Let's share a thought: divide every natural number by the sum of the squares of its digits (for single-digit numbers, divide by the square of the number).

Is there a smallest quotient among the quotients obtained this way, and if so, which one is it?
If we examine some natural numbers, we conjecture that among the quotients, $9 / 81 = 1 / 9$ is the smallest. In the following, we will prove this conjecture.

For single-digit numbers, the quotient in question is $n / n^{2} = 1 / n$, and among these, indeed the smallest is $1 / 9$.

If now the number $n$ is at least two digits and its decimal representation is $a_{k} a_{k-1} \ldots a_{1} a_{0}\left(a_{k} \neq 0, k \geq 1\right)$, then we need to prove that

$$
\frac{n}{\sum_{i=0}^{k} a_{i}^{2}}=\frac{10^{k} a_{k}+10^{k-1} a_{k-1}+\ldots+10 a_{1}+a_{0}}{a_{k}^{2}+a_{k-1}^{2}+\ldots+a_{1}^{2}+a_{0}^{2}}>\frac{1}{9}
$$

After identical transformations,

$$
a_{k}\left(9 \cdot 10^{k}-a_{k}\right)+a_{k-1}\left(9 \cdot 10^{k-1}-a_{k-1}\right)+\ldots+a_{0}\left(9-a_{0}\right)>0
$$

Since $a_{0}, \ldots, a_{k}$ are all digits between 0 and 9, each term on the left side is non-negative, and the first term is positive because $a_{k} \neq 0$ and $k \geq 1$; thus, the inequality in question also holds.

Therefore, $1 / 9$ is indeed the smallest among the quotients.

Bognár Gabriella (Miskolc, Földes F. Gimn., II. o. t.)

Note. In the problem statement, some may find the part "is there a smallest among the quotients obtained this way" somewhat strange, as we are accustomed to the fact that among given numbers, there is always a smallest. However, this is not always the case. For example, there is no smallest integer, rational number, or even real number. There is also no smallest positive rational number: for any positive rational number $g$, $g / 2$ is smaller and also a positive rational number. Similarly, if we consider the reciprocals of all natural numbers, there will be no smallest among the resulting quotients. Therefore, it was not superfluous to include this part in the question.

But when can we be sure that there is a smallest among certain numbers, what tells us that in a set of numbers, there is definitely a smallest element? Well, this is indicated by the number of elements in the set, the cardinality of the set. If the set contains a finite number of numbers, that is, if we can count the numbers and say how many numbers are in the set (ten, a hundred, etc.), then there is definitely a smallest (and largest as well) among the numbers in the set. However, if the set contains an infinite number of numbers, then it is possible, but not certain, that there is a smallest among the numbers.

We would like to draw attention to the fact that in mathematics, the word "infinite" is not always used in a finite sense. In the above example, a set is finite if we can count its constituent numbers and say how many elements the set contains. A set is infinite if we cannot count its constituent elements, as in the case of integers or real numbers.