Problem 1240

1. Define the set and its properties:
Let $A = \{x_1, x_2, x_3, x_4, x_5\}$ be a set of five distinct positive integers. Denote by $S_A$ the sum of its elements:
$$S_A = x_1 + x_2 + x_3 + x_4 + x_5$$
Denote by $T_A$ the number of triples $(i, j, k)$ with $1 \le i < j < k \le 5$ for which $x_i + x_j + x_k$ divides $S_A$.

2. Calculate the number of possible triples:
The number of ways to choose 3 elements out of 5 is given by the binomial coefficient:
$$\binom{5}{3} = 10$$
Therefore, there are 10 possible triples $(i, j, k)$.

3. Example with specific values:
Consider the set $A = \{1, 2, 3, 4, 494\}$:
$$S_A = 1 + 2 + 3 + 4 + 494 = 504$$
We need to check which triples $(i, j, k)$ satisfy the condition that $x_i + x_j + x_k$ divides $S_A$.

4. Check each triple:
- For $(1, 2, 3)$:
$$x_1 + x_2 + x_3 = 1 + 2 + 3 = 6 \quad \text{and} \quad 504 \div 6 = 84 \quad \text{(divides)}$$
- For $(1, 2, 4)$:
$$x_1 + x_2 + x_4 = 1 + 2 + 4 = 7 \quad \text{and} \quad 504 \div 7 = 72 \quad \text{(divides)}$$
- For $(1, 2, 5)$:
$$x_1 + x_2 + x_5 = 1 + 2 + 494 = 497 \quad \text{and} \quad 504 \div 497 \quad \text{(does not divide)}$$
- For $(1, 3, 4)$:
$$x_1 + x_3 + x_4 = 1 + 3 + 4 = 8 \quad \text{and} \quad 504 \div 8 = 63 \quad \text{(divides)}$$
- For $(1, 3, 5)$:
$$x_1 + x_3 + x_5 = 1 + 3 + 494 = 498 \quad \text{and} \quad 504 \div 498 \quad \text{(does not divide)}$$
- For $(1, 4, 5)$:
$$x_1 + x_4 + x_5 = 1 + 4 + 494 = 499 \quad \text{and} \quad 504 \div 499 \quad \text{(does not divide)}$$
- For $(2, 3, 4)$:
$$x_2 + x_3 + x_4 = 2 + 3 + 4 = 9 \quad \text{and} \quad 504 \div 9 = 56 \quad \text{(divides)}$$
- For $(2, 3, 5)$:
$$x_2 + x_3 + x_5 = 2 + 3 + 494 = 499 \quad \text{and} \quad 504 \div 499 \quad \text{(does not divide)}$$
- For $(2, 4, 5)$:
$$x_2 + x_4 + x_5 = 2 + 4 + 494 = 500 \quad \text{and} \quad 504 \div 500 \quad \text{(does not divide)}$$
- For $(3, 4, 5)$:
$$x_3 + x_4 + x_5 = 3 + 4 + 494 = 501 \quad \text{and} \quad 504 \div 501 \quad \text{(does not divide)}$$

5. Count the valid triples:
The valid triples are $(1, 2, 3)$, $(1, 2, 4)$, $(1, 3, 4)$, and $(2, 3, 4)$. Thus, there are 4 valid triples.

Conclusion:
$$\boxed{4}$$