Question 48
Let $m, n$ be positive integers. There are $n$ piles of gold coins, where the $i^{th}$ pile contains $a_i (\gt 0)$ coins ($i = 1, \dots, n$). Consider the following game:
**Step 1** Bob selects sets $B_1, B_2, \dots, B_n$ such that each set $B_i$ is a nonempty subset of $\{1, 2, \dots, m\}$.
**Step 2** Alice selects a set $S$ such that $S$ is a nonempty subset of $\{1, 2, \dots, m\}$, knowing the sets $B_1, B_2, \dots, B_n$ that Bob has selected in Step 1.
**Step 3** The gold coins in the $i^{th}$ pile ($i = 1, 2, \dots, n$) is given to Bob if the number of elements in $B_i \cap S$ is even. Otherwise, they will be given to Alice.
Show that no matter how Bob selects the sets $B_1, B_2, \dots, B_n$, Alice will be able to select a set $S$ so as to receive more gold coins than Bob in total.
**Step 1** Bob selects sets $B_1, B_2, \dots, B_n$ such that each set $B_i$ is a nonempty subset of $\{1, 2, \dots, m\}$.
**Step 2** Alice selects a set $S$ such that $S$ is a nonempty subset of $\{1, 2, \dots, m\}$, knowing the sets $B_1, B_2, \dots, B_n$ that Bob has selected in Step 1.
**Step 3** The gold coins in the $i^{th}$ pile ($i = 1, 2, \dots, n$) is given to Bob if the number of elements in $B_i \cap S$ is even. Otherwise, they will be given to Alice.
Show that no matter how Bob selects the sets $B_1, B_2, \dots, B_n$, Alice will be able to select a set $S$ so as to receive more gold coins than Bob in total.