Question 19
In a mathematical competition some competitors are friends; friendship is always mutual, that is to say that when $A$ is a friend of $B$, then also $B$ is a friend of $A$. We say that $n \geq 3$ different competitors $A_{1}, A_{2}, \ldots, A_{n}$ form a weakly-friendly cycle if $A_{i}$ is not a friend of $A_{i+1}$, for $1 \leq i \leq n\left(A_{n+1}=A_{1}\right)$, 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 weakly-friendly cycle $\mathcal{S}$ of competitors not including $C$, the set of competitors $D$ in $\mathcal{S}$ 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 room are friends.
Problem (Romanian duplicate in the source, translated):
In a mathematical competition some students are friends; friendship is always mutual, that is to say that when $A$ is a friend of $B$, then also $B$ is a friend of $A$. We say that $n \geq 3$ different students $A_{1}, A_{2}, \ldots, A_{n}$ form a weakly-friendly cycle if $A_{i}$ is not a friend of $A_{i+1}$, for $1 \leq i \leq n\left(A_{n+1}=A_{1}\right)$, and there are no other pairs of non-friends among the components of this cycle.
The following property is satisfied:
for every student $C$, and every weakly-friendly cycle $\mathcal{S}$ of students not including $C$, the set of students $D$ in $\mathcal{S}$ which are not friends of $C$ has at most one element.
Prove that all students of this mathematical competition can be arranged into three rooms, such that every two students that are in the same room are friends.
The following property is satisfied:
for every competitor $C$, and every weakly-friendly cycle $\mathcal{S}$ of competitors not including $C$, the set of competitors $D$ in $\mathcal{S}$ 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 room are friends.
Problem (Romanian duplicate in the source, translated):
In a mathematical competition some students are friends; friendship is always mutual, that is to say that when $A$ is a friend of $B$, then also $B$ is a friend of $A$. We say that $n \geq 3$ different students $A_{1}, A_{2}, \ldots, A_{n}$ form a weakly-friendly cycle if $A_{i}$ is not a friend of $A_{i+1}$, for $1 \leq i \leq n\left(A_{n+1}=A_{1}\right)$, and there are no other pairs of non-friends among the components of this cycle.
The following property is satisfied:
for every student $C$, and every weakly-friendly cycle $\mathcal{S}$ of students not including $C$, the set of students $D$ in $\mathcal{S}$ which are not friends of $C$ has at most one element.
Prove that all students of this mathematical competition can be arranged into three rooms, such that every two students that are in the same room are friends.