Determine all polynomials $P(x)$ with integer coefficients such that, for any positive integer $n$, the equation $P(x)=2^n$ has an integer root.
1. Let \( P(x) \) be a polynomial with integer coefficients such that for any positive integer \( n \), the equation \( P(x) = 2^n \) has an integer root. Let \( m = \deg P \) and let \( a \) be the leading coefficient of \( P \). Let \( x_n \) be an integer solution to \( P(x) = 2^n \).

2. Since \( \lim_{n \to \infty} |x_n| = +\infty \), we have:
   \[
   \lim_{n \to \infty} \frac{a |x_n|^m}{2^n} = 1 \implies \lim_{n \to \infty} \left| \frac{x_{n+1}}{x_n} \right| = \sqrt[m]{2}.
   \]

3. Since \( (x_{n+1} - x_n) \mid P(x_{n+1}) - P(x_n) \), we have \( |x_{n+1} - x_n| = 2^{k_n} \) for some \( k_n \geq 0 \). Thus:
   \[
   \left| \frac{x_{n+1}}{x_n} \right| = \frac{2^{k_n}}{|x_n|} + \epsilon_n,
   \]
   where \( \epsilon_n = \pm 1 \).

4. It follows that:
   \[
   \sqrt[m]{2} = \lim_{n \to \infty} \left( \frac{2^{k_n}}{|x_n|} + \epsilon_n \right) = \lim_{n \to \infty} \left( 2^{k_n} \cdot \sqrt[m]{\frac{a}{2^n}} + \epsilon_n \right).
   \]

5. Note that \( \epsilon_n \) equals \( 1 \) or \( -1 \) for infinitely many \( n \). Without loss of generality, consider the case where \( \epsilon_n = 1 \). Let \( s_i \) be the set of indices for which \( \epsilon_{s_i} = 1 \). Then:
   \[
   \sqrt[m]{2} - 1 = \sqrt[m]{a} \lim_{i \to \infty} 2^{k_{s_i} - s_i},
   \]
   so \( k_{s_i} - s_i \) converges to some integer \( \ell \).

6. It follows that:
   \[
   \left( \sqrt[m]{2} - 1 \right)^m = a 2^{m \ell}
   \]
   is an integer. By Eisenstein's criterion, \( x^m - 2 \) is irreducible, so \( (x + 1)^m - 2 \) is the minimal polynomial of \( x = \sqrt[m]{2} - 1 \). Hence:
   \[
   (x + 1)^m - 2 = x^m - a 2^{m \ell} \implies m = 1.
   \]

7. Let \( P(x) = ax + b \). Then \( a(x_2 - x_1) \mid 2 \), so it follows that:
   \[
   P(x) = a(x + b),
   \]
   where \( a \in \{\pm 1, \pm 2\} \) and \( b \in \mathbb{Z} \).

The final answer is \( \boxed{ P(x) = a(x + b) } \), where \( a \in \{\pm 1, \pm 2\} \) and \( b \in \mathbb{Z} \).