Progetto Olimpiadi della Matematica

Solution:

The answer is (A). Let us consider the polynomial $q(x) = (x+1)^5 - 1$. For every integer $n$ we have
$$q\left(5^{n}-1\right) = \left(5^{n}-1+1\right)^5 - 1 = 5^{5 n} - 1 = p\left(5^{n}-1\right)$$
so the difference polynomial $p(x)-q(x)$ vanishes for infinitely many values of $x$ (all those of the form $5^{n}-1$ with $n$ an integer). It follows that the difference $p(x)-q(x)$ is identically zero, that is $p(x)=q(x)$, so that
$$p(3) = q(3) = 4^5 - 1 = 1023.$$