Question 13
Let $n$ and $k$ be two integers with $n\gt k \geqslant 1$. There are $2 n+1$ students standing in a circle. Each student $S$ has $2 k$ neighbours - namely, the $k$ students closest to $S$ on the right, and the $k$ students closest to $S$ on the left.
Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbours.
Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbours.