Question 21
In a mathematical competition some competitors are friends; friendship is always mutual, that is to say that is when $A$ is a friend of $B$, then also $B$ is a friend of $A$. We say that $n \ge 3$ different competitors $A_1, A_2, \dots, A_n$ form a weekly-friendly circle if $A_i$ is not a friend of $A_{i+1}$, for $1 \le i \le n$ ($A_{n+1} = A_1$), and there are no other pairs of non-friends among the components of this cycle.
The following property is satisfied:
For every competitor $C$, and every weekly-friendly cycle $\mathcal{A}$ of competitors not including $C$, the set of competitors $D$ in $\mathcal{A}$ which are not friends of $C$ has at most one element.
Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors that are in the same rooms are friends.
The following property is satisfied:
For every competitor $C$, and every weekly-friendly cycle $\mathcal{A}$ of competitors not including $C$, the set of competitors $D$ in $\mathcal{A}$ which are not friends of $C$ has at most one element.
Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors that are in the same rooms are friends.