Problems

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

4652-S: Student Problem - only undergraduate and pre-college students are eligible to submit a solution.
Proposed by Murray S. Klamkin, Edmonton, AB, Canada.
Determine all roots of the equation (z + 1)⁷ = z⁷ + 1.

4653-OBG: Oldie But Goodie
Proposed by H.E. Trefethen, Kent's Hill, ME.
Show that the squares of the sides of a triangle are in arithmetical progression, if the cotangents of the angles are in arithmetical progression.

4654: Proposed by Charles Ashbacher, Hiawatha, IA.

4655: Proposed by Richard L. Francis, Cape Girardeau, MO.
For any positive integer n, the positive integers 100n + 1 through 100n + 100 form by definition a century. If p is any prime greater than 5, show that p is a divisor of the first element of some century.

4656: Proposed by Richard L. Francis, Cape Girardeau, MO.
Obviously, no square number is a prime. How many (positive) triangular numbers are prime?

4657: Proposed by Ghunaym M. Ghunaym, Albany, GA.
If a and b are real numbers satisfying

find a^1998+k + b^l998+k for k = 0, 1, ..., 5.

4658: Proposed by Stanley Rabinowitz, Westford, MA.