Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing the last three decimal digits of $n$.
Answer: 32768 is the only such integer.

If $n=m^{3}$ is a solution, then $m$ satisfies $1000 m \leqslant m^{3}<1000(m+1)$. From the first inequality, we get $m^{2} \geqslant 1000$, or $m \geqslant 32$. By the second inequality, we then have

$$
m^{2}<1000 \cdot \frac{m+1}{m} \leqslant 1000 \cdot \frac{33}{32}=1000+\frac{1000}{32} \leqslant 1032,
$$

or $m \leqslant 32$. Hence, $m=32$ and $n=m^{3}=32768$ is the only solution.