Problems

The following problems are printed in the January 1998 issue. Solutions will be printed in future issues.
Solutions for theses problems should be mailed before April 30, 1998. (Solutions sent after that date will be considered as deadlines permit.)

4598: Proposed by Michael Brozinsky, West Hempstead, NY.

Editors' Note: This problem appeared originally in the January, 1997 issue. Since no correct solutions were received, the problem remains "open" until April 30, 1998.

4638S: Student Problem - only undergraduate and pre-college students are eligible to submit a solution.
Proposed by V. C. Bailey, Naples, FL.
4639-OBG: Oldie But Goodie
Proposed by F. Eugene Seymour, Trenton, NJ
If similar triangles be circumscribed about and inscribed in a given triangle, the area of the given triangle is the mean proportional between the areas of the circumscribed and the inscribed triangles.

4640: Proposed by Richard L. Francis, Cape Girardeau, MO.
Show how to construct (with unmarked straightedge and compass, in a finite number of steps) the regular polygon of 255 sides if given the construction steps for the regular 3-, 5- and 17-sided polygons.

4641: Proposed by Richard L. Francis, Cape Girardeau, MO.
On what days of the week can Leap Year Day occur in a year whose last two digits are zeros? Consider both Julian and Gregorian calendar forms.

4642: Proposed by Murray S. Klamkin, Edmonton, AB, Canada.
Determine all pairs of integers (a, b) such that all the roots of the equation

are also integers.

4643: Proposed by Ghunaym M. Ghunaym, Albany, GA.

4644: Proposed by Heinz-Jiirgen Seiffert, Berlin, Germany.

Note: for a related problem, see Problem 4546 in the Solutions section.