imo预选题1987(shortlist)

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1、ProblemsShortlistedtothe1987IMOJuryHavana,Cuba1.Letfbeafunctionthatsatisfiesthefollowingconditions:(a)ifx>yandf(y)¡y¸v¸f(x)¡xthenf(z)=v+zforsomez2(x;y).(b)Theequationf(x)=0hasatleastonesolutionandamongallsolutionsthereisonethatismaximal.(c)f(0)=1.(d)f(19

2、87)·1988.(e)f(x)f(y)=f(xf(y)+yf(x)¡xy)forallx;y2R.Findf(1987).2.Atapartyattendedbyncoupleseachpersontalkstoeveryoneelseatthepartyexcepthisorherspouse.TheconversationsinvolvecliquesC1;:::;Ckwiththepropertythatnocoupleisinasameclique,butforeveryotherpairo

3、fpersonsthereisacliquetowhichitbelongs.Provethatn¸4;k¸2n.3.Doesthereexistaseconddegreepolynomialp(x;y)intwovariablessothatforeverynonnegativeintegernthereareuniquenonnegativek;msothatp(k;m)=n?4.LetABCDEFGHbeaparallelepipedwithAEjjBFjjCGjjDH.ProvethatAF+

4、AH+AC·AB+AD+AE+AG:Whendoesequalityhold?5.FindPinsidetheacute-angledtriangleABCforwhichBL2+CM2+AN2isminimal,whereL;M;NarethefeetoftheperpendicularsfromPtoBC;CA;AB.6.Showthatisa;b;carethelengthsofthesidesofatriangleandpisthesemiperimeterthenµ¶anbncn2n¡2n¡

5、1++¸p;b+cc+aa+b3(n¸1).7.Considerfiverealnumbersu1;u2;u3;u4;u5.ProvethatonecanfindfiverealnumbersPv;v;:::;vsothatu¡varenonnegativeintegersforalliand(v¡v)2<4.125iii

6、fthetypeaibj(0·i·m;0·j·k)givethesameremaindermodulomk.(b)Let(m;k)>1.Provethatforanyintegersa1;a2;:::;amandb1;b2;:::;bkthereexisttwodistinctpairs(i;j);(u;v)(0·i;u·m;0·j;v·k)sothataibj´aubv(modmk).19.DoesthereexistasetSinR3sothatanyplaneintersectsSinanone

7、mptybutfinitesetofpoints?10.LetS1;S2betwospheresofdistinctradiiwhichareexternallytangent.Thespheresareinsideaconeandaretangenttoitsothattheintersectionbetweentheconeandanyofthespheresisacircle(andnotonlyapoint).Therearensolidspheresarrangedinaringsothate

8、achlittlespheretouchestheconeandbothS1andS2andanytwoneighboringlittlespheresareexternallytangent.Findthepossiblevaluesofn.11.Findthenumberofpartitionsofthesetf1;:::;ngintothreesubsetsA1;A2;A3(notnecessarilynonempty)sothat:(a)Afte