Herzog Competition - 1983

THE ELEVENTH ANNUAL HERZOG PRIZE EXAMINATION

November 5, 1983

Problem 1

(Fritz Herzog) Show that the numbers

4^{n} + n^{4}, n = 2,3,4,5,…

are all composite.

Problem 2

(John Kinney) Suppose we are given a loop of string of length sufficient to enclose S, a point set in the plane. A point of a pencil now draws a generalized ellipse keeping the string taut all the while. Will the tangent make equal angles with the two string segments at each point?

Problem 3

(L. Kelly) If the opposite edges of a tetrahedron are of equal length, then all the face angles are acute.

Problem 4

(T. Yen) Suppose we are given 2n + 3 points in the plane, no three collinear, and no four on a circle. Then there is a circle through three of these points containing exactly half the remaining points.

Problem 5

Do there exist differentiable functions h with

\cos h (x) = h (\sin x)

for all x?

Problem 6

(C.R. MacCluer) Consider the ellipse

r = \frac{p e}{1 + e \cos θ}

Compute

\int_{0}^{2 π} r d θ, \int_{0}^{2 π} r^{2} d θ, and \int_{0}^{2 π} r^{3} d θ,

though not necessarily in that order.