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marked p cells in (k+1)x n table - All-Russian MO 2004 Regional (R4) 10.8

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September 27, 2024
combinatoricsgeometryrectangle

Problem Statement

Given natural numbers p<k<np < k < n. On an endless checkered plane some cells are marked so that in any rectangle (k+1)×n(k + 1) \times n (nn cells horizontally, k+1k + 1 vertically) marked exactly pp cells. Prove that there is a k×(n+1)k \times (n + 1) rectangle (n+1n + 1 cell horizontally, kk - vertically), in which no less than p+1p + 1 cells.
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