Putnam 1977 B3

Problem B-3 An (ordered) triple ( $x_{1}, x_{2}, x_{3}$ ) of positive irrational numbers with $x_{1}+x_{2}+x_{3}=1$ is called "balanced" if each $x_{i}<1 / 2$. If a triple is not balanced, say if $x_{j}>1 / 2$, one performs the following "balancing act" $$B\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}^{\prime}, x_{2}^{\prime}, x_{3}^{\prime}\right),$$ where $x_{i}^{\prime}=2 x_{i}$ if $i \neq j$ and $x_{j}^{\prime}=2 x_{j}-1$. If the new triple is not balanced, one performs the balancing act on it. Does continuation of this process always lead to a balanced triple after a finite number of performances of the balancing act?