Question 28
For a finite non-empty set $A$ of real numbers, let $\max(A)$ denote its maximum element, and define:
$$
P(A) = \sum_{\substack{B \subseteq A \\ |B| \text{ odd}}} m(B), \quad Q(A) = \sum_{\substack{\emptyset \neq B \subseteq A \\ |B| \text{ even}}} m(B),
$$
where $m(B)$ is the median of finite non-empty set $B$: if $B = \{b_1, b_2, \dots, b_n\}$ ($b_1 \lt b_2 \lt \dots \lt b_n$), then $m(B) = \frac{1}{2}(b_{\lfloor \frac{n+1}{2} \rfloor} + b_{\lceil \frac{n+1}{2} \rceil})$.
Find the smallest real number $c$ such that for any set $A$ of 2025 distinct positive real numbers,
$$
P(A) - Q(A) \leq c \cdot \max(A).
$$
$$
P(A) = \sum_{\substack{B \subseteq A \\ |B| \text{ odd}}} m(B), \quad Q(A) = \sum_{\substack{\emptyset \neq B \subseteq A \\ |B| \text{ even}}} m(B),
$$
where $m(B)$ is the median of finite non-empty set $B$: if $B = \{b_1, b_2, \dots, b_n\}$ ($b_1 \lt b_2 \lt \dots \lt b_n$), then $m(B) = \frac{1}{2}(b_{\lfloor \frac{n+1}{2} \rfloor} + b_{\lceil \frac{n+1}{2} \rceil})$.
Find the smallest real number $c$ such that for any set $A$ of 2025 distinct positive real numbers,
$$
P(A) - Q(A) \leq c \cdot \max(A).
$$