5. (20 points) Students' written work is evaluated on a two-point grading system, i.e., a work will either be credited if it is done well, or not credited if it is done poorly. The works are first checked by a neural network, which gives an incorrect answer in $10 \%$ of cases, and then all works deemed uncredited are rechecked manually by experts, who do not make mistakes. The neural network can both incorrectly classify a good work as uncredited and a bad work as credited. It is known that among all the works submitted by students, $20 \%$ are actually bad. What is the smallest percentage of bad works that can be among those that experts recheck after the neural network's selection? Provide the integer part of the number.
Answer: 66.

Solution. We will represent the students' works in the diagram below.

Neural network errors can:
a) classify all 10% of good works as bad;

b) classify part of the good works as bad and part of the bad works as good;

c) classify all 10% of bad works as good.

![](https://cdn.mathpix.com/cropped/2024_05_06_dc247db770ca7d1ba777g-3.jpg?height=425&width=1219&top_left_y=433&top_left_x=401)

Let's analyze the general case. Let \( x \) be the percentage of all works that are incorrectly classified as good (these are the bad works that will not be reviewed by experts); \( 0 \leqslant x \leqslant 10 \). Then the experts will receive for rechecking (in percentages of all works): bad works that the neural network did not misclassify - these are \((20-x) \% \);

good works that were incorrectly classified as bad - these are \((10-x) \% \).

Thus, the percentage of bad works to be rechecked is

\[
\frac{(20-x) \times 100}{(20-x)+(10-x)} = \frac{(20-x) \times 100}{30-2x} = 50 + \frac{500}{30-2x}
\]

This function is increasing over its entire domain, so under the conditions of the problem, it takes the smallest value when \( x = 0 \). This value is \( 66 \frac{2}{3} \), and its integer part is 66.