Herzog Competition - 1995

Problem 1

Find two non-trivial factors of $x^{10}+x^5+1$.

Problem 2

Prove or disprove: The binomial coefficient $(\genfrac{}{}{0ex}{}{2n}{n})$ is always an even number for $n\ge 1$.

Problem 3

Consider the plane curve defined by the parametric equations

$$\{\begin{array}{l}x=\sin \pi t\\ y=\cos t\end{array},t\in [0,\infty ).$$

The points on this curve all lie in the square $\{(x,y):-1\le x\le 1,-1\le y\le 1\}$. Do all of the points in this square lie on this curve, or are there some that are not on the curve?

Problem 4

Prove that $2^{\mathrm{mn}}-1$ is divisible by $2^m-1$ for every pair of positive integers $m$ and $n$.

Problem 5

A wide-beam spotlight can cast its light on all points lying within and on the boundary of a 90° angle. Four such lamps are placed at arbitrary points in the plane. Prove that it is always possible to direct the beams so that the entire plane will be illuminated.

Problem 6

Find all continuous real-valued functions that satisfy $f(x)f(y)=f(x-y)$ for every pair of real numbers $x$ and $y$.