8
Problems(2)
some classical combinatorial geometry
Source: II Caucasus Mathematical Olympiad
3/20/2018
points are marked in the plane so that no three of marked points are collinear. One of marked points is red, and the others are blue. A triangle with vertices at blue points is called good if the red point lies inside it. Determine if it is possible that the number of good triangles is not less than the half of the total number of traingles with vertices at blue points.
combinatorial geometrygeometry
hard combinatorics
Source: II Caucasus Mathematical Olympiad
3/20/2018
Given a table in a form of the regular -gon with sidelength . A Beetle initially is in one of its vertices. All vertices are numbered in some order by numbers , , , so that initially the Beetle is in the vertex . The Beetle can move only along the edges of -gon and only clockwise. He starts to move from vertex and he is moving without stops until he reaches vertex where he has a stop. Then he continues his journey clockwise from vertex until he reaches the vertex where he has a stop, and so on. The Beetle finishes his journey at vertex . Find the number of ways to enumerate all vertices so that the total length of the Beetle's journey is equal to .
combinatorics