Nemotron-IMO-Bench 2026 A4

Let $n>1$ be an integer. Two polynomials $P,Q\in \mathbb{R}[X]$ are called block-similar if for each $i \in \{1, 2, \ldots, n\}$ the sequences $$P(2015i), P(2015i - 1), \ldots, P(2015i - 2014) \, \text{and}\, Q(2015i), Q(2015i - 1), \ldots, Q(2015i - 2014)$$ 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 $n$.