Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing its last three decimal digits.
1. Let \( a = \sqrt[3]{n} \) and \( b \) be the number formed from the last three digits of \( n \). Then \( b \in \mathbb{Z}^+ \cup \{0\} \).
2. We have \( n = 1000a + b \) and \( a = \frac{n - b}{1000} \in \mathbb{Z}^+ \).
3. Since \( 0 \leq b \leq 999 \), we can write:
   \[
   0 \leq a^3 - 1000a \leq 999
   \]
4. From \( 0 \leq a^3 - 1000a \), we get:
   \[
   a^3 \geq 1000a
   \]
   Dividing both sides by \( a \) (assuming \( a \neq 0 \)):
   \[
   a^2 \geq 1000
   \]
   Taking the square root of both sides:
   \[
   a \geq \sqrt{1000} \approx 31.62
   \]
   Since \( a \) is an integer:
   \[
   a \geq 32
   \]
5. From \( a^3 - 1000a \leq 999 \), we get:
   \[
   a^3 \leq 1000a + 999
   \]
   Adding 1 to both sides:
   \[
   a^3 + 1 \leq 1000a + 1000
   \]
   Factoring the right-hand side:
   \[
   a^3 + 1 \leq 1000(a + 1)
   \]
   Using the identity \( a^3 + 1 = (a + 1)(a^2 - a + 1) \):
   \[
   (a + 1)(a^2 - a + 1) \leq 1000(a + 1)
   \]
   Dividing both sides by \( a + 1 \) (assuming \( a \neq -1 \)):
   \[
   a^2 - a + 1 \leq 1000
   \]
6. Solving the quadratic inequality \( a^2 - a + 1 \leq 1000 \):
   \[
   a^2 - a + 1 \leq 1000
   \]
   Rearranging:
   \[
   a^2 - a - 999 \leq 0
   \]
   Solving the quadratic equation \( a^2 - a - 999 = 0 \) using the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
   \[
   a = \frac{1 \pm \sqrt{1 + 4 \cdot 999}}{2} = \frac{1 \pm \sqrt{3997}}{2}
   \]
   Approximating \( \sqrt{3997} \approx 63.25 \):
   \[
   a = \frac{1 \pm 63.25}{2}
   \]
   This gives two solutions:
   \[
   a \approx \frac{64.25}{2} \approx 32.125 \quad \text{and} \quad a \approx \frac{-62.25}{2} \approx -31.125
   \]
   Since \( a \) must be a positive integer:
   \[
   a \leq 32
   \]
7. Combining the results from steps 4 and 6:
   \[
   32 \leq a \leq 32
   \]
   Therefore:
   \[
   a = 32
   \]
8. Substituting \( a = 32 \) back into the expression for \( n \):
   \[
   n = a^3 = 32^3 = 32768
   \]
   Verifying, the last three digits of 32768 are 768, and removing them gives 32, which is indeed the cube root of 32768.

Thus, \( n = 32768 \) is the only solution.

The final answer is \( \boxed{ n = 32768 } \).