# Task 7. (14 points)

In a company, there are 168 employees. Among any four people, at least one can be chosen who is acquainted with the other three. What is the minimum possible number of people who are acquainted with everyone? #
# Solution.

If there are no strangers, then the number of people who know everyone is 168. Let $A$ and $B$ not know each other, then all other people know each other (if $C$ does not know $D$, then in the group $A, B, C, D$ no one knows the other three). If $A$ and $B$ know all the others, then 166 people know everyone. If $A$ also does not know $C$, where $C \neq D$, then $A$, $B$, and $C$ know all the other 165 people (since in any group $A, B, C, D$ only $D$ can be known by the other three), who also know each other. Thus, the minimum number of people who know everyone is 165.

Answer. 165

| Criterion Content  Full solution. | Evaluation | Points |
| :---: | :---: | :---: |
| The answer is correct. All main logical steps of the solution are provided. Some rigorous justifications of individual conclusions are missing. | $\pm$ | 11 |
| The answer is correct. The solution is missing or incorrect. | $\mp$ | 14 |
| The main idea of the solution is found. The answer is missing or |  | 3 |

All-Russian School Olympiad "Mission Possible. Your Calling - Financier!"

| incorrect. |  |  |
| :---: | :---: | :---: |
| The solution does not meet any of the criteria described above. | $-/ 0$ | 0 |
| Maximum score |  | 14 |