Problems

The following problems are printed in the May 1998 issue. Solutions will be printed in future issues.
Solutions for theses problems should be mailed before August 31, 1998. (Solutions sent after that date will be considered as deadlines permit.)

4666-S: Student Problem - only undergraduate and pre-college students are eligible to submit a solution.
Proposed by Russell Euler and Jawad Sadek, Maryville, MO.

4667-OBG: Oldie But Goodie
Proposed by J.A. vangroos, Portland, OR.
Of the three inscribed squares on the three sides of a triangle, the largest square is on the smallest side.

4668: Proposed by Charles Ashbacher, Cedar Rapids, IA.

4669: Proposed by Michael Brozinsky, West Hempstead, NY.
An archeological expedition uncovered N houses. The floor of each of these houses was a square area covered by n² tiles. Each tile was a 1 unit by 1 unit square. The tiles in each house were white, except for a (non-empty) rectangular configuration of blue tiles. Among the N houses, all possible rectangular configurations of blue tiles appeared once and only once. The total number of white tiles in the N houses was 68544. Find N and n..

4670: Proposed by Michael Brozinsky, West Hempstead, NY.
Let C be
(a) an ellipse
(b) a hyperbola
in the plane.
Let the tangents at two distinct points P, Q of C intersect at M. Suppose that the line through M and the center of C intersects C at N. Show that N is a "mean value point" of P and Q, i.e., the tangent to C at N is parallel to the line through P and Q.

4671: Proposed by Richard L. Francis, Cape Girardeau, MO.
Can consecutive Independence Day Sundays (July 4th) occur exactly twelve years apart? If so, when did it last happen, when will it happen again, and with what frequency does it occur?

4672: Proposed by Richard L. Francis, Cape Girardeau, MO.
Show that an integer is divisible by 41 if and only if the result of subtracting 4 times the last digit from the number formed by the preceding digits is divisible by 41.