PROBLEMS           Vol. 99(5), May 1999

4716-OBG: Oldie But Goodie - Proposed by  Michael Goldberg, Philadelphis, PA.
Tie a flat closed knot in a strip of paper. Prove that a regular pentagon is formed.

4717: Proposed by  Richard L. Francis, Cape Girardeau, MO.
A super-millennium is a period of 10,000 years. Hence the second skuper-millennium will begin on January 1, 10001. On what days of the week can s super-millennium begin?

4718:Proposed by Richard L. Francis, Cape Girardeau, MO.
If x an y are perfect numbers of the same parity and greater that 6, show that  is not perfect.

4719: Proposed by  Joe Howard, Las Vegas, NM (generalized by the Editors).
Let be real numbers satisfying  Prove that
 
 

On the left-hand side, for k = 0, the inner sum is defined to be 1, and for k > 0, the inner sum is taken over all
indices  On the right-hand side, the inner sum is taken over all indices 
Note: This problem is generalization of Problem 4922-S, which appeared in the January, 1999 issue.

4720: Proposed by Monte J. Zerger, Alamosa, CO.
(a) Find all triangular number T which lie between a prime pair (i.e., such that T - 1 and T + 1 are both prime).
(b) Find all non-trival powers (positive integers of the form where n > 1 and k > 1) which lie between a prime pair.
4721-S: Student Problem - only undergraduate and pre-college students are eligible to submit a solution.
Proposed by the Editors, Durango, CO.
The line given by 13x + 8y = 1999 does not intersect the curve given by  Which point on the curve is closest to the line?