Question 82
Let $n$ and $k$ be positive integers with $n \gt k \ge 1$. There are $2n+1$ students standing in a circle. For each student, his $2k$ neighbors are the $k$ students closest to him on his left-hand side and the $k$ students closest to him on his right-hand side.
It is known that exactly $n+1$ of the students are girls. Prove that there is a girl such that at least $k$ of her $2k$ neighbors are girls.
It is known that exactly $n+1$ of the students are girls. Prove that there is a girl such that at least $k$ of her $2k$ neighbors are girls.