2016 IMO Shortlist N8

Find all polynomials $P(x)$ of odd degree $d$ and with integer coefficients satisfying the following property: for each positive integer $n$, there exist $n$ positive integers $x_{1}, x_{2}, \ldots, x_{n}$ such that $\frac{1}{2}<\frac{P\left(x_{i}\right)}{P\left(x_{j}\right)}<2$ and $\frac{P\left(x_{i}\right)}{P\left(x_{j}\right)}$ is the $d$-th power of a rational number for every pair of indices $i$ and $j$ with $1 \leq i, j \leq n$.

找到所有奇数次 $d$ 且整数系数满足以下性质的多项式 $P(x)$：对于每个正整数 $n$，存在 $n$ 个正整数 $x_{1}, x_{2}, \ldots, x_{n}$ 使得 $\frac{1}{2}<\frac{P\left(x_{i}\right)}{P\left(x_{j}\right)}<2$ 且$\frac{P\left(x_{i}\right)}{P\left(x_{j}\right)}$ 是每对索引 $i$ 和 $j$ 的有理数的 $d$ 次幂，其中 $1 \leq i, j \leq n$。