RO-TST 2004 Problem 9

Let $n\geq 2$ be a positive integer, and $X$ a set with $n$ elements. Let $A_{1},A_{2},\ldots,A_{101}$ be subsets of $X$ such that the union of any $50$ of them has more than $\frac{50}{51}n$ elements.

Prove that among these $101$ subsets there exist $3$ subsets such that any two of them have a common element.