Problems

The following problems are printed in the February 1997 issue. Solutions will be printed in future issues.
Solutions for theses problems should be mailed before May 31, 1997. (Solutions sent after that date will be considered as deadlines permit.)

With the January issue we began a new feature of the SSM Problem Department - an Oldie But Goodie problem. We have selected some of our favorite problems that have appeared in this department since it began in 1905 and will feature one in each issue.

4602S: Student Problem - only undergraduate and pre-college students are eligible to submit a solution.
Proposed by Gerald Thompson, Augusta, GA.

46030BG: Oldie But Goodie - Proposed by P.F. Gaehr, Ithaca, NY.
Construct an equilateral triangle whose vertices shall be on three given parallel lines.

4604: Proposed by Charles Ashbacher, Cedar Rapids, IA.

For any natural number n, the Smarandache function S(n) is defined as the smallest natural number m such that n divides m!.
(a) Show that the equation S(n)S(n + 1) = n has no solution.
(b) Does the equation S(n)S(n + 1) = 2n have a solution?

4605: Proposed by V.C. Bailey, Naples, FL (edited).

4606: Proposed by Richard L. Francis, Cape Girardeau, MO.
Can the product of the first n Fermat primes be perfect?

4607: Proposed by Richard L. Francis, Cape Girardeau, MO.
Show that a cube can be duplicated with compass and unmarked straightedge if infinitely many steps are allowed.