Question 42
Let $n$ be a positive integer. Let $P_n = \{2^n, 2^{n-1} \cdot 3, 2^{n-2} \cdot 3^2, \dots, 3^n\}$. For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_\emptyset = 0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \le y \le 3^{n+1} - 2^{n+1}$.
Prove that there is a subset $Y$ of $P_n$ such that $0 \le y - S_Y \lt 2^n$.
Prove that there is a subset $Y$ of $P_n$ such that $0 \le y - S_Y \lt 2^n$.