PROBLEMSAND SOL{JTIONS
Editedby Gerald A. Edgar, Doug Hensley,Douglas B. West
with the collaborationof Paul T. Bateman,Mario Benedicty,Itshak Borosh, Paul
Bracken, Ezra A. Brown, Randall Dougherty,Dennis Eichhorn, Tam6s Erd6lyi,
ZacharyFranco,ChristianFriesen,Ira M. Gessel,JerroldR. Griggs,JerroldGrossman,
Frederick W. Luttmann, Vania Mascioni, Frank B. Miles, Richard Pfiefer, Cecil C.
LeonardSmiley,JohnHenry Steelman,KennethStolarsky,RichardStong,
Rousseau,
WalterStromquist,DanielUllman, CharlesVandenEynden,andFuzhenZhang.
PROBLEMS
11284. Proposedby Greg Oman, Ohio State University, Columbus,OH. Let R be
an infinite commutativering with identity. Supposethat every proper ideal of R has
smallercardinalitythanR. Provethat R is a field.
11285.Proposedby YakubAliyev, QofqozUniversityand Baku StateUniversity,Baku,
Azerbaijan Let six pointsbe chosenin cyclic orderon the sidesof triangleABC: A1
andA2 on BC , 81 and 82 on C A, andC1 and C2 on AB . Let K denotethe intersection of A182 andCrAz, L the intersectionof B1C2and A1 82, etrdM the intersection
of ArBz and ,B142, B1C2
of C1A2 and BtCz.Let T, U, and V be the intersections
Provethat lines AK , B L, and C M arc
and B2C1,and C1A2 andCzAr respectively.
concurrentif andonly if pointsT , U , and V arecollinear.
11286.Proposedby Marc LeBrun, Fixpoint Inc., Larkspur,CA, and David Applegate
and N.J.A.Sloane,AT&T ShannonLabs,Florham Park,NJ. When a andb are positive
integerswith b > 10, wite a6 (or a : b inline) for the integerwhosebaseb expansion
is thedecimalexpansion
of a. Thatis, if a : II:o a;10twith eachai in {0, 1, . . . , 9},
t h e na 6 : a : b : Il =o a ;b t.T hus,
l 0 tttzr: : 10: ( 11 : ( 12: 13) ): 16'
Considerthe "dungeonsequences"
1 0 , l 0 : 1 1 , (1 0 : 11): 12, ( ( 10: 11) : 12): 13... ,
1 0 , 1 0 :1 1 , 1 0 : ( 11 : l2) , l0: ( 11: ( L2:13) ) ... ,
1 0 , 1 1 :1 0 , (L 2:11) :10, ( ( 13:12) :11) :10...
,
1 0 , 1 1 :1 0 , 1 2 : ( 11 :10) , 13:( 12 ( 11:10) ) ... .
Show that loglogs"l(nloglogn)
Let s, be the nth term in any of thesesequences.