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:


Alternatively: Count the total number of ways to arrange $0$s and $1$s around a circle ($2n$ binary digits in total), consider the number of Dyck words of length $2n$, i.e., the $n$th Catalan number, and then use the Pigeonhole Principle. "QED"
