Problem

Source: Romanian JBTST VI 2007, problem 1

Tags: analytic geometry, geometry, rectangle, combinatorics proposed, combinatorics



Consider an 8x8 board divided in 64 unit squares. We call diagonal in this board a set of 8 squares with the property that on each of the rows and the columns of the board there is exactly one square of the diagonal. Some of the squares of this board are coloured such that in every diagonal there are exactly two coloured squares. Prove that there exist two rows or two columns whose squares are all coloured.