Let $A = \{a_1, a_2, \ldots, a_I\}$ be a set of real numbers, where for all $i \in \{1,2,\ldots,I\}$, $0<a_i\le1$ and $\sum_{j \ne i} a_j \ge 1$.
I am interested in the set (or sets) $A$ satisfying the following condition:
For every subset $S \subset \{1,2,\ldots,I\}$ where $\sum_{i \in S} a_i \ge 1$, there exists a subset $S' \subseteq S$ where $\sum_{i \in S'} a_i = 1$, i.e. that exactly sums to 1.
Is it true that the only set $A$ satisfying this condition is the one where, for all $i \in \{1,2,\ldots,I\}$:
$$a_i = a \quad\land\quad \frac{1}{a} \text{ is an integer}$$
No, it is not true. Consider the following counter-example where $a_i \ne a, \forall i$:
$$A = \{a_1, a_2, a_3, a_4\}, \text{ where } a_1 = a_2 = \frac{1}{4} \text{ and } a_3 = a_4 = \frac{1}{2}$$
This satisfies the condition that all the subsets $$S = \{a_1, a_2, a_3\}, \{a_1, a_2, a_4\}, \{a_3, a_4\}, \{a_1, a_3, a_4\}, \{a_2, a_3, a_4\}$$ where $\sum_{i \in S} a_i \ge 1$, have subsets $$S' = \{a_1, a_2, a_3\}, \{a_1, a_2, a_4\}, \{a_3, a_4\}$$ where $\sum_{i \in S'} a_i = 1$.