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2020 Macedonian Junior BMO TST- P1

Source: 2020 Junior Macedonian Mathematical Olympiad

September 7, 2020
set of positive integersjmmo2020composites

Problem Statement

Let SS be the set of all positive integers nn such that each of the numbers n+1n + 1, n+3n + 3, n+4n + 4, n+5n + 5, n+6n + 6, and n+8n + 8 is composite. Determine the largest integer kk with the following property: For each nāˆˆSn \in S there exist at least kk consecutive composite integers in the set {n,n+1,n+2,n+3,n+4,n+5,n+6,n+7,n+8,n+9n, n +1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9}.
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