Find all polynomials of odd degree and with integer coefficients satisfying the following property: for each positive integer , there exists positive integers such that and is the -th power of a rational number for every pair of indices and with .
Find all polynomials $P(x)$ of odd degree $d$ and with integer coefficients satisfying the following property: for each positive integer $n$, there exists $n$ positive integers $x_1, x_2, \ldots, x_n$ such that $\frac12 < \frac{P(x_i)}{P(x_j)} < 2$ and $\frac{P(x_i)}{P(x_j)}$ is the $d$-th power of a rational number for every pair of indices $i$ and $j$ with $1 \leq i, j \leq n$.
Školjka 
of odd degree 
and with integer coefficients satisfying the following property: for each positive integer 
, there exists 
such that 
and 
is the 
and 
with 
.