An odd question about induction. Given $n$ $0$'s and $n$ $1$'s distributed in any manner whatsoever around a circle, show, using induction on $n$, that it is possible to start at some number and proceed clockwise around the circle to the original starting position so that, at any point during the cycle, we have seen at least as many $0$'s as $1$'s:


There is at least one instance of a $0$ followed by a $1$. Remove these two numbers, find (by induction hypothesis) a good starting point for the reduced circle, put the $0$ and $1$ back in and use the same starting position. It is easily verified that this is a valid starting position.
It is also not difficult to show the result without induction: A person walking around the circle and dropping a dollar at each $0$ seen and collecting a dollar (out of nowhere) at each $1$ seen will end up with the same amount he started with. And he will manage to complete the tour provided he starts with enough money in the briefcase. For an arbitrary starting point, chose the minimal valid starting amount of dollars. Then at some point of the tour the briefcase is empty. Starting the round trip at that place instead gives the desired result.
