Question 182
A nonempty set $A$ of integers is called a “beautiful set” if for any $a \in A$ and $k \in \{1, 2, \dots, 2023\}$, the set
$$
\{b \in A \mid \lfloor \frac{b}{3^k} \rfloor = \lfloor \frac{a}{3^k} \rfloor \}
$$
has exactly $2^k$ elements.
Prove that: if the intersection of an integer set $S$ and any beautiful set is not empty, then $S$ contains a beautiful set.
*Note:* Here $\lfloor x \rfloor$ represents the greatest integer not exceeding $x$.
$$
\{b \in A \mid \lfloor \frac{b}{3^k} \rfloor = \lfloor \frac{a}{3^k} \rfloor \}
$$
has exactly $2^k$ elements.
Prove that: if the intersection of an integer set $S$ and any beautiful set is not empty, then $S$ contains a beautiful set.
*Note:* Here $\lfloor x \rfloor$ represents the greatest integer not exceeding $x$.