PROBLEMS           Vol. 99(4), April 1999

Solutions should be mailed before July 30, 1999. (Solutions sent after that date will be considered as deadlines permit.)
 

4710-S: Student Problem - only undergraduate and pre-college students are eligible to submit a solution.
Propsed by V. C. Bailey, Naples, FL.
In the figure, right triangle ABC is inscribed in a semicircle. Circle  is inscribed in the right triangle. Circle  is inscribed in the circular segment as shown, touching side AC it its midpoing . Circle andhave the same radious. Determine  the ratio AC/BC.
 

4711-OBG: Oldie But Goodie - Proposed by JR. T. McGregor, Elk Grove,CA.
The centers of four circles circumscribed about the triangles formed by four straight lines are concyclic.

4712: Proposed by  Herta T. Freitag, Roanoke, VA.
A right triangle ABC with is given. denote the incircle and the excircle to side BC, respectively.
Letbe the triangle whose vertices are the points of tangency ofto the three sides of 
Letbe the triangle whose vertices are the points of tangency of to BC and the to the extensions of sides AB and CA.
Define triangels similarly to , by considering the excircles to sides CA and AB, respectively.
Finally, let denote the areas of triangels respectively.
Prove that 
Note: For two related problems, see the solution of Problem 4153 in the October, 1988 issue, and the solution of Problem 4223 in the January, 1990 issue.

4713:Proposed by Richard L. Francis, Cape Girardeau, MO.
Of al the right triangles inscribed in the unit circle, which has the Morley triangle of greatest area?

4714: Proposed by Richard L. Francis, Cape Girardeau, MO.
Let X be a product of (not necessarily distinct) even perfect numbers.
(a) Prove that X + 2 is not perfect.
(b) Can X + 2 be deficient?
(c) Can X + 2 be abundant?

4715: Proposed by  Herbert Wils III, Tallahassee, FL.
A Fermat prime is a prime of the form  where n is a nonnegative integer. A Sophie Germain prime is a prime p such that 2p + 1 is also a prime. Neither the cardinality of the set of Fermat primes, not the cardinality of the set of Sophie Germain primes is known. What is the cardinality of the intersection of the set of Germat primes and the set of Sophie Germain primes?