RO-TST 2004 Problem 2

Let $\{R_i\}_{1\leq i\leq n}$ be a family of disjoint closed rectangular surfaces with total area 4 such that their projections of the $Ox$ axis is an interval. Prove that there exist a triangle with vertices in $\bigcup_{i=1}^n R_i$ which has an area of at least 1.

[Thanks Grobber for the correction]