APMO 2014 - Asian Pacific Mathematics Olympiad Question 4

For positive integers $n,b$, say that $n$ is $b$-discerning if there is a set of $n$ distinct positive integers less than $b$ whose different subsets always have different sums.

(a) Prove that 8 is 100-discerning.

(b) Prove that 9 is not 100-discerning.

Proposed by the Senior Problems Committee of the Australian Mathematical Olympiad Committee.