Question 75
Let $n$ be a fixed integer with $n \geqslant 2$. We say that two polynomials $P$ and $Q$ with real coefficients are block-similar if for each $i \in\{1,2, \ldots, n\}$ the sequences
$$
\begin{aligned}
& P(2015 i), P(2015 i-1), \ldots, P(2015 i-2014) \quad \text{ and } \\
& Q(2015 i), Q(2015 i-1), \ldots, Q(2015 i-2014)
\end{aligned}
$$
are permutations of each other.
a. Prove that there exist distinct block-similar polynomials of degree $n+1$.
b. Prove that there do not exist distinct block-similar polynomials of degree $n$.
$$
\begin{aligned}
& P(2015 i), P(2015 i-1), \ldots, P(2015 i-2014) \quad \text{ and } \\
& Q(2015 i), Q(2015 i-1), \ldots, Q(2015 i-2014)
\end{aligned}
$$
are permutations of each other.
a. Prove that there exist distinct block-similar polynomials of degree $n+1$.
b. Prove that there do not exist distinct block-similar polynomials of degree $n$.