Find all positive integers $n$ with the following property: 
there exists a polynomial $P_n(x)$ of degree $n$, with integer coefficients, such that $P_n(0)=0$ and $P_n(x)=n$ for $n$ distinct integer solutions.
To find all positive integers \( n \) such that there exists a polynomial \( P_n(x) \) of degree \( n \) with integer coefficients, satisfying \( P_n(0) = 0 \) and \( P_n(x) = n \) for \( n \) distinct integer solutions, we proceed as follows:

1. **Form of the Polynomial**:
   Since \( P_n(0) = 0 \), we can write \( P_n(x) \) in the form:
   \[
   P_n(x) = xQ(x)
   \]
   where \( Q(x) \) is a polynomial of degree \( n-1 \) with integer coefficients.

2. **Condition for \( P_n(x) = n \)**:
   We need \( P_n(x) = n \) to have \( n \) distinct integer solutions. Let these solutions be \( a_1, a_2, \ldots, a_n \). Then:
   \[
   P_n(a_i) = n \quad \text{for} \quad i = 1, 2, \ldots, n
   \]
   This implies:
   \[
   a_i Q(a_i) = n \quad \text{for} \quad i = 1, 2, \ldots, n
   \]

3. **Form of \( Q(x) \)**:
   Since \( Q(x) \) must be an integer polynomial, we can write:
   \[
   Q(x) = c(x - b_1)(x - b_2) \cdots (x - b_{n-1})
   \]
   where \( c \) is a non-zero integer and \( b_1, b_2, \ldots, b_{n-1} \) are distinct integers.

4. **Product of Roots**:
   Substituting \( Q(x) \) into the equation \( a_i Q(a_i) = n \), we get:
   \[
   a_i c (a_i - b_1)(a_i - b_2) \cdots (a_i - b_{n-1}) = n
   \]
   Since \( a_i \) are distinct integers, none of them can be zero. Therefore, we have:
   \[
   n = c a_1 (a_1 - b_1)(a_1 - b_2) \cdots (a_1 - b_{n-1})
   \]
   and similarly for \( a_2, a_3, \ldots, a_n \).

5. **Bounding \( n \)**:
   For \( n \geq 3 \), consider the product of \( n \) distinct integers. The smallest product of \( n \) distinct non-zero integers is:
   \[
   1 \cdot 2 \cdot 3 \cdots n = n!
   \]
   However, we need to consider the absolute value of the product of \( n \) distinct integers, which can be much larger. For simplicity, we use the inequality:
   \[
   n \geq 2^{n-2}
   \]
   This inequality holds for \( n \geq 3 \). Solving \( n \leq 4 \) for \( n \geq 3 \), we get \( n = 3 \) or \( n = 4 \).

6. **Verification for Small \( n \)**:
   - For \( n = 1 \):
     \[
     P_1(x) = x
     \]
     This satisfies \( P_1(0) = 0 \) and \( P_1(1) = 1 \).

   - For \( n = 2 \):
     \[
     P_2(x) = 2 - (x-1)(x-2)
     \]
     This satisfies \( P_2(0) = 0 \) and \( P_2(1) = 2 \), \( P_2(2) = 2 \).

   - For \( n = 3 \):
     \[
     P_3(x) = 3 + (x-1)(x+1)(x+3)
     \]
     This satisfies \( P_3(0) = 0 \) and \( P_3(1) = 3 \), \( P_3(-1) = 3 \), \( P_3(-3) = 3 \).

   - For \( n = 4 \):
     \[
     P_4(x) = 4 - (x^2 - 1)(x^2 - 4)
     \]
     This satisfies \( P_4(0) = 0 \) and \( P_4(1) = 4 \), \( P_4(-1) = 4 \), \( P_4(2) = 4 \), \( P_4(-2) = 4 \).

Conclusion:
The only positive integers \( n \) that satisfy the given conditions are \( n = 1, 2, 3, 4 \).

The final answer is \( \boxed{ n = 1, 2, 3, 4 } \)