ASU 247 All Soviet Union MO 1977 unique node inside a 100x100 square
Source:
July 6, 2019
square tablecombinatoricscombinatorial geometry
Problem Statement
Given a square on the sheet of cross-lined paper. There are several broken lines drawn inside the square. Their links consist of the small squares sides. They are neither pairwise- nor self-intersecting (have no common points). Their ends are on the big square boarder, and all the other vertices are in the big square interior. Prove that there exists (in addition to four big square angles) a node (corresponding to the cross-lining family, inside the big square or on its side) that does not belong to any broken line.