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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

1.  $6$
2.  $5$
3.  $7$
4.  $4$

Given that, $A>0, B>0, C>0$

And, $A + \frac{B+C}{2} = 5$

$\Rightarrow 2A + B + C = 10 \quad \longrightarrow (1)$

And, $B + \frac{A+C}{2} = 7$

$\Rightarrow 2B + A + C = 14 \quad \longrightarrow (2)$

Subtract equation $(2)$ from $(1),$ we get

$\require{cancel} \begin{array} {cccc} 2A + B + \cancel{C} = 10 \\ A + 2B + \cancel{C} = 14 \\\hline \boxed{A – B = -4} \end{array}$

$\textsf{Case 1:}$ The least value of $B$ is  $1.$

Then, $A – 1 =\; – 4$

$\Rightarrow \boxed{ A = 5}$

So, $A + B = 5 + 1 = 6.$

$\textsf{Case 2:}$ The least value for $A$ is  $1.$

Then, $1 – B = \;– 4$

$\Rightarrow \boxed{ B = 5}$

So, $A + B = 1 + 5 = 6.$

$\therefore$ The sum of $A$ and $B$ is $6.$

Correct Answer$: \text{A}$
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