This is a problem where you have to represent some numbers as a sum of some other numbers. We also need any partial sum to be different. This means that we need the $b_i$ to be a sequence of numbers so that adding any subsequence gives a distinct sum, and not only that, we need to be able to represent the $a_i$ as a sum of some $b_i$ too. If you think about this, you should be reminded of using bases to represent the numbers. So let the $b_i$ be $2^1.2^2,2^3,\cdots$ respectively. Then any partial sum is just a base $2$ number and is therefore distinct. However, there is no guarantee that $m<2^n$. This is only half the battle. The rest of the solution is pretty much combinatorial (and a lot more challenging, so try it!).