PROBLEMS AND SOLUTIONS
Editedby Geratd A. Edgar, Doug Hensley DouglasB. 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, Jerrold Grossman,Andre Ktindgen,FrederickW.Luttman,VaniaMascioni,FrankB. Miles, Richard
LeonardSmiley,JohnHenry Steelman,KennethStolarsky,
Pfiefer,Cecil C. Rousseau,
Richard Stong, Walter Stromquist,Daniel Ullman, Charles Vanden Eynden, and
FuzhenZhang.
PROBLEMS
11166. Proposed by Greg Oman, The Ohio State University, Columbus, OH. Let Rc
denote the ring of continuous functions /: IR + R, and let Rp denote the subring of
R. consisting of those elements of R6 that are differentiable on IR. Are the rings R6'
and Rp isomorphic?
11167. Proposed by Vicenliu Rddulescu, University of Craiova, Craiova, Romania. Let
Q be the set of all complex numbers z satisfying 0 < lzl < 1. Fix a positive integer n,
and for arbitrary distinct elements 21,. . . , zn of Q, define
f( zr , -.. , z ,) :
fl] :, l, ilz0 - lz il\ _'flr=i.o=nlzil ' lz r , l' lz i- z , , l' '
+ ( l - lz ilz ) ( t- lz tl\ f .
fl rs ;=r<,l z i l 'lz r , l' llr t - z r , l2
(a) For n :2 prove that the maximum of / is attained by a unique configuration (up
to a rotation) that consists of two points symmetric with respect to the origin.
(b) For n :3 prove that the maximal configuration for / is also unique (up to a
rotation) and consists of three points that are the vertices of an equilateral triangle
centered at the origin.
1,1L68.Proposed by Kent Holing, Trondheim,Norway. Let a, b, and c be the sides of a
Pythagoreantriangle, with c the hypotenuse.
(a) Show that c - a and c * a cannot both be sides of a single Pythagorean triangle.
(b) Show that none of c2 * 4ab, c2 - 4ab, or c2 - 9az canbe square.
11169. Proposed by Mohammad Hossein Mehrabi, University of Science and Technology, Tehran, Iran. Let @ be the function on the positive real numbers given by
6@) - (e/x)"f (x), where f is the unique log convex function on the positive real
numbers satisfying f (n) - (n - I)l for positive integers n. Prove that Q is strictly
decreasingon (0, oo).