There are 3 clubs. Each club includes exactly k students and each student in one club has atleast $\dfrac{3k}{4}$ friends and in each of the 2 other clubs (i.e. $\frac{3k}{4}$ in one club and $\frac{3k}{4}$ in other one as well).
Prove that we can divide $3\cdot k$ students in k groups of size $3$ ,such that in each group all of the students be friend with each other.
My attempt:
I tried counting in two ways , and my other attempt was to consider two groups and prove there is a perfect matching between each of them(there is and its proven) and try to build those k groups which leads to nothing:)
Use Hall's theorem to find a perfect matching $M$ between two clubs.
Then, form a bipartite graph where one side is the third club, and the other side is the $k$ edges of $M$, where a student of the third club is adjacent to an edge of $M$ if the student is friends with both endpoints.
Apply Hall's theorem again, this time to this bipartite graph.