Problems

The following problems are printed in the December 1997 issue. Solutions will be printed in future issues.
Solutions for theses problems should be mailed before March 31, 1998. (Solutions sent after that date will be considered as deadlines permit.)

4584S: Student Problem - only undergraduate and pre-college students are eligible to submit a solution.
Proposed by Joe Dan Austin, Houston, TX.
The usual formula for the area of a triangle, A = (1/2)bh, where the base b is a side of the triangle and h is the corresponding altitude, presupposes that the formula yields the same value for A, regardless of which side is taken as the base b. Prove that this is in fact the case.
Editors' Note: This problem appeared originally in the November, 1996 issue. There were a number of partial solutions submitted but, due to a failure to consider all cases, none was completely correct. Thus the problem remains "open" until March 31, 1998.

4586: Proposed by John P. Hoyt, Lancaster, PA.
A quadrilateral has consecutive sides of constant lengths a, b, c and d. The longest side has length a and a + c = b + d. The endpoints of the longest side are fixed while the other two vertices are free to move about. Determine the locus of the center of the inscribed circle.
Editors' Note: This problem appeared originally in the November, 1996 issue. No complete solution was received and thus the problem remains 'open" until March 31, 1998.

4587: Proposed by Murray S. Klamkin, Edmonton, AB, Canada.
Let ABC be an acute-angled, non-equilateral triangle. Prove that there is an interior point P other than the circumcenter such that

Editors' Note: This problem appeared originally in the November, 1996 issue. No complete solution was received, and thus the problem remains "open" until March 31, 1998.

4632-OBG: Oldie But Goodie
Proposed by H.E. Trefethen, Waterville, ME
Prove that the length of the angle bisector in any triangle is less than the arithmetic mean of the two adjacent sides.

4633: Proposed by V. C. Bailey, Naples, FL

4634: Proposed by William D. Markel, Hanover, IN
A beer vendor for a company picnic buys beer at $10 per gallon and sells it at $40 per gallon. Past experience indicates that the demand for beer sold is exponentially distributed with a mean of 50 gallons. Any quantitiy of beer which is ordered but not sold must be thrown away at a loss. How many gallons should the vendor buy in order to maximize her expected profit?

4635: Proposed by Richard L. Francis, Cape Girardeau, MO.
Powers may be permutations of other like powers (such as the squares 256 and 625 or the cubes 125 and 512). Show that there are infinitely many pairs of squares, not ending in zero, in which each square is a different permutation of the other.

4636: Proposed by Herta T. Freitag, VA.
Show that there are infinitely many pairs of triangular numbers whose sum is equal to the square of their dfference.
Note: For a related problem, see Problem 4568, whose solution appeared in the April, 1996 issue.

4637: Proposed by Ghunaym M. Ghunaym, Albany, GA.