## 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 m-1 rings of the first set, the bullseye and the n-1 rings of the second set, and finally, the background region, which is the region not in any of the drawn circles.

1. 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.
2. 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:

1. Answers must be written neatly on 8.5 by 11 inch paper.
2. Answers must be submitted to Dr. Buchanan either at his office (Wickersham 113) or placed in his department mailbox by the department secretary.
3. The contest will open on September 3, 1996.
4. The first complete and correct answer will be awarded the sum of \$5.00 (five American dollars).
5. 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

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