KöMaL problem A, 2015

We call a bar of width $w$ on the surface $S^2$ of the unit sphere in $3$-dimension, centered at the origin a spherical zone which has width $w$ and is symmetric with respect to the origin. Prove that there exists a constant $c>0$ such that for every positive integer $n$ the surface $S^2$ can be covered with $n$ bars of the same width so that every point is contained in no more than $c\sqrt{n}$ bars.
Miklós Schweitzer competition, 2015
(5 pont)