Herzog Competition - 1982

THE TENTH ANNUAL HERZOG PRIZE EXAMINATION

November 6, 1982

Problem 1:

(CWRU 1979 MC) For which positive integers N can you completely pack a 10 x 10 x 20 rectangular box with 1 x 1 x N bricks?

Problem 2:

(CWRU 1979 MC) Starting with the point (1,1), construct a sequence of points $P_{n}$ as follows: Draw the arc of the circle with center at the origin from $P_{k}$ down to the positive x-axis. $P_{k + 1}$ is the midpoint of the chord of this arc. Find

lim_{n \to \infty} {P_{n}} .

Problem 3:

(Fritz Herzog) Let ABC be a spherical triangle of area $π$ on the unit sphere. Can the sphere be completely covered using four congruent copies of ABC?

Problem 4:

(L.M. Kelly) Show that for some side E of a given closed planar polygon and some vertex V the foot of the perpendicular from V to the line containing E falls on E.

Problem 5:

Find all positive continuous functions f(x) such that

\int_{0}^{1} \frac{d x}{f (x)} = 1 / \int_{0}^{1} f (x) d x .

Problem 6:

(AMM E. 2928) Compute

lim_{x \to \infty} \sum_{0}^{\infty} \frac{x^{n}}{n^{n}}

(if it exists).