1. At the competition, there were 24 multiple-choice problems. If a contestant did not circle any answer or circled more than 1 answer, they received 0 points. For circling the correct answer, the contestant received 1 point, and for circling an incorrect answer, they had $\frac{1}{4}$ of a point deducted. What is the maximum number of problems on which the contestant circled the correct answer if they scored 13 points?
I/1. 1. way. Let $P$ be the number of tasks for which the contestant received 1 point and $N$ be the number of tasks for which $\frac{1}{4}$ points were deducted. Then we have $P - \frac{1}{4} N = 13$ or $N = 4P - 52$. Since there are 24 tasks in total, it follows that $P + N \leq 24$ or $5P - 52 \leq 24$. Therefore, $5P \leq 76$ and thus $P \leq 15$. The maximum value of $P$ is 15 and is possible if $N = 8$.

Introduction of the notation $P$ and $N$ (and possibly $S$ for tasks not attempted) ................. 1 point

![](https://cdn.mathpix.com/cropped/2024_06_07_2dd3bccc19bc96d6d7dfg-06.jpg?height=63&width=1633&top_left_y=1096&top_left_x=220)

Writing the inequality $P + N \leq 24$ (or the equation $P + N + S = 24$) . . . . . . . . . 1 point

![](https://cdn.mathpix.com/cropped/2024_06_07_2dd3bccc19bc96d6d7dfg-06.jpg?height=57&width=1633&top_left_y=1194&top_left_x=220)

![](https://cdn.mathpix.com/cropped/2024_06_07_2dd3bccc19bc96d6d7dfg-06.jpg?height=54&width=1633&top_left_y=1252&top_left_x=220)

2. way. The contestant scored 13 points if he correctly solved 15 tasks, incorrectly 8 tasks, and either did not answer or circled more than one answer for one task.

If the contestant had correctly solved at least 16 tasks and scored 13 points, he would have to have received at least 3 negative points, so he would have to have solved at least $3 \cdot 4 = 12$ tasks incorrectly. Since there are 24 tasks in total and $16 + 12 = 28 > 24$, this is not possible. Therefore, he correctly solved at most 15 tasks.[^0]

3. way. The contestant scored 13 points if he correctly solved 15 tasks, incorrectly 8 tasks, and either did not answer or circled more than one answer for one task.

If the contestant had correctly solved at least 16 tasks, he would have solved at most $24 - 16 = 8$ tasks incorrectly and would have received at least $16 - \frac{1}{4} \cdot 8 = 14$ points. Since he received 13 points, this is not possible. Therefore, he correctly solved at most 15 tasks.

![](https://cdn.mathpix.com/cropped/2024_06_07_2dd3bccc19bc96d6d7dfg-06.jpg?height=57&width=1636&top_left_y=2336&top_left_x=216)

Conclusion that from at least 16 correct answers follows at most 8 incorrect answers ............... 2 points Conclusion that he then received at least 14 points .......................................... 2 points

![](https://cdn.mathpix.com/cropped/2024_06_07_2dd3bccc19bc96d6d7dfg-06.jpg?height=57&width=1633&top_left_y=2487&top_left_x=220)
(If the case with 16 correct answers is considered, but the cases with 17, 18, ..., 24 correct answers are not, award a maximum of 4 points.)