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Maximum size of a set with integer means

High school · ●●●○○ 50 pts

Let SS be a set of positive integers with the property that for any x∈Sx \in S, the arithmetic mean of the remaining numbers in SS after removing xx is a positive integer. Suppose 1∈S1 \in S and 20162016 is the largest element of SS. Find the maximum value of ∣S∣|S|.

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divisibilityarithmetic meannumber theoryextremal construction