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Part of 2020 Junior Macedonian National Olympiad

Problems(1)

2020 Macedonian Junior BMO TST- P1

Source: 2020 Junior Macedonian Mathematical Olympiad

S

be the set of all positive integers

n

such that each of the numbers

n + 1

n + 3

n + 4

n + 5

n + 6

n + 8

is composite. Determine the largest integer

k

with the following property: For each

n \in S

there exist at least

k

consecutive composite integers in the set {

n, n +1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9

set of positive integersjmmo2020composites