PROBLEMS           Vol. 98(7), November 1998

4633: Proposed by V. C. Bailey, Naples, FL.
Show that there are infinitely many (non-similar) quadrilaterals ABCD, with sides AB = a, BC = b, CD = c, DA = d, satisfying
(a) a² + b² + c² = d².
(b) area of ABCD = 
(c) cos A = a/(a + 2), and
(d) 0 < A < 60^0.
Editors' Note: This problem appeared originally in the December, 1997 issue. Since no solutions were received, the problem remains "open" until February 28, 1999.

4680-S: Student Problem - only undergraduate and pre-college students are eligible to submit a solution.
Proposed by Ghunaym M. Ghunaym, Albany, GA.
A familiar approximation to   is 22/7, whose decimal representation is the repeating decimal       = 3.14285714... . It is therefore natural to consider the points

in the Euclidean plane. Show that these points lie on a conic section and classify the conic section.

4681-OBG: Oldie But Goodie - Proposed by Daniel Kreth, Wellman, IA
If x be any prime number except 2, the integral part of (1 +  )^x, diminished by 2, is divisible by 4x.

4682: Proposed by V. C. Bailey, Naples, FL
The ellipse b²x² + a²y² = c² is given, with foci A(-c, 0) and B(c, 0). For -a < x < a, define a point D = D(x) and a function d(x) as follows:
(i) Let P = P(x) be the point of intersection of the vertical line through (x, 0) and the ellipse in the upper half plane.
(ii) Let D = D(x) be the point of intersection of the bisector of angle APB and the x-axis.
(iii) Finally, let d(x) be the distance DA.
Find the extreme values of d(x) and the extreme positions of D(x) for -a < x < a.

4683: Proposed by Herta T. Freitag, Roanoke, VA
For any positive integer x, define
(i) u(x) to be the unit's digit of x, and
(ii) t(x) to be the number obtained by dropping the unit's digit of x (if x < 10 then t(x) = 0).
Let d be a positive integer such that u(d) = 9.
(a) Find a positive integer k(d) such that, for all n divisible by d, t(n) + k(d)u(n) is divisible by d.
(b) Find (t(n) + k(d)u(n))/d as a function of d and n.

4684: Proposed by Richard L. Francis, Cape Girardeau, MO.
A perfect polygon is one of integral side measure such that its perimeter is numerically equal to the area (e.g., the triangle of sides 9, 10, and 17). Find all perfect polygons which are also regular.

4685: Proposed by Richard L. Francis, Cape Girardeau, MO.
Let x be an exact square and y the number formed by reversing the digits of x. Show that x + y cannot be an even perfect number.