoelementsx=ysuchthatx|y.Provethatthereexistsasubset{x1,x2,...,xn+1}∈Xsuchthatxi|xi+1foralli=1,2,...,n.3.Provethatifthedistancefromapointinsideaconvexpolyhedrawithnfacestotheverticesofthepolyhedraisatmost1,thenthesumofthedistancesfromthispointtothefacesofthepolyhedraissmallerthann−2.

Workingtime:4hours

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RomanianIMOTST2005

DayII

4.Provethatinanyconvexpolygonwith4n+2sides(n≥1)thereexisttwoconsecutive

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sideswhichformatriangleofareaatmostoftheareaofthepolygon.

6n5.Letm,nbeco-primeintegers,suchthatmisevenandnisodd.Provethatthefollowingexpressiondoesnotdependonthevaluesofmandn:

󰀂mk1+(−1)[n]2n

k=1n−1

󰀃

mkn

󰀁

.

6.Asequenceofrealnumbers{an}niscalledabssequenceifan=|an+1−an+2|,foralln≥0.Provethatabssequenceisboundedifandonlyifthefunctionfgivenbyf(n,k)=anak(an−ak),foralln,k≥0isthenullfunction.

Workingtime:4hours

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RomanianIMOTST2005

DayIII

7.LetA0A1A2A3A4A5beaconvexhexagoninscribedinacircle.DefinethepointsA󰀂0,

󰀂A󰀂2,A4onthecircle,suchthat

A0A󰀂0󰀌A2A4,

A2A󰀂2󰀌A4A0,

A4A󰀂4󰀌A2A0.

󰀂󰀂LetthelinesA󰀂0A3andA2A4intersectinA3,thelinesA2A5andA0A4intersectin

󰀂󰀂A󰀂5andthelinesA4A1andA0A2intersectinA1.

ProvethatifthelinesA0A3,A1A4andA2A5areconcurrentthenthelinesA0A󰀂3,󰀂A4A󰀂1andA2A5arealsoconcurrent.8.LetABCbeatriangle,andletD,E,Fbe3pointsonthesidesBC,CAandAB

respectively,suchthattheinradiiofthetrianglesAEF,BDFandCDEareequalwithhalfoftheinradiusofthetriangleABC.ProvethatD,E,FarethemidpointsofthesidesofthetriangleABC.9.LetPbeapolygon(notnecessarilyconvex)withnvertices,suchthatallitssidesanddiagonals√arelessorequalwith1inlength.Provethattheareaofthepolygon

3islessthan.

2

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RomanianIMOTST2005

DayIV

10.Leta∈R−{0}.Findallfunctionsf:R→Rsuchthatf(a+x)=f(x)−xforall

x∈R.

DanSchwartz

11.Ontheedgesofaconvexpolyhedrawedrawarrowssuchthatfromeachvertexat

leastanarrowispointinginandatleastoneispointingout.Provethatthereexistsafaceofthepolyhedrasuchthatthearrowsonitsedgesformacircuit.

DanSchwartz

12.Letn≥0beanintegerandletp≡7(mod8)beaprimenumber.Provethat

p−1󰀃2n󰀂kk=1

1

−p2

󰀁

=

p−1

.2

C˘alinPopescu

13.a)Provethatthereexistsasequenceofdigits{cn}n≥1suchthatoreachn≥1no

matterhowweinterlacekndigits,1≤kn≤9,betweencnandcn+1,theinfinitesequencethusobtaineddoesnotrepresentthefractionalpartofarationalnumber.

b)Provethatfor1≤kn≤10thereisnosuchsequence{cn}n≥1.

DanSchwartz

Workingtime:4hours

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RomanianIMOTST2005

DayV

14.Ona2004×2004chesstablethereare2004queenssuchthatnotwoareattacking

eachother1.

Provethatthereexisttwoqueenssuchthatintherectangleinwhichthecenterofthesquaresonwhichthequeensliearetwooppositecorners,hasasemiperimeterof2004.

15.Letn≥2beaninteger.Findthesmallestrealvalueρ(n)suchthatforanyxi>0,

i=1,2,...,nwithx1x2···xn=1,theinequality

nn󰀂󰀂1

≤xrixii=1

i=1

istrueforallr≥ρ(n).

16.LetN={1,2,...}.Findallfunctionsf:N→Nsuchthatforallm,n∈Nthe

numberf2(m)+f(n)isadivisorof(m2+n)2.17.Weconsiderapolyhedrawhichhasexactlytwoverticesadjacentwithanoddnumber

ofedges,andthesetwoverticesarelyingonthesameedge.

Provethatforallintegersn≥3thereexistsafaceofthepolyhedrawithanumberofsidesnotdivisiblebyn.

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twoqueensattackeachotheriftheylieonthesamerow,columnordirectionparallelwithonofthemaindiagonalsofthetable

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