Determine all polynomials $p(x)$ with real coefficients such that

$$
p\left((x+1)^{3}\right)=(p(x)+1)^{3}
$$

and

$$
p(0)=0
$$

Answer: $p(x)=x$.
Consider the sequence defined by

$$
\left\{\begin{array}{l}
a_{0}=0 \\
a_{n+1}=\left(a_{n}+1\right)^{3}
\end{array}\right.
$$

It follows inductively that \( p(a_{n}) = a_{n} \). Since the polynomials \( p \) and \( x \) agree on infinitely many points, they must be equal, so \( p(x) = x \).