Question 6
Let $X$ be a finite set of positive integers such that no two subsets have the same sum. Prove that the sum of the inverses of the elements of $X$ satisfy
$$
\sum_{x \in X} \frac{1}{x} \lt \frac{5}{2}.
$$
For example, for $X = \{1, 2, 5\}$ the sums $1, 2, 5, 1+2, 1+5, 2+5, 1+2+5$ are all distinct and $1/1 + 1/2 + 1/5 = 1.7 \lt 2.5$ holds.
$$
\sum_{x \in X} \frac{1}{x} \lt \frac{5}{2}.
$$
For example, for $X = \{1, 2, 5\}$ the sums $1, 2, 5, 1+2, 1+5, 2+5, 1+2+5$ are all distinct and $1/1 + 1/2 + 1/5 = 1.7 \lt 2.5$ holds.