Problems

The following problems are printed in the April 1998 issue. Solutions will be printed in future issues.
Solutions for theses problems should be mailed before July 31, 1998. (Solutions sent after that date will be considered as deadlines permit.)

4614-S: Student Problem - only undergraduate and pre-college students are eligible to submit a solution.
Proposed by Joe Flowers, Kirksville, MO.
The following methods are well-known for testing an equation f(x, y) = 0 for symmetry with respect to (i) the x-axis (ii) the line y = x (iii) the origin:

(i) replacing y with -y yields an equivalent equation

(ii) interchanging x and y yields an equivalent equation

(iii) replacing x with -x and y with -y yields an equivalent equation.

(a) the point (a, b)

(b) the line y = mx + b.

Editors' Note: This problem appeared originally in the April, 1997 issue. Since no solutions were received, the problem remains "open" until July 31, 1998.

4659-OBG: Oldie But Goodie - Proposed by Norman Anning, Chilliwack, BC, Canada.
In any scale of notation any multiple of m which has n digits remains a multiple after cyclic permutation of the digits, provided that m is a factor of the number formed by placing n ones in a row.

4660: Proposed by Russell Euler and Jawad Sadek, Maryville, MO.

4661: Proposed by the late Jack Garfunkel, Flushing, NY.

4662: Proposed by Roger Izard, Dallas, TX.

4663: Proposed by Stanley Rabinowitz, Westford, MA.
The numerical identity cos² 12^° - cos 6^° cos 18^° = sin² 6^° is a special case of the more general identity cos² 2x - cos x cos 3x = sin² x. Find and prove a general identity for the numerical identity

4664: Proposed by Richard L. Francis, Cape Girardeau, MO.
If 1 is inserted as a digit anywhere within the decimal representation of a repdigit prime of more than one digit, show that the resulting number is not a prime.

4665: Proposed by Richard L. Francis, Cape Girardeau, MO.
Show that for any even perfect number r there is a unique integer x such that r = 2x² - x.