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Solutions should be mailed before April 30, 1999. (Solutions sent after that date will be considered as deadlines permit.)
4644: Proposed
by Heinz-Jürgen Seiffert, Berlin, Germany
Note: for a related problem, see the solution
of Problem 4546 in the January, 1998 issue.
Editors' Note: This problem appeared originally
in the January, 1998 issue. Since no solutions were received, the
problem remains "open" until April 30, 1999.
4692-S: Student
Problem - only undergraduate and pre-college students are eligible
to submit a solution.
Proposed by Joe
Howard, Las Vegas, NM, and generalized by the editors.
Suppose that each of the real numbers a, b, and
c
is greater than or equal to 1. Prove that
4693-OBG: Oldie
But Goodie - Proposer anonymous.
A tells the truth 3 times out of 4; B tells the truth
2 times out of 3; C tells the truth 4 times out of 5. C makes an
assertion and A and B deny it. What is the probability of the assertion
being true?
4694: Proposed
by Richard L. Francis, Cape Girardeau, NO.
Though 19, 199, and 1999 are prime, the number 19999
is composite (7 times 2857). Show that the set of composites of the
form 1999...999 is infinite.
4695: Proposed
by Richard L. Francis, Cape Girardeau, NO.
A prime triangle
is
a triangle having the properties (i)
each side is a prime, (ii)
the perimeter is a prime, and (iii)
the area is likewise a prime. If only (i)
and (ii) are satisfied, the triangle
is semi-prime.
(a) Does a prime triangle exist?
(b) Give an example of a semi-prime triangle.
(c*) Is the set of semi-prime triangles an infinite set?
4696: Proposed by Ghunaym M. Ghunaym, Albany,
GA.
For what positive integer values m,
n is
(a) rational (b) irrational algebraic (c) transcendental?
(Here all of the infinitely many "stacked up" exponents not explicitly
show are 
).
4697: Proposed
by V. C. Bailey, Naples, FL.
Suppose that 
has integral side lengths a = BC,
b
= CA, c = AB, and side AB
is its longest side. Construct a square ABDE
on the side of AB
remote
from C. Suppose
furthermore that side DE
of the square
ABDE is
tangent to the circumcircle of .
(a) Express c
as a function of a and
b.
(b) Find two (non-similar) triangles which satisfy the
hypotheses of the problem.