5
Part of 2016 BAMO
Problems(2)
Chessboard parallelograms
Source: 2016 BAMO-8 #5
2/24/2016
For consider an chessboard and place identical pieces at the centers of different squares.
[*] Show that no matter how identical pieces are placed on the board, that one can always find pieces among them that are the vertices of a parallelogram.
[*] Show that there is a way to place identical chess pieces so that no of them are the vertices of a parallelogram.
geometryparallelogramcombinatorics
The "MOAB" Problem
Source: 2016 BAMO-12 #5
2/25/2016
The corners of a fixed convex (but not necessarily regular) -gon are labeled with distinct letters. If an observer stands at a point in the plane of the polygon, but outside the polygon, they see the letters in some order from left to right, and they spell a "word" (that is, a string of letters; it doesn't need to be a word in any language). For example, in the diagram below (where ), an observer at point would read "," while an observer at point would read "."Diagram to be added soonDetermine, as a formula in terms of , the maximum number of distinct -letter words which may be read in this manner from a single -gon. Do not count words in which some letter is missing because it is directly behind another letter from the viewer's position.
B8Perspectivecombinatorial geometry