PL-OM 2021 Problem 6

Given an integer $d \ge 2$ and a circle $\omega$ . Hansel drew a finite number of chords of circle $\omega$ . The following condition is fulfilled: each end of each chord drawn is at least an end of $d$ different drawn chords. Prove that there is a drawn chord which intersects at least $\tfrac{d^2}{4}$ other drawn chords. Here we assume that the chords with a common end intersect.

Note: Proof that a certain drawn chord crosses at least $\tfrac{d^2}{8}$ other drawn chords will be awarded two points.