Sum of positive integers A, B, C

We are given:

1. $A + \frac{B + C}{2} = 5 \implies 2A + B + C = 10$
2. $B + \frac{A + C}{2} = 7 \implies A + 2B + C = 14$

Subtract equation (1) from equation (2): $(A + 2B + C) - (2A + B + C) = 14 - 10$ $B - A = 4 \implies B = A + 4$

Substitute $B = A + 4$ into equation (1): $2A + (A + 4) + C = 10 \implies 3A + C = 6$

Since $A$ and $C$ are positive integers: If $A = 1$, $C = 3$, which gives $B = 1 + 4 = 5$. Then $A + B = 1 + 5 = 6$.

(If $A \ge 2$, $C \le 0$, which contradicts $C$ being a positive integer).