5
Part of 2006 May Olympiad
Problems(2)
triangular grid of 28 points
Source: XII May Olympiad (Olimpiada de Mayo) 2006 L2 P5
9/19/2022
With points, a “triangular grid” of equal sides is formed, as shown in the figure.
One operation consists of choosing three points that are the vertices of an equilateral triangle and removing these three points from the grid. If after performing several of these operations there is only one point left, in what positions can that point remain?
Give all the possibilities and indicate in each case the operations carried out.
Justify why the remaining point cannot be in another position.
https://cdn.artofproblemsolving.com/attachments/f/c/1cedfe0e1c5086b77151538265f8e253e93d2e.gif
combinatoricscombinatorial geometry
pieces on a 10x10 board
Source: XII May Olympiad (Olimpiada de Mayo) 2006 L1 P5
9/22/2022
In some squares of a board, a piece is placed in such a way that the following property is satisfied: For each square that has a piece, the number of pieces placed in the same row must be greater than or equal to the number of pieces placed in the same column. How many tiles can there be on the board? Give all chances.
combinatorics