PROBLEMS           Vol. 98(8), December 1998

4598: Proposed by Michael Brozinsky, West Hempstead, NY.
According to legend, the only way that a famous pirate's chest could be opened was to put keys, numbered from 1 to m (m 2), into the lock in consecutive order. All the keys were kept together in a jar to prevent any of the keys from getting lost. Due to corrosion, the numbers on the keys had become obliterated and the keys could in no way be marked (making it impossible to decide whether a given key has been tried). What is the expected number of keys that will be tried before the chest is opened?
Editors' Note: This problem has been restated. See the Solutions section for the solution to a related problem, also numbered 4598.

4639-OBG: Oldie But Goodie
Proposed by F. Eugene Seymour, Trenton, NJ
Triangles similar to a given triangle, with corresponding sides parallel, are circumscribed about and inscribed in the given triangle. Prove that the area of the given triangle is the mean proportional between the areas of the circumscribed and the inscribed triangles.
Editors' Note: Murray Klamkin, Edmonton, AB, Canada points out that the problem was incorrect as originally stated since there are infinitely many similar triangles that can be circumscribed about and inscribed in the given triangle. The above is a restatement of the problem.

4685-S: Student Problem-only undergraduate and pre-college students are eligible to submit a solution.
Proposed by William R. Klinger, Upland, IN.
Let f and g be real-valued functions such that f(x) + g(x) = 1 and f(x - a) = g(x) for all real numbers x and some constant real number a.
(a) Determine 
(b) Give and example of two non-constant functions f and g and a real number a that satisfy the hypotheses.

4686-OBG: Oldie But Goodie - Proposed by Wm. B. Campbell, Philadelphia, PA.
Starting with two equal piles of material, A and B, one-third of that in A is transferred to B, then one-third of B's new total is transferred to A, etc., indefinitely. A condition is manifestly approached where each pile alternates between 4/5 and 6/5 of its original condition, but give the general expression stating the fraction in each pile after n transfers.

4687: Proposed by V. C. Bailey, Naples, FL.
A 1-parameter family of triangles is given, with side lengths a(x) = sin x, b(x) = sin 2x, and c(x) = sin 3x, satisfying a < b  c. Determine the domain of x and show that the circumradius of the triangles is a constant (independent of x).

4688: Proposed by Kuo-Tsai Chang, Hsin-chu, Taiwan.
Let * be a binary operation on a set S with an identify element e  S    such that a * e = e * a = a for all a  S  . Suppose that there is a permutation   (other than the identity mapping) of (a, b, c, d) such that 
(a) Prove that * is commutative.
(b) Suppose furthermore that   Prove that * is also associative.

4689: Proposed by Richard L. Francis, Cape Girardeau, MO.
Find all Mersenne primes M such that M is 1 more or 1 less than a perfect number p.

4690: Proposed by Richard L. Francis, Cape Girardeau, MO.
Suppose (a₁, a₂, ..., a_n) and [a₁, a₂, ..., a_n] denote respectively the greatest common divisor and least common multiple of a₁, a₂, ... , a_n. For what values of n is the formula (a₁, a₂, ... , a_n){a₁, a₂, ..., a_n] = a₁a₂...a_n valid?