Herzog Competition - 1980

The Eight Annual Herzog Prize Examination

November 8, 1980

Problem 1:

Find all positive integral solutions of

x^{y} = y^{x - y}

Problem 2:

(T. Butts) Prove that

\int_{0}^{\infty} \frac{\sin x}{x} d x = \int_{0}^{\infty} \frac{\sin^{2} x}{x^{2}} d x .

Problem 3:

(T. Butts) If k different integers are chosen at random from the set {1,2,3,...p} where p is prime, find the probability that their sum is divisible by p.

Problem 4:

(L.M. Kelly) Two wire tracks follow along the curves

$y = \frac{1}{x}$ and $y = \frac{−1}{x}$ , $x > 0$ . A sphere of radius 1 rolls toward the origin along these tracks starting at a great distance from the origin. Describe the path followed by the center of the sphere.

Problem 5:

(M.A.A. E 1336) Let $f (0) > 0$ , $f (1) < 0$ . Suppose that there is a continuous function g such that f + g is non-decreasing on [0,1]. Show that f(x) = 0 for some 0 < x < 1.

Problem 6:

(L.M. Kelly) Is there a polyhedron with exactly 10 faces, each a pentagon?