imo预选题1989(shortlist)

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1、ProblemsShortlistedtothe1989IMOJury1.AliBabahasarectangularpieceofcarpet.Hefindsthatifhelaysifflatonthefloorofeitherofhisstoreroomstheneachcornerofthecarpettouchesadifferentwalloftheroom.Heknowsthatthestoreroomshavewidths38and50feetandthesame(unknownbutintegral)leng

2、th.Whatarethedimensionsofthecarpet?2.Provethatforn>1thepolynomial1n1n¡111P(x)=x+x+¢¢¢+x+n!(n¡1)!1!0!hasnorationalroots.3.ThepolynomialP(x)=xn+nxn¡1+axn¡2+¢¢¢+ahasnrootswhose16thpowers20havesumn.Findtheseroots.4.TheanglebisectorsofthetriangleABCmeetthecircumcircl

3、eagainatA0B0C0.Showthat0003416[ABC]¸27[ABC]RwhereRisthecircumradiusofABCand[XYZ]representstheareaofXYZ.5.ShowthatanytwopointsPandQinsidearegularn-goncanbejoinedbytwocirculararcsPQdwhichlieinsidethen-gonandmeetatanangle¸(1¡2)¼.n6.TherectangleRiscovereddisjointly(

4、disjointinteriors)withafinitenumberofrect-anglesR1;R2;:::;RnsuchthatRi½RforalliandthesidesofRiareparalleltothesidesofRforalli.IfeachRihasasideofintegrallengthshowthatRhasasideofintegrallength.pppp33n337.Foreachn>0write(1+42¡44)=an+bn2+cn4,wherean;bn;cn2Z.Showthat

5、cn6=0.8.Letg:C¡!Candlet!6=1beacuberootof1.Forv2Cfindafunctionf:C¡!Csuchthatf(z)+f(wz+v)=g(z)forallzandshowthatitisunique.9.Considerthesequencea1;a2;:::sothatXn2=addjnShowthatnjan.10.Therearencarswaitingatdistinctpointsofacircularracetrack.Atthestartingsignaleachc

6、arstarts.Eachcarmaychoosearbitrarilywhichofthetwopossibledirectionstogo.Eachcarhasthesameconstantspeed.Whenevertwocarsmeettheybothchangedirection(butnotspeed).Showthatatsometimeeachcarisbackatitsstartingpoint.111.Considerancycliccircumscribedquadrilateral.Showth

7、atintersectionpointofthediagonalsliesonthelinejoiningthecentersoftheincircleandcircumcircle.12.Leta;b;c;d;m;nbepositiveintegerssuchthata2+b2+c2+d2=1989,a+b+c+d=m2andthelargestofmax(a;b;c;d)=n2.Findmandn.13.TherealnumbersPa0;a1;a2;:::;ansatisfya0=an=0andforallkwe

8、haveak=n¡11c+i=kai¡k(ai+ai+1).Showthatc·4n14.Findthesmallestnumberofsegmentsbetween7pointsintheplanesothatinanygroupof3points(ofthe7)atleast2arejoinedbyasegment.15.Gi