Competition problem — Combinatorics

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Combinatorics Difficulty 5.2 AIME, harder Prove it Belarus

$N$ cells are marked on an $n \times n$ table so that at least one marked cell is among any four cells of the table which form the figure $\Box\Box$ (the figure may be rotated). Find the smallest possible value of $N$ .

Solution

Answer: $(n - 3)([n/3] + 1) + 1$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.