1. Find such non-zero numbers $n$ and $m$ that the equation $\left(n \cdot 5^{n}\right)^{n}=m \cdot 5^{9}$ holds true.
# Solution.

There are infinitely many such pairs of numbers.

Let's show one of the most natural solutions. We need $n^{n} \cdot 5^{n^{2}}=m \cdot 5^{9}$. If $n=3$, then $5^{n^{2}}=5^{9}$. Now let's calculate $m$: $m=3^{3}=27$. The pair $n=3, m=27$ is a solution.

Let's show how to obtain other solutions. Take an arbitrary $n$. For example, $n=6$.

Then in the left part of the equation, we get: $6^{6} \cdot 5^{36}$, so $m=6^{6} \cdot 5^{27}$.

Grading criteria.

- A correct pair of numbers $n$ and $m$ is provided, and it is shown that for such $n$ and $m$ the equation is correct (or how to find $n$ and $m$, or a check is performed) - 7 points.
- A correct pair of numbers $n$ and $m$ is provided without any explanations - 5 points.
- It is shown that $n$ can be 3, but for this $n$, $m$ is incorrectly found - 2 points.
- Only the correct answer is provided - 1 point.
- The correct answer is provided with incorrect reasoning - 0 points.