ASU 133 All Soviet Union MO 1970 91 castle rooms and triangle split
Source:
June 24, 2019
geometrycombinatoricscombinatorial geometry
Problem Statement
a) A castle is equilateral triangle with the side of metres. It is divided onto triangle rooms. Each wall between the rooms is metres long and contain one door. You are inside and are allowed to pass through every door not more than once. Prove that you can visit not more than room (not exiting the castle). b) Every side of the triangle is divided onto parts by the lines parallel to the sides. And the triangle is divided onto small triangles. Let us call the "chain" such a sequence of triangles, that every triangle in it is included only once, and the consecutive triangles have the common side. What is the greatest possible number of the triangles in the chain?