IMO 1983 LL ISR36

The set X has 1983 members. There exists a family of subsets

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IMO 1983 LL ISR36

Origin: ISR

Problem

The set X has 1983 members. There exists a family of subsets {S1, S2, . . . , Sk} such that: (i) the union of any three of these subsets is the entire set X, while (ii) the union of any two of them contains at most 1979 members. What is the largest possible value of k?