2021 IMO Shortlist C5

Let $n$ and $k$ be two integers with $n>k \geq 1$. 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.

令$n$ 和$k$ 为两个整数且$n>k \geq 1$。有 $2 n+1$ 学生围成一圈。每个学生 $S$ 有 $2 k$ 个邻居 - 即，右边最接近 $S$ 的 $k$ 学生，以及左边最接近 $S$ 的 $k$ 学生。假设 $n+1$ 名学生是女生，其余 $n$ 是男生。证明有一个女孩的邻居中至少有 $k$ 个女孩。