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A 1 x 3 or 3 x 1 array with three different colours

Source: Romanian Master Of Mathematics 2012

March 3, 2012
floor functionceiling functiongeometryrectanglecombinatorics proposedcombinatoricsdouble counting

Problem Statement

Given a positive integer n≥3n\ge 3, colour each cell of an n×nn\times n square array with one of ⌊(n+2)2/3⌋\lfloor (n+2)^2/3\rfloor colours, each colour being used at least once. Prove that there is some 1×31\times 3 or 3×13\times 1 rectangular subarray whose three cells are coloured with three different colours.
(Russia) Ilya Bogdanov, Grigory Chelnokov, Dmitry Khramtsov
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