2006 Romanian NMO Problems/Grade 7/Problem 2

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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:

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.

See also

• 2006 Romanian NMO Problems