Question 31
There are three distinct positive integers written on the blackboard. When real numbers $a$, $b$, $c$ are on the blackboard, consider the procedure that replaces those three numbers with $\frac{b+c}{2}$, $\frac{c+a}{2}$, $\frac{a+b}{2}$ at the same time. After this procedure is done $2021$ times, all three numbers on the blackboard are positive integers. Determine the minimum value of the sum of initial three positive integers on the blackboard.