A theater company performed thirty "Evenings of Improvisation" during the season as planned. Fan, an admirer of the main protagonist, calculated at the beginning of the season how much she would spend in total on admission if she attended every performance. After a few performances, however, the admission price unexpectedly increased by 60 Kč. Later, the company gained a significant sponsor and reduced the new admission price by 85 Kč. At the end of the season, Fan could say that she did not miss a single performance of the program and spent exactly as much on admission as she had calculated at the beginning of the season. How many times did Fan attend the performance at the original admission price?

(L. Šimünek)
The number of shows where the entrance fee was increased by 60 Kč is denoted by $a$. Fans spent $60a$ Kč more than they had planned for these shows. After the reduction, the entrance fee was lower than at the beginning of the season, by $85-60=25$ Kč. The number of shows with this entrance fee is denoted by $b$. Fans paid $25b$ Kč less than originally planned for these shows. The total amount spent corresponds exactly to the plan, so the following must hold:

$$
60a = 25b.
$$

The unknowns $a$ and $b$ are natural numbers. We can rearrange the equation to a form that clearly shows the ratio of these numbers:

$$
\frac{a}{b} = \frac{25}{60} = \frac{5}{12}
$$

The unknown $a$ must therefore be a multiple of five, and the unknown $b$ must be the corresponding multiple of twelve, as shown in the table:

| $a$ | 5 | 10 | 15 | $\ldots$ |
| :---: | :---: | :---: | :---: | :---: |
| $b$ | 12 | 24 | 36 | $\ldots$ |

Since the sum $a + b$ must not exceed 30 according to the problem statement, only the first option is valid. The total number of shows with the changed entrance fee was $5 + 12 = 17$. Therefore, the number of shows with the original entrance fee was $30 - 17 = 13$.

Evaluation. 4 points for recognizing that the numbers of shows $a$ and $b$ are in the ratio 5:12 (award 2 points based on the completeness of the explanation); 1 point for discussing that the solution is unique; 1 point for a correct and clearly formulated answer.

Note. A longer approach can also be chosen, where we substitute natural numbers for $a$ into the equation $60a - 25b = 0$ and each time check whether $b$ also comes out as a natural number. Full points can be awarded for such work if it shows that the problem has a unique solution.