Proposed by Bill Livingston, Missouri Southern State University. The product of three real numbers is
-1/2 and the sum of their squares is 57/4. If the three numbers are taken two at a time and multiplied,
the sum of the reciprocals of these products is -7. Determine these three numbers.
S-81.
Proposed by Efim I. Cherkassy, Rasba Institute, Israel, and Alexander A. Katz, St. John's University, NY.
In a triangle ABC, AB is perpendicular to AC and the incircle touches BC at D. Prove that the area of ABC
is equal to the product of BD and CD.
S-82.
Proposed by U. I. Lydna, Beloretsk, Russia. In triangle ABC, BC^2 = AB(AB+AC). Prove that angle A is twice angle
C without using trigonometry.
S-83.
Proposed by Jerry Slocum, Beverly Hills. I had a set of seven tables arranged to form a square as in the standard
Tangram puzzle depicted in the diagram on the left (see Math Horizons). One day, I found my tables rearranged to
form a square with one-ninth of it missing, as depicted in the diagram on the right. At first, I thought someone
had stolen the small square table, until I realized that the proportions were wrong. In fact, all seven tables
were there. How were they arranged?
