Problems

Vol. 97(4), April 1997

Volume 97(4)

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

4571: Proposed by Thomas Leong, Staten Island, NY.

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.
Determine similar methods for testing f(x, y) = 0 for symmetry with respect to
(a) the point (a, b)
(b) the line y = mx + b.

4615-OBG: Oldie But Goodie - Proposer anonymous.

4616: Proposed by Charles Ashbacher, Cedar Rapids, IA.
For any positive integer n, the Smarandache function S(n) is defined as the smallest integer m such that n|m!. Prove that, for any even perfect number r, S(r) is prime.

4617: Proposed by Ernesto B. Cossi, Porto Alegre RS, Brasil.

4618: Proposed by Richard L. Francis, Cape Girardeau, MO.
Which of the following angles, given in degree measure, are constructible with an unmarked straightedge and compass, in a finite number of steps:

1/7, 2/7, 3/7, 4/7, 5/7, and 6/7 ?

4619: Proposed by Jerry Brown, a student at Mesa Community College, and Jorge Castillo, Mesa, AZ.
Find a recursive formula for each of the following sequences:
(a) the sequence of Smarandache crescendo subsequences:

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6,...

(b) the sequence of Smarandache crescendo pyramidal subsequences:

1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1,...

1,2,1,3,4,2,1,3,5,6,4,2,1,3,5,7,8,6,4,2,1,3,5,7,9,10,8,6,4,2,... .