Determine all pairs of positive integers $(m,n)$ for which an $m\times n$ rectangle can be tiled with (possibly rotated) L-shaped trominos.
To determine all pairs of positive integers \((m, n)\) for which an \(m \times n\) rectangle can be tiled with (possibly rotated) L-shaped trominos, we need to consider the properties of L-shaped trominos and the constraints they impose on the dimensions of the rectangle.

1. **Understanding the L-shaped tromino:**
   An L-shaped tromino is a shape consisting of three unit squares arranged in an L pattern. Each tromino covers exactly 3 unit squares.

2. **Area consideration:**
   For an \(m \times n\) rectangle to be tiled with L-shaped trominos, the total area of the rectangle must be divisible by 3. This is because each tromino covers 3 unit squares. Therefore, we must have:
   \[
   m \times n \equiv 0 \pmod{3}
   \]

3. **Parity consideration:**
   Each L-shaped tromino can be rotated, but it always covers an odd number of squares. Therefore, the product \(m \times n\) must be even. This implies that at least one of \(m\) or \(n\) must be even.

4. **Constructing the tiling:**
   We claim that the only solutions are \((m, n) = (3a, 2b)\) and its permutations, where \(a\) and \(b\) are positive integers. This means that \(m\) must be a multiple of 3 and \(n\) must be a multiple of 2, or vice versa.

5. **Proof by construction:**
   To construct a tiling, we can use \(3 \times 2\) rectangles repeatedly. Each \(3 \times 2\) rectangle can be perfectly tiled with two L-shaped trominos. Therefore, if we can cover the entire \(m \times n\) rectangle with \(3 \times 2\) rectangles, we can tile it with L-shaped trominos.

6. **Verification:**
   - If \(m = 3a\) and \(n = 2b\), then the area of the rectangle is \(m \times n = 3a \times 2b = 6ab\), which is divisible by 3 and even.
   - If \(m = 2b\) and \(n = 3a\), then the area of the rectangle is \(m \times n = 2b \times 3a = 6ab\), which is also divisible by 3 and even.

7. **Necessity of the condition:**
   - If \(m\) is not a multiple of 3, then there will be leftover cells that cannot be covered by L-shaped trominos.
   - If \(n\) is not a multiple of 2, then there will be leftover cells that cannot be covered by L-shaped trominos.

Therefore, the only pairs \((m, n)\) that allow an \(m \times n\) rectangle to be tiled with L-shaped trominos are those where \(m\) is a multiple of 3 and \(n\) is a multiple of 2, or vice versa.

\(\blacksquare\)

The final answer is \( \boxed{ (m, n) = (3a, 2b) } \) and its permutations, where \(a\) and \(b\) are positive integers.