Question 90
Suppose that $m$ is a positive integer. Let $P_m = \{2^m, 2^{m-1}3, 2^{m-2}3^2, \dots, 3^m\}$. If $X$ is a subset of $P_m$, 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 $0 \le y \le 3^{m+1} - 2^{m+1}$. Prove that there is a subset $Y$ of $P_m$ such that $0 \le y - S_Y \lt 2^m$.