Question 4
The number $1$ is written $10$ times on a whiteboard. Every minute, Simon chooses two distinct numbers $a$ and $b$ on the whiteboard which may be of equal value, erases them, and writes down two new numbers $a+\dfrac1b$ and $b+\dfrac1a$.
(a) Show that, regardless of Simon's sequence of operations, after $1000$ minutes there will be at least one number on the board which is greater than $20$.
(b) Show that there is some sequence of operations for Simon to ensure that after $1000$ minutes, all numbers on the board are less than $21$.
(a) Show that, regardless of Simon's sequence of operations, after $1000$ minutes there will be at least one number on the board which is greater than $20$.
(b) Show that there is some sequence of operations for Simon to ensure that after $1000$ minutes, all numbers on the board are less than $21$.