Problems

Vol. 97(3), March 1997

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The following problems are printed in the March 1997 issue. Solutions will be printed in future issues.
Solutions for theses problems should be mailed before June 30, 1997. (Solutions sent after that date will be considered as deadlines permit.)

ANNOUNCEMENT

With the January issue we began a new feature of the SSM Problem Department - an Oldie But Goodie problem. We have selected some of our favorite problems that have appeared in this department since it began in 1905 and will feature one in each issue.

4560-S: Student Problem - only undergraduate and pre-college students are eligible to submit a solution.
Proposed by J. Sriskandarajah, Richland Center, WI.
An equilateral triangle ABC is inscribed in a rectangle APQR so that B and C are interior points of the sides PQ and QR, respectively. Show that the area of one of the three triangles APB, BQC, and CRA equals the sum of the areas of the other two.

4561: Proposed by V.C. Bailey, Naples, FL.

4608-OBG: Oldie But Goodie - Proposed by John W. Scoville, Syracus e, NY.
A prison consists of 36 cells arranged like the squares of a chessboard. There are doors between all adjoining cells. A prisoner in one of the corner cells is told that he can have his freedom, if he can get into the diagonally opposite corner cell, by passing through each of the cells once and only once. Can the prisoner win his freedom?

4609: Proposed by V. C. Bailey, Naples, FL.
Find and sketch the locus of the midpoint of a line segment of length 6 which moves so that one end traverses the circle x2 + y2 = 9 while the other end traverses the circle (x + 2)2 + y2 = 25.

4610: Proposed by Richard L. Francis, Cape Girardeau, MO.
Find all perfect numbers which are of the form p + 1 where p is a prime.

4611: Proposed by Richard L. Francis, Cape Girardeau, MO.
It is known that if an algebraic equation of degree 3 (with integral coefficients) has no rational roots, neither can it have a constructible root. Does the same conclusion hold for an algebraic equation of degree 13?

4612: Proposed by Kandasamy Muthuvel, Oshkosh, WI.

4613: Proposed by Herbert Wills III, Tallahassee, FL.
For positive integers a and n, define S(a, n) = an + (a + 1)n + (a + 2)n + (a + 3)n + (a + 4)n. Suppose that, for some particular a and n, the unit's digit of S(a, n) is 9. What is the unit's digit of S(a, n + 1997)?