2015 IMO Shortlist A6

Let $n$ be a fixed integer with $n \geq 2$. 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 $n$.

设 $n$ 为固定整数，$n \geq 2$。我们说两个具有实数系数的多项式 $P$ 和 $Q$ 是块相似的，如果对于每个 $i \in\{1,2, \ldots, n\}$ 序列 $$\begin{aligned} & P(2015 i), P(2015 i-1), \ldots, P(2015 i-2014) \quad \text { 和 } \\ & Q(2015 i), Q(2015 i-1), \ldots, Q(2015 i-2014) \end{aligned}$$ 是彼此的排列。 (a) 证明存在$n+1$次的不同块相似多项式。 (b) 证明不存在$n$次的不同块相似多项式。