ISL 1988 Problem 46

$A_1, A_2, \ldots, A_{29}$ are $29$ different sequences of positive integers. For $1 \leq i < j \leq 29$ and any natural number $x,$ we define $N_i(x) =$ number of elements of the sequence $A_i$ which are less or equal to $x,$ and $N_{ij}(x) =$ number of elements of the intersection $A_i \cap A_j$ which are less than or equal to $x.$ It is given for all $1 \leq i \leq 29$ and every natural number $x,$ $$N_i(x) \geq \frac{x}{e},$$ where $e = 2.71828 \ldots$ Prove that there exist at least one pair $i,j$ ( $1 \leq i < j \leq 29$ ) such that $$N_{ij}(1988) > 200.$$