International Mathematical Olympiad 2017 Problem 5

An integer $N \geqslant 2$ 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.