Question 118
After a volleyball tournament (every two teams played exactly once) with $n$ teams it turned out that for any two teams $A$ and $B$, such that $B$ wins over $A$, there exist positive integer $t$ and teams $C_{1}, C_{2}, \ldots, C_{t}$, such that $A$ wins over $C_{1}, C_{1}$ wins over $C_{2}, \ldots, C_{t}$ wins over $B$.
Prove that for any $k=3,4, \ldots, n$ there exist $k$ teams $A_{1}, A_{2}, \ldots, A_{k}$, such that $A_{1}$ wins over $A_{2}, A_{2}$ wins over $A_{3}, \ldots, A_{k-1}$ wins over $A_{k}$ and $A_{k}$ wins over $A_{1}$.
Prove that for any $k=3,4, \ldots, n$ there exist $k$ teams $A_{1}, A_{2}, \ldots, A_{k}$, such that $A_{1}$ wins over $A_{2}, A_{2}$ wins over $A_{3}, \ldots, A_{k-1}$ wins over $A_{k}$ and $A_{k}$ wins over $A_{1}$.