Let A, B and C be three positive integers such that the sum of A and the mean of B and C is 5. In addition, the sum of B and the mean of A and C is 7. Then the sum of A and B is

Given that A, B, and C are positive integers.

The sum of A and the mean of B and C is 5:
$A+\frac{B+C}{2}=5$
$2A+B+C=10$ — (Equation 1)

The sum of B and the mean of A and C is 7:
$B+\frac{A+C}{2}=7$
$A+2B+C=14$ — (Equation 2)

Subtracting Equation 1 from Equation 2:
$(A+2B+C)-(2A+B+C)=14-10$
$B-A=4$
$B=A+4$

Since A, B, and C are positive integers, let us substitute $B=A+4$ into Equation 1:
$2A+(A+4)+C=10$
$3A+C=6$

Since A and C must be positive integers (so $A\ge 1$ and $C\ge 1$):
If $A=1$, then $3\left(1\right)+C=6\Rightarrow C=3$. This is a valid positive integer solution.
If $A\ge 2$, then $3A\ge 6$, which would force $C\le 0$, violating the condition that C is a positive integer.

Therefore, the unique solution is:
$A=1$
$C=3$
$B=A+4=5$

The sum of A and B is:
$A+B=1+5=6$