Problem

Source: Slovenia National MO 2005 4th Grade P4

Tags: combinatorics, extremal principle, algebra



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.