BY-MO 2001 Problem 5

In the increasing sequence of positive integers $a_1$ , $a_2$ ,. . . , the number $a_k$ is said to be funny if it can be represented as the sum of some other terms (not necessarily distinct) of the sequence. (a) Prove that all but finitely terms of the sequence are funny. (b) Does the result in (a) always hold if the terms of the sequence can be any positive rational numbers?