CMO 2014 Problem 5

Let $f:X\rightarrow X$ , where $X=\{1,2,\ldots ,100\}$ , be a function satisfying:

1. $f(x)\neq x$ for all $x=1,2,\ldots,100$ ;
2. for any subset $A$ of $X$ such that $|A|=40$ , we have $A\cap f(A)\neq\emptyset$ . Find the minimum $k$ such that for any such function $f$ , there exist a subset $B$ of $X$ , where $|B|=k$ , such that $B\cup f(B)=X$ .