PROBLEMS           Vol. 99(2), February 1999

Solutions should be mailed before May 31, 1999. (Solutions sent after that date will be considered as deadlines permit.)

4650: Proposed by Murray S. Klamkin, Edmonton, AB, Canada.
If a rectangular hyperbola and a circle intersect in four points, prove that the segment joining the centers of the two curves is bisected by the centroid of the four points of intersection.

Note: For a related problem, see the solution of Problem 4548-S in the January, 1998 issue.
Editors' Note: This problem appeared orininally in the February, 1998 issue. Since no solutions were received, the problem remains "open" until May 31, 1999.
 
 

4698-S: Student Problem - only undergraduate and precollege students are eligible to submit a solution.  Proposed by Ghunaym M. Ghunaym, Albany, GA Let  and   be two triangles whose sides intersect in points M, N, P, Q, R, and S, as shown in the figure.  Prove that

Figure for Problem 4698-S

4699-OBG: Oldie But Goodie - Proposed by Michael Goldberg, Philadelphia, PA.
Find a point D in base AB of    such that the circles inscribed in     and in    will touch CD at the same point.

4700: Proposed by V. C. Bailey, Naples, FL.
(a) Determine the range of real numbers x, 0 < x <    , for which there is a triangle ABC with sides a = BC, b = CA and c = AB, satisfying a = sin x < b = sin 2x < c = sin 3x.
(b) Express the angles A, B, and C of the triangle in terms of x.

4701: Proposed by Richard L. Francis, Cape Girardeau, MO.
The occurrence of the sequence 1234567890 in   or e may be called "accidental order".
(a) Show that this sequence appears in the digital interior of infinitely many prime numbers.
(b) Does the sequence consisting of the first billion digits of  appear in the digital interior of infinitely many prime numbers?