Question 2
a. Given a quadruple $(a, b, c, d)$ of positive reals, transform to the new quadruple $(ab, bc, cd, da)$. Repeat arbitrarily many times. Prove that you can never return to the original quadruple unless $a = b = c = d = 1$.
b. Given $n$ a power of $2$, and an $n$-tuple $(a_1, a_2, \ldots, a_n)$ transform to a new $n$-tuple $(a_1 a_2, a_2 a_3, \ldots, a_{n-1} a_n, a_n a_1)$. If all the members of the original $n$-tuple are $1$ or $-1$, prove that with sufficiently many repetitions you obtain all $1$s.
b. Given $n$ a power of $2$, and an $n$-tuple $(a_1, a_2, \ldots, a_n)$ transform to a new $n$-tuple $(a_1 a_2, a_2 a_3, \ldots, a_{n-1} a_n, a_n a_1)$. If all the members of the original $n$-tuple are $1$ or $-1$, prove that with sufficiently many repetitions you obtain all $1$s.