At the Elementary School U Tří dubů, where Zikmund also attends, they annually organize a knowledge competition where each participant can earn a maximum of 15 points. This year, the average score of the participants, rounded to tenths, was 10.4. After the competition, Zikmund realized that he had misread one question and answered something else. He could have had 4 more points, and the average score, rounded to tenths, would have increased to 10.6.

Determine the minimum and maximum number of children who participated in the competition this year at U Tří dubů. (M. Petrová)
We are working with rounded numbers, which means the actual average point gain could have been in the range from 10.35 inclusive to 10.45 exclusive (this number is already rounded to 10.5). If \( n \) denotes the number of participants in the competition and \( c \) the total sum of points earned by all participants, then the previous condition can be written as:

$$
10.35 \leqq \frac{c}{n} < 10.45 \\
\frac{c+4}{n} > 10.55 \\
\frac{c}{n} - \frac{4}{n} < 10.35 \\
\frac{4}{n} > \frac{10}{3} - 10.35 \\
40 > n > \frac{40}{3}
$$

These constraints are satisfied by all natural numbers from 14 to 39.

The competition was attended by at least 14 and at most 39 children.

Notes. Notice that both inequalities in (3) are strict: for the lower bound, we subtract from 10.55 the largest possible value of \(\frac{c}{n}\), which is strictly less than 10.45; for the upper bound, we subtract from a value strictly less than 10.65 the smallest possible value of \(\frac{c}{n}\), which is 10.35. Information about the maximum number of points that each participant can earn is redundant.

Hint for another solution. The possible number of participants can also be found by trial and error. Assuming that the values of the average point gains are exact, conditions (1) and (2) would have been replaced by equalities

$$
\frac{c}{n} = 10.4, \quad \frac{c+4}{n} = 10.6.
$$

Substituting the first equation into the second and simplifying, we get \(\frac{4}{n} = 0.2\), hence \( n = 20 \). This is a valid number of participants, and other valid possibilities can be found by testing nearby numbers and verifying the conditions from the problem. It is not necessary to proceed completely systematically; it is sufficient to find the boundary values for which the conditions hold, but for the successor or predecessor, they do not. For example, verifying the upper limit of the number of participants looks like this:

- Condition (1) for \( n = 39 \) and its step-by-step transformations give

$$
\begin{aligned}
10.35 & \leqq \frac{c}{39} < 10.45, \\
403.65 & \leqq c < 407.55 \\
407.65 & \leqq c+4 < 411.55 \\
10.4525541 & \frac{c+4}{39} < 10.5525641 .
\end{aligned}
$$

These constraints do not contradict the constraints (2); they only refine them. Thus, the number of participants could have been 39.

- Condition (1) for \( n = 40 \) and its step-by-step transformations give

$$
\begin{gathered}
10.35 \leqq \frac{c}{40} < 10.45 \\
414 \leqq c < 418 \\
418 \leqq c+4 < 422 \\
10.45 \leqq \frac{c+4}{40} < 10.55 .
\end{gathered}
$$

These constraints are in conflict with constraints (2) — no number is strictly less than 10.55 and at the same time greater than or equal to 10.55. Thus, the number of participants could not have been 40.