William was bored at the math lesson, so he drew a circle and $n\ge3$ empty cells around the circumference. In every cell he wrote a positive number. Later on he erased the numbers and in every cell wrote the geometric mean of the numbers previously written in the two neighboring cells. Show that there exists a cell whose number was not replaced by a larger number.
Consider the largest number in the cell say $b$ with its surrounding elements being $a$ and $c$, let $a\geq c$. $b$ gets replaced with $\sqrt{ac} \leq \frac{a+c}{2} \leq a \leq b$. Hence $b$ does not get replaced by a larger number.