Question 58
Consider $m \ge 3$ positive real numbers $g_1, \dots, g_m$, each number being less than the sum of the others. For any subset $M \subseteq \{1, \dots, m\}$, denote
$$
S_M = \sum_{k \in M} g_k.
$$
Find all $m$ for which it is always possible to partition the indices $1, \dots, m$ into three sets $A, B, C$, with the property that
$$
S_A \lt S_B + S_C, \quad S_B \lt S_A + S_C \quad \text{and} \quad S_C \lt S_A + S_B.
$$
$$
S_M = \sum_{k \in M} g_k.
$$
Find all $m$ for which it is always possible to partition the indices $1, \dots, m$ into three sets $A, B, C$, with the property that
$$
S_A \lt S_B + S_C, \quad S_B \lt S_A + S_C \quad \text{and} \quad S_C \lt S_A + S_B.
$$