Let's determine the smallest and the largest value of the natural number $n$, for which there are 100 different natural numbers between $0.5 n$ and $0.6 n$.
Let $n$ be a natural number such that the natural numbers between $0.5n$ and $0.6n$, different from them, are $k, k+1, \ldots, k+99$, so $k-1 \leq 0.5n$ and $k+100 \geq 0.6n$. Then the length of the segment between the points $0.5n$ and $0.6n$ on the number line is equal to the sum of the lengths of the segments between the points $0.5n$ and $k$, $k$ and $k+99$, and $k+99$ and $0.6n$. The length of the first segment is clearly 0.5 if $n$ is odd and 1 if $n$ is even; the second is 99, and the third is 1 if $n$ is divisible by 5, since then $0.6n$ is an integer, and if the remainder of $n$ divided by 5 is $1, 2, 3, 4$, then it is respectively $0.6, 0.2, 0.8$, or $0.4$, since, for example, $0.6(5m+1) = 3m+0.6$.

The length of the mentioned segment is also $0.1n$; thus, we get upper and lower bounds for $n$ by considering the lengths of the two extreme segments as large as possible and as small as possible:

$$
\begin{aligned}
& 0.1n \leq 1+99+1=101, \quad n \leq 1010, \text{ or } \\
& 0.1n \geq 0.5+99+0.2=99.7, \quad n \geq 997
\end{aligned}
$$

For $n=1010$, indeed 100 natural numbers fall between $0.5n$ and $0.6n$, because $n$ is even and divisible by 5, so the largest desired value of $n$ is 1010. Similarly, $n=997$ is the smallest desired value, because it is odd and gives a remainder of 2 when divided by 5 (The natural numbers in question are $506, 507, \ldots, 605$, and $499, 500, \ldots, 598$.)

Frey Julianna (Budapest, Kölcsey F. Gymnasium, 2nd grade)

Note. Our result does not mean that for every integer value of $n$ between 997 and 1010, 100 natural numbers fall between $0.5n$ and $0.6n$; counterexamples: $n=1000, 1007$.