2017 IMO Shortlist C4

Let $N \geq 2$ be an integer. $N(N+1)$ soccer players, no two of the same height, stand in a row in some order. Coach Ralph wants to remove $N(N-1)$ people from this row so that in the remaining row of $2 N$ players, no one stands between the two tallest ones, no one stands between the third and the fourth tallest ones, ..., and finally no one stands between the two shortest ones. Show that this is always possible. (Russia)

设 $N \geq 2$ 为整数。 $N(N+1)$ 个足球运动员，没有两个身高相同，按某种顺序站成一排。拉尔夫教练想要从这一排中删除 $N(N-1)$ 人，这样在剩下的一排 $2 N$ 球员中，没有人站在两个最高的人之间，没有人站在第三个和第四个最高的人之间，......，最后没有人站在两个最矮的人之间。证明这总是可能的。 (俄罗斯)