Herzog Competition - 1984

THE TWELFTH ANNUAL HERZOG PRIZE EXAMINATION

November 3, 1984

Problem 1

(L.M. Kelly) Assume that the points of the plane are each colored red, white, or blue.

Prove that there are two points 1 unit distance apart of the same color.

Problem 2

(J.S. Frame) Evaluate

\int_{0}^{\frac{π}{2}} ℓ n (2 \sin 2 θ) d θ

Problem 3

(Alan Parks) Let $a_{1} < a_{2} < \dots < a_{n}$ be a list of positive integers such that for every $i \neq j$ , at least one of

a_{i} - a_{j}, a_{j} - a_{i},

or $a_{i} + a_{j}$ is again among the list. An example is $a_{i} = i k$ for a constant k. Show that if the list is not of this form, then n = 3.

Problem 4

(Kevin McCurley) Show that $f (n) = n^{6} + 671$ is composite for n = -83, -82, -81, ..., 0, 1, 2, ..., 82, 83.

Problem 5

(L.M. Kelly) If x and y are positive numbers and m and n positive integers, prove

\frac{x^{m} y^{n}}{{(x + y)}^{m + n}} \leq \frac{m^{m} n^{n}}{{(m + n)}^{m + n}} .

Problem 6

(Chris Bishop) Given a disc of radius R and n discs of radius r < R inside, prove that there exists a line segment intersecting at least [nr/R] of the smaller discs. ([x] denotes the greatest integer less than or equal to x).