1. Determine for which pairs of positive integers $m$ and $n$ the following inequality holds:

$$
\sqrt{m^{2}-4}<2 \sqrt{n}-m<\sqrt{m^{2}-2} .
$$
1. If the natural numbers $m, n$ satisfy the given inequalities, it must clearly hold that

$$
m \geqq 2 \quad \text { and } \quad 2 \sqrt{n}-m>0
$$

otherwise, the represented square roots would not be defined, or the middle expression $2 \sqrt{n}-m$ would be non-positive, and thus could not be greater than the non-negative expression $\sqrt{m^{2}-4}$.

Assume that conditions (1) are satisfied and equivalently transform each of the given inequalities in one column (in each of the four squaring operations, both sides are defined and have non-negative values, and both divisions by the positive number $n$ are valid):

$$
\begin{array}{rlrlrl}
2 \sqrt{n}-m & 0, \\
2 \sqrt{m^{2}-2} & >m,\left.\quad\right|^{2} \\
4\left(m^{2}-2\right) & >m^{2}, \\
3 m^{2} & >8 .
\end{array}
$$

The last inequality, however, is already satisfied for any $m \geqq 2$.

Answer. The sought pairs are precisely the pairs of the form $(m, n)=\left(m, m^{2}-2\right)$, where $m \geqq 2$ is any natural number.

For a complete solution, award 6 points. For finding the interval for the value of $m^{2}$ (or solving the system of given inequalities with the unknown $m$ and parameter $n$ correctly), award 2 points for correct consequential transformations. Award another 2 points for the consideration of the integer nature leading to the relation $n=m^{2}-2$; the remaining 2 points are for the verification, where it is necessary to justify the validity of the original inequalities for the found pairs, explaining why the performed squaring operations are correct. If the value $m=1$ is not excluded in the verification, deduct 1 point.

If the solver establishes general conditions under which all transformations of the given inequalities are equivalent, then award 5 points for deriving the relation $n=m^{2}-2$ and 1 point for the subsequent verification that for every $m \geqq 2$ the pair $(m, n)=\left(m, m^{2}-2\right)$ satisfies the established conditions.