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Belarus — Selection and Training Session

Combinatorics Difficulty 5.1 AIME, harder Prove it Belarus

Is it possible to arrange the numbers $1, 2, \ldots, 2011$ over the circle in some order so that among any $25$ successive numbers at least $8$ numbers are multiples of $5$ or $7$ (or both $5$ and $7$ )?

Solution

Answer: it is not possible.

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