# Task 7. (14 points)

In a certain company, no two employees have jobs of the same difficulty, and no two employees receive the same salary. On April 1, each employee made two statements:

(a) There will not be 12 employees with more difficult jobs.

(b) At least 30 employees have a higher salary. How many employees are in the company if some employees told the truth twice, and the rest lied twice.

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# Solution.

If on April 1st all employees of the company told the truth, then the second statement about the highest salary is false, which cannot be. If, however, all of them lied, then the first statement for the employee with the highest salary would be true, again leading to a contradiction.

Thus, there is at least one liar and at least one truth-teller.

Let's take the employee who told the truth with the highest salary among all truthful employees. Since from his second statement it follows that at least 30 "liars" have a higher salary than he does. The second statement of the lying employee with the lowest salary among all "liars" is false, thus, no more than 29 "liars" have a higher salary, and no more than 30 "liars" in total. That is, there are 30 liars in total.

The first statement for the "liar" with the most difficult job among all "liars" is false, so there are at least 12 truthful employees with more difficult jobs.

The first statement for the truthful employee with the least difficult job among all truthful employees is true, so there are no more than 12 truthful employees in total. That is, there are exactly 12 truthful employees.

Ultimately, we get that the company employs 42 employees. Answer: 42

| Content of the criterion | Evaluation | Points |
| :---: | :---: | :---: |
| Complete solution. | + | 14 |
| The answer is correct. There is no strict proof that there are 30 liars or that there are 12 truthful employees. | $\pm$ | 11 |
| The answer is correct. It is claimed that there are 30 liars and 12 truthful employees, but neither of these facts is proven. | $+/ 2$ | 7 |
| The correct answer is provided. The solution does not specify that there are 30 liars and 12 truthful employees. | $\mp$ | 3 |
| The solution does not meet any of the criteria described above. | $-/ 0$ | 0 |
| Maximum score | 14 |  |