Herzog Competition - 1979

Problem 1

(D. Moran) Find N, given that N is a prime which can be written as an alternating sequence of 0’s and 1’s.

Problem 2

(L. Kelly) Let f(x) be a continuous real-valued function mapping [0,1] into [0,1] with f(0) = 0. If for every x ∈ [0,1] there is a positive integer n such that

$$f^{\left(n\right)}\left(x\right)=x$$

(n-th iterate) show that f(x) = x for all x.

(Note: f⁽²⁾(x) = f(f(x)), f⁽³⁾(x) = f(f(f(x))) etc.).

Problem 3

(J. Kinney) Let A, B and C be the vertices of an equilateral triangle of side 1. Let R be the intersection of the closed discs of radius 1 with centers A, B and C. Can R be covered by two closed discs of diameter 1?

Problem 4

(A. Baily) Recall that $e^{\pi i}=-1$ . Justify

$$e^{\pi \left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)}=-\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$$

Problem 5

(A.M.M. E 2791) Suppose ${\displaystyle \sum _1^{\infty }{a}_n}$ converges. Must ${\displaystyle \sum _1^{\infty }{a}_n^3}$ converge?

Problem 6

(L. Kelly) A square matrix has the property that the interchange of any two distinct rows produces a symmetric matrix. Characterize such matrices.