USAMO 1983 Problem 3

$\mathrm{S}_{1}, \mathrm{~S}_{2}, \ldots, \mathrm{S}_{\mathrm{n}}$ are subsets of the real line. Each $\mathrm{S}_{\mathrm{i}}$ is the union of two closed intervals. Any three $\mathrm{S}_{\mathrm{i}}$ have a point in common. Show that there is a point which belongs to at least half the $\mathrm{S}_{\mathrm{i}}$.