2006 Romanian NMO Problems/Grade 7/Problem 2
Problem
A square of side 
is formed from 
unit squares, each colored in red, yellow or green. Find minimal , such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or column).
Solution
For 
, consider this coloring for a 6x6 board:
RYGRYG GRYGRY YGRYGR RYGRYG GRYGRY YGRYGR
We can take the top -by- grid of this board as a coloring not satisfying the conditions.
For 
, we note that each row or column must have at least one color with 3 or more squares by the pigeonhole principle, so our answer is 7.
