Denote by $P^{(n)}$ the set of all polynomials of degree $n$ the coefficients of which is a permutation of the set of numbers $\{2^0, 2^1,..., 2^n\}$. Find all pairs of natural numbers $(k,d)$  for which there exists a $n$ such that for any polynomial  $p \in P^{(n)}$,  number $P(k)$ is divisible by the number $d$.

(Oleksii Masalitin)
1. **Define the set of polynomials and the problem constraints:**
   Let \( P^{(n)} \) be the set of all polynomials of degree \( n \) whose coefficients are a permutation of the set \( \{2^0, 2^1, \ldots, 2^n\} \). We need to find all pairs of natural numbers \( (k, d) \) such that there exists an \( n \) for which any polynomial \( p \in P^{(n)} \) satisfies \( p(k) \equiv 0 \pmod{d} \).

2. **Consider two specific polynomials:**
   Assume \( p(x) = 1 + 2x + \cdots \) and \( q(x) = 2 + x + \cdots \) are two polynomials in \( P^{(n)} \) with the same coefficients except for the first two terms.

3. **Evaluate the polynomials at \( k \):**
   Since \( d \mid p(k) \) and \( d \mid q(k) \), it follows that \( d \mid (p(k) - q(k)) \).

4. **Calculate the difference:**
   \[
   p(k) - q(k) = (1 + 2k + \cdots) - (2 + k + \cdots) = 2k + 1 - 2 - k = k - 1
   \]
   Therefore, \( d \mid (k - 1) \), which implies \( k \equiv 1 \pmod{d} \).

5. **Evaluate \( p(k) \) for any polynomial \( p \in P^{(n)} \):**
   For any \( p \in P^{(n)} \), we have:
   \[
   p(k) \equiv p(1) \pmod{d}
   \]
   Since \( k \equiv 1 \pmod{d} \).

6. **Sum of the coefficients:**
   The sum of the coefficients of any polynomial in \( P^{(n)} \) is:
   \[
   p(1) = 1 + 2 + 2^2 + \cdots + 2^n = 2^{n+1} - 1
   \]
   We need \( 2^{n+1} - 1 \equiv 0 \pmod{d} \).

7. **Condition for divisibility:**
   This means \( d \) must divide \( 2^{n+1} - 1 \) for some \( n \). Since \( 2^{n+1} - 1 \) is always odd, \( d \) must be odd.

8. **General form of \( k \) and \( d \):**
   Therefore, all possible pairs \( (k, d) \) are of the form \( (b(2a+1) + 1, 2a+1) \) for any nonnegative integers \( a \) and \( b \).

The final answer is \( \boxed{ (b(2a+1) + 1, 2a+1) } \) for any nonnegative integers \( a \) and \( b \).