Question 170
For a polynomial $P(x)$ with integer coefficients, $r(2i - 1)$ (for $i = 1, 2, 3, \dots, 512$) is the remainder obtained when $P(2i - 1)$ is divided by $1024$. The sequence
$$
(r(1), r(3), \dots, r(1023))
$$
is called the *remainder sequence* of $P(x)$. A remainder sequence is called *complete* if it is a permutation of $(1, 3, 5, \dots, 1023)$. Prove that there are no more than $2^{35}$ different complete remainder sequences.
$$
(r(1), r(3), \dots, r(1023))
$$
is called the *remainder sequence* of $P(x)$. A remainder sequence is called *complete* if it is a permutation of $(1, 3, 5, \dots, 1023)$. Prove that there are no more than $2^{35}$ different complete remainder sequences.