Question 5
There are $1993$ points arranged on a circle. Initially each is assigned either $+1$ or $-1$, arbitrarily. At steps $n=1,2,3,\ldots$, the labels are changed: at step $n$ a point is assigned $+1$ if its two neighboring points have equal labels at step $n-1$. Prove that at some step $n$ all labels are $1$.