3. Determine the natural numbers $\overline{a b}$ with the property that $\breve{a}\left[\overline{a, b}^{3}\right]=\overline{a b}$. We denote $[x]$ as the integer part of $x$.


Solution.

We write the relation from the statement in the form $\left[\left(\frac{\overline{a b}}{10}\right)^{3}\right]=\overline{a b}$, and from the definition of the integer part, we have $\overline{a b} \leqslant\left(\frac{\overline{a b}}{10}\right)^{3}<\overline{a b}+1$; $1000 \cdot \overline{a b} \leqslant \overline{a b}^{3}<1000 \cdot \overline{a b}+1000$. (1p)

Noting $x=\overline{a b} \in N$, the relation becomes $1000 x \leqslant x^{3}<1000 x+1000$.

From $x^{3}-1000 x=x\left(x^{2}-1000\right) \geqslant 0$, we have $x^{2}-1000 \geqslant 0$ and $x \geqslant 32$. (2p)

In $x^{3}<1000 x+1000, x^{3}-1000 x-1000<0$, for $x=32$, we have $32 \cdot 24<1000$, true. (2p)

For $x=33$, from the same relation we get $1937<0$, false, so values $x \geqslant 33$ cannot be solutions. (2p)

Thus, $x=32$ is the unique solution.