Question 57
Given $X = \{1, 2, \dots, 100\}$, consider function $f: X \to X$ satisfying both the following conditions:
(1) $f(x) \neq x$ for all $x \in X$;
(2) $A \cap f(A) \neq \emptyset$ for all $A \subseteq X$ with $|A| = 40$.
Find the smallest positive integer $k$, such that for any such function $f$ there exists a set $B \subseteq X$ satisfying $|B| = k$ and $B \cup f(B) = X$.
**Remark.** For a subset $T$ of $X$, we define $f(T) = \{x \mid \text{there exists } t \in T \text{ such that } x = f(t)\}$.
(1) $f(x) \neq x$ for all $x \in X$;
(2) $A \cap f(A) \neq \emptyset$ for all $A \subseteq X$ with $|A| = 40$.
Find the smallest positive integer $k$, such that for any such function $f$ there exists a set $B \subseteq X$ satisfying $|B| = k$ and $B \cup f(B) = X$.
**Remark.** For a subset $T$ of $X$, we define $f(T) = \{x \mid \text{there exists } t \in T \text{ such that } x = f(t)\}$.