Question 31
For any set $A=\{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\}$ of five distinct positive integers denote by $S_{A}$ the sum of its elements, and denote by $T_{A}$ the number of triples $(i, j, k)$ with $1 \leqslant i\lt j\lt k \leqslant 5$ for which $x_{i}+x_{j}+x_{k}$ divides $S_{A}$.
Find the largest possible value of $T_{A}$.
Find the largest possible value of $T_{A}$.