Question 5
Let $n\ge3$ and $a_1,\ldots,a_n\in\{0,1\}$. Let $S_i=(a_i,a_{i+1},\ldots,a_n,a_1,\ldots,a_{i-1})$ be the $n$ cyclic shifts. For a binary tuple $r=(b_1,\ldots,b_n)$ define $\omega(r)=\sum_{j=1}^n b_j2^{n-j}$. Suppose $\omega(S_1),\ldots,\omega(S_n)$ take exactly $k$ distinct values.
1. Prove $k\mid n$ and $\frac{2^n-1}{2^k-1}\mid\omega(S_i)$ for every $i$.
2. If $M$ and $m$ are their maximum and minimum, prove
$$M-m\ge\frac{(2^n-1)(2^{k-1}-1)}{2^k-1}.$$
1. Prove $k\mid n$ and $\frac{2^n-1}{2^k-1}\mid\omega(S_i)$ for every $i$.
2. If $M$ and $m$ are their maximum and minimum, prove
$$M-m\ge\frac{(2^n-1)(2^{k-1}-1)}{2^k-1}.$$