MathDB

In a mathematical competition, some competitors are friends; friendship is mutual, that is, when

A

is a friend of

B

, then

B

is also 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

(where

A_{n+1} = A_1

), and there are no other pairs of non-friends among the components of the 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

\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 in the same room are friends.
(Serbia)

Problem Statement