8.2. The work was divided equally among the workers in the team. After the first day, they counted how many people had completed no less than 30 percent of their share; it turned out that 70 percent of all workers had done so. When they counted only those who had completed no less than 70 percent of their share, it turned out that 30 percent of the workers had done so. Can we be sure that at least a third of the work has been completed? Justify your answer.
Solution: Let's provide an example where the condition of the problem is met, but a third of the work is not done. Suppose there were 10 workers, and the work involves creating 300 identical parts. Then each worker needs to make 30 parts, and a third of the work consists of 100 parts. Suppose three workers made 21 parts each (exactly 70%), another four made 9 parts each (exactly 30%), and the remaining three did not start working. Then the condition of the problem is met, but only 99 parts were made in total, which is less than a third.

Answer: No.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| A correct counterexample to the question of the problem is provided | 7 points |
| A specific case is considered, showing that at least a third of the work could have been done | 0 points |
| Incorrect counterexample or incorrect justification of the absence of a counterexample | 0 points |