To solve this problem, we are tasked with finding all polynomials $P(x)$ of odd degree $d$ with integer coefficients satisfying a specific condition. The condition states that for each positive integer $n$, there exist $n$ positive integers $x_1, x_2, \ldots, x_n$ such that the ratio $\frac{P(x_i)}{P(x_j)}$ lies strictly between $\frac{1}{2}$ and $2$ and is a $d$-th power of a rational number for every pair of indices $i, j$.

### Analysis

1. Polynomial Structure:

Since $P(x)$ is of odd degree $d$, we express it in the form:
$$P(x) = a_d x^d + a_{d-1} x^{d-1} + \cdots + a_1 x + a_0$$

The degree $d$ being odd ensures that the leading coefficient $a_d \neq 0$.

2. Condition on Ratios:

The condition that $\frac{1}{2} < \frac{P(x_i)}{P(x_j)} < 2$ and $\frac{P(x_i)}{P(x_j)}$ is a $d$-th power indicates certain divisibility and growth controls on $P(x)$. Rewriting this condition implies:

$$P(x_i) = \left(\frac{p}{q}\right)^d P(x_j)$$

where $\left(\frac{p}{q}\right)$ is a reduced rational number and $(p/q)^d$ indicates that the ratio is indeed a $d$-th power.

3. Implications on Form:

For the above to hold for arbitrary $n$, particularly as $n$ grows, implies that the polynomial $P(x)$ must retain a consistent ratio property. This strongly suggests a form based on scaled and shifted integer variables.

4. Determining the Polynomial:

A suitable candidate satisfying these conditions is:
$$P(x) = a(rx + s)^d$$

Here, $a, r, s$ are integers, with $a \neq 0$, $r \geq 1$, and $(r, s) = 1$ ensuring that the transformation and scaling do not introduce any non-integer terms or additional roots that disrupt the integer coefficient condition.

### Validation:

- Integer Coefficients:
By the form $(rx+s)^d$, expansion ensures integer coefficients since $r$ and $s$ are integer and relatively prime.

- Degree Check:
The degree of $P(x)$ remains $d$ as desired.

- Condition Satisfaction:
For $\frac{P(x_i)}{P(x_j)} = \left(\frac{rx_i+s}{rx_j+s}\right)^d$, the ratios naturally scale as $d$-th powers of rational numbers, which also lie in the (1/2, 2) interval for sufficiently close choices of $x_i$ and $x_j$.

With these considerations, we conclude that the polynomials satisfying all conditions are indeed of the form:
$$P(x) = a(rx + s)^d$$
where $a, r, s$ are integers with $a \neq 0$, $r \geq 1$, and $(r, s) = 1$.

### Final Answer:
$$\boxed{P(x) = a(rx + s)^d \text{ where } a, r, s \text{ are integers with } a \neq 0, r \geq 1 \text{ and } (r, s) = 1.}$$