PL-OM 2022 Problem 3

One has marked $n$ points on a circle and has drawn a certain number of chords whose endpoints are the marked points. It turned out that the following property is satisfied: whenever any $2021$ drawn chords are removed one can join any two marked points by a broken line composed of some of the remaining drawn chords. Prove that one can remove some of the drawn chords so that at most $2022n$ chords remain and the property described above is preserved.