Herzog Competition - 1985

THE THIRTEENTH ANNUAL HERZOG PRIZE EXAMINATION

November 9, 1985

Problem 1

Suppose

S = f (1) + f (2) + f (3) + \dots

where f(mn) = f(m)f(n) > 0 for all m and n. Compute

f(1) + f(3) + f(5) + f(7) + \dots

f(2) + f(4) + f(6) + f(8) + \dots

and

f(1) - f(2) + f(3) - f(4) + f(5) - \dots

(M.J. Winter) What is the expected number of throws of a fair coin until heads occur twice in succession.

(J.G. Hocking) Given two circles and a point P on one, construct a circle through P tangent to both given circles.

(Wade C. Ramey) Let P(x,y) be a polynomial where

\int_{D} \int P (x, y) dxdy = 0

over every disk D of radius 1 containing the origin. Prove that P(x,y) is the 0 polynomial.

(L.M. Kelly) Is the volume of a tetrahedron a function of the areas of its (four) faces?

(J.G. Hocking) Suppose f is differentiable on [a,b] with f(a) = a and f(b) = b. Do there exist a < x₁ < x₂ < b with

\frac{1}{f^{'} (x_{1})} + \frac{1}{f^{'} (x_{2})} = 2 ?