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VIII OBM

Brazil counting and probability

Problem

A number is written in each square of a chessboard, so that each number not on the border is the mean of the 4 neighboring numbers. Show that if the largest number is , then there is a number equal to in the border squares.
Solution
Take the leftmost . Suppose it is not in the border. Then it must be the mean of the 4 neighboring numbers. Hence each of the 4 neighboring numbers must also be , but one of them is to the left of . Contradiction.

Techniques

Coloring schemes, extremal argumentsLogic