KöMaL problem A, 2016

Combinatorics Difficulty 6.0 National Olympiad Prove it Hungary

We have colour pearls placed on an $n\times n$ board; a square may contain more than one pearl. Altogether we used $2n-1$ colours and $n$ pearls from each colour. The pearls are arranged in such a way that no row or column contains more than one pearl of the same colour. Prove that it is possible to select $n$ pearls with distinct colours such that no two of them are in the same row or column.
(5 pont)

We don't reproduce this publisher's solutions. Their own solution is here — work the problem first.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: KöMaL, licensed Rights held by KöMaL and the MATFUND Foundation. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.