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A\cap l_j =\emptyset = B\cap l_j, A\cap D_j \ne \emptyset \ne B\cap D_j

Source: Czech And Slovak Mathematical Olympiad, Round III, Category A 1986 p4

February 17, 2020
SubsetscombinatoricsconvexSets

Problem Statement

Let C1,C2C_1,C_2, and C3C_3 be points inside a bounded convex planar set MM. Rays l1,l2,l3l_1,l_2,l_3 emanating from C1,C2,C3C_1,C_2,C_3 respectively partition the complement of the set M∪l1∪l2∪l3M \cup l_1 \cup l_2 \cup l_3 into three regions D1,D2,D3D_1,D_2,D_3. Prove that if the convex sets AA and BB satisfy A∩lj=∅=B∩ljA\cap l_j =\emptyset = B\cap l_j and A∩Dj≠∅≠B∩DjA\cap D_j \ne \emptyset \ne B\cap D_j for j=1,2,3j = 1,2,3, then A∩B≠∅A\cap B \ne \emptyset
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