IMO 2017 Problem 5

An integer $N \ge 2$ is given. A collection of $N(N + 1)$ soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove $N(N - 1)$ players from this row leaving a new row of $2N$ players in which the following $N$ conditions hold: ( $1$ ) no one stands between the two tallest players, ( $2$ ) no one stands between the third and fourth tallest players, $\;\;\vdots$ ( $N$ ) no one stands between the two shortest players.

Show that this is always possible.

Proposed by Grigory Chelnokov, Russia