SCHWEITZER 2015 Problem 7

We call a bar of width ${w}$ on the surface of the unit sphere ${\Bbb{S}^2}$ , a spherical segment, centered at the origin, which has width ${w}$ and is symmetric with respect to the origin. Prove that there exists a constant ${c>0}$ , such that for any positive integer ${n}$ the surface ${\Bbb{S}^2}$ can be covered with ${n}$ bars of the same width so that any point is contained in no more than ${c\sqrt{n}}$ bars.