Question 39
Aerith writes $100$ positive numbers on a blackboard. Every minute, Bob either replaces one number $x$ with $\frac{1}{x}$, or replaces two numbers $x, y$ with $\frac{x y+1}{x+y}$. Given that after $2021$ minutes, there is only one number left, show that this number only depends on Aerith's initial numbers.