APMO 2014 Problem 4

Let $n$ and $b$ be positive integers. We say $n$ is $b$-discerning if there exists a set consisting of $n$ different positive integers less than $b$ that has no two different subsets $U$ and $V$ such that the sum of all elements in $U$ equals the sum of all elements in $V$. (a) Prove that 8 is a 100 -discerning. (b) Prove that 9 is not 100-discerning. (Senior Problems Committee of the Australian Mathematical Olympiad Committee)