Target Practice
An amazingly accurate archer found that the bullseye
of an old target was wearing out, so the archer decided to make
a new target by
starting with a circular board of radius one foot and drawing two
disjoint sets of concentric circles.
The figure below was produced by drawing one set of three concentric
circles and a second disjoint set of two concentric circles.
Drawing one set of m concentric circles and a second disjoint
set of n concentric circles gives the final target a total of
m+n+1 regions  the bullseye and the m1
rings of the first set, the bullseye and the n1
rings of the second set, and finally, the background region, which is
the region not in any of the drawn circles.

Find, with proof, all of the values of m and n for
which each of the m+n+1 regions can have exactly the
same area.
Give your answer by listing all possible pairs (m,
n) with 1<=m<=n.

Find, with proof, all of the values of m and n for
which all regions can have the same area except for the background
region, which will have five times the area of any other single
region.
Again give your answer by listing all possible pairs (m,
n) with 1<=m<=n.
Rules:

Answers must be written neatly on 8.5 by 11 inch paper.

Answers must be submitted to Dr. Buchanan
either at his office
(Wickersham 113) or placed in his department mailbox by the department
secretary.

The contest will open on September 3, 1996.

The first complete and correct answer will be awarded the sum of $5.00
(five American dollars).

All complete and correct answers will be listed on a "Mathematical
Puzzle List of
Distinction" to be posted outside Wickersham 113 as well as on a
"Mathematical Puzzle List of Distinction" web page.
Page maintained by: Robert.Buchanan
Robert.Buchanan@millersville.edu
Last updated: