IMO 1988 LL ISR51

Let A1, A2, . . . , A29 be 29 different sequences of positive integers.

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IMO 1988 LL ISR51

Origin: ISR

Problem

Let A1, A2, . . . , A29 be 29 different sequences of positive integers. For 1 \leqi < j \leq29 and any natural number x, we define Ni(x) to be the number of elements of the sequence Ai that are less than or equal to x, and Nij(x) to be the number of elements of the intersection Ai \capAj that are less than or equal to x. It is given that for all 1 \leqi \leq29 and every natural number x, Ni(x) \geqx e , where e = 2.71828 . . . . Prove that there exists at least one pair i, j (1 \leqi < j \leq29) such that Nij(1988) > 200.