Herzog Competition - 1992

THE TWENTIETH HERZOG PRIZE EXAMINATION

November 14, 1992

Problem 1

(See the figure.) The plane is closely packed with circles of unit radius, each circle touching six others, and a point is chosen at random. Find the probability that the point lies in none of the circles.

Close packing of equal circles in a triangular arrangement, each circle touching six neighboring circles in the infinite pattern. — Circle packing illustration for Problem 1.

Problem 2

Find all possible p, q and r for which

\frac{\log p}{\log q} = r,

where p and q are prime and r is rational.

Problem 3

Find a closed form expression for the product

(1 - \frac{1}{2^{2}}) (1 - \frac{1}{3^{2}}) \dots (1 - \frac{1}{n^{2}})

and prove the correctness of your answer.

Problem 4

Three circles of equal radius intersect in a common point. Show that the remaining three intersection points lie on a circle of the same radius.

Problem 5

A Pythagorean triple is a set of three positive integers (a, b, c) for which $a^{2} + b^{2} = c^{2}$ . For example, (3,4,5) and (5,12,13) are Pythagorean triples. Prove that in every Pythagorean triple, at least one of the numbers is a multiple of 5.

Problem 6

Prove or disprove: The graph of the equation

x^{4} + x^{3} y + x^{2} y^{2} + x y^{3} + y^{4} = 1

is a bounded subset of the plane.