Question 59
Let $a_{1}, a_{2}, \ldots, a_{m}$ be arbitrary positive integers. Prove that there exist distinct positive integers $b_{1}, b_{2}, \ldots, b_{n}$, $n \leq m$, such that the following two conditions are satisfied:
(1) all subsets of $\{b_{1}, b_{2}, \ldots, b_{n}\}$ have distinct sums of elements;
(2) every number $a_{1}, a_{2}, \ldots, a_{m}$ is the sum of the elements of some subset of $\{b_{1}, b_{2}, \ldots, b_{n}\}$.
(1) all subsets of $\{b_{1}, b_{2}, \ldots, b_{n}\}$ have distinct sums of elements;
(2) every number $a_{1}, a_{2}, \ldots, a_{m}$ is the sum of the elements of some subset of $\{b_{1}, b_{2}, \ldots, b_{n}\}$.