PROBLEMS           Vol. 98(6), October 1998

4673-S: Student Problem - only undergraduate and pre-college students are eligible to submit a solution.
Proposed by Douglas Joy, Guelph, ON, Canada.
Replace the letters in the following cryptarithm by base 10 digits in a one-to-one manner so as to obtain a valid addition operation in base 10. Also show that your solution is unique, i.e., that your solution is the only possible solution.

4674-OBG: Oldie But Goodie - Proposed by Murray J. Levanthal, NY.
Given an equilateral triangle ABC, and a point P so that PA = PB + PC. Show that the locus of P is a circle. (From a New York City examination for teachers.)

4675*: Proposed by Michael Brozinsky, West Hempstead, NY.
If N points are chosen at random on the surface of a sphere, find the probability that they are on a hemisphere.
Note: If N points are chose on the circumference of a circle, the probability that they are on a semicircle is known to be n/(2^n-1).

4676: Proposed by Richard L. Francis, Cape Girardeau, MO.
Suppose p is a repunit prime.
(a) Show that 10p + 1 is necessarily composite.
(b) Is 10000p + 1111 necessarily composite?

4677: Proposed by Richard L. Francis, Cape Girardeau, MO.
(a) Is the centroid of a triangle the same as the centroid of its Morley triangle?
(b) Is the incenter of a triangle the same as the incenter of its Morley triangle?

4678: Proposed by Ghunaym M. Ghunaym, Albany, GA.
Determine the unit's digit of the number

4679: Proposed by Charles P. Howerton, Denver, CO.
Suppose that A, B, C, D are points in 3-dimensional space such that
(i) AB = BC = CD = DA, and
(ii) AC = BD = 
Prove that the points A, B, C, D are coplanar and are the vertices of a square.