IMO 1971 LL POL37

Origin: POL

Problem

Let S be a circle, and \alpha = {A1, . . . , An} a family of open arcs in S. Let N(\alpha) = n denote the number of elements in \alpha. We say that \alpha is a covering of S if %n k=1 Ak \supsetS. Let \alpha = {A1, . . . , An} and \beta = {B1, . . . , Bm} be two coverings of S. Show that we can choose from the family of all sets Ai \capBj, i = 1, 2, . . . , n, j = 1, 2, . . ., m, a covering \gamma of S such that N(\gamma) \leqN(\alpha) + N(\beta).