Problem 1472

1. **Define the polynomials $Q(x)$ and $R(x)$:**
$$Q(x) = P(P(P(x))) - P(x)$$
$$R(x) = P(P(x)) - x$$
Note that all roots of $R(x)$ are also roots of $Q(x)$.

2. Identify the roots:
Let $r_1, r_2, \dots, r_{n^2} \in \mathbb{R}$ be the roots of $R(x)$ and let $r_1, r_2, \dots, r_{n^3} \in \mathbb{R}$ be the roots of $Q(x)$.

3. **Express $P(x)$ in standard polynomial form:**
$$P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$$
Let $t = \frac{-a_{n-1}}{a_n}$.

4. **Compute $R(x)$:**
By some computation, we get:
$$R(x) = P(P(x)) - x = a_n^{n+1} x^{n^2} + a_n^n n a_{n-1} x^{n^2 - 1} + A(x)$$
where $\deg(A(x)) \leq n^2 - 2$.

5. **Sum of the roots of $R(x)$:**
By Vieta's formulas, the sum of the roots of $R(x)$ is:
$$r_1 + r_2 + \dots + r_{n^2} = tn$$

6. **Compute $Q(x)$:**
Again, after some computation, we have:
$$Q(x) = P(P(P(x))) - P(x) = a_n^{n^2 + n + 1} x^{n^3} + a_n^{n^2 + n} n^2 a_{n-1} x^{n^3 - 1} + B(x)$$
where $\deg(B(x)) \leq n^3 - 2$.

7. **Sum of the roots of $Q(x)$:**
By Vieta's formulas, the sum of the roots of $Q(x)$ is:
$$r_1 + r_2 + \dots + r_{n^3} = tn^2$$

8. Arithmetic mean of the roots:
The arithmetic mean of the roots of $R(x)$ is:
$$\frac{r_1 + r_2 + \dots + r_{n^2}}{n^2} = \frac{tn}{n^2} = \frac{t}{n}$$
The arithmetic mean of the remaining roots of $Q(x)$ is:
$$\frac{r_{n^2 + 1} + \dots + r_{n^3}}{n^3 - n^2} = \frac{tn^2 - tn}{n^3 - n^2} = \frac{tn(n-1)}{n^2(n-1)} = \frac{t}{n}$$

Thus, the roots can be split into two groups with equal arithmetic mean.