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Oleg always choose two cells

Source: ARO 2005 - problem 9.2

April 30, 2005
pigeonhole principlenumber theory unsolvednumber theory

Problem Statement

Lesha put numbers from 1 to 22222^2 into cells of 22×2222\times 22 board. Can Oleg always choose two cells, adjacent by the side or by vertex, the sum of numbers in which is divisible by 4?
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