11. The cellular figure "corner" consists of a central cell, to which horizontal and vertical rectangles are attached (the figure shows one of the four possible types of corners, the side of each cell is 1, and there are 21 cells in total in the figure). Prove that for any coloring of the cells of a square in 120 colors, it is possible to cut out a corner containing two cells of the same color. ( ( Karpov)
Problem 914
Official solution
11. Consider an square located "deep inside" a square (for example, a square whose central cell coincides with the central cell of the square). Since the considered square contains 121 cells, and there are only 120 colors, it must contain two cells of the same color. It is not difficult to understand that these two cells can always be covered by one corner piece that protrudes beyond the boundaries of the square but lies entirely within the larger square.