OIMU 2006 Problem 5

A regular $n$ -gon is inscribed in a circle of radius $1$ . Let $a_1,\cdots,a_{n-1}$ be the distances of one of the vertices of the polygon to all the other vertices. Prove that $$(5-a_1^2)\cdots(5-a_{n-1}^2)=F_n^2$$ where $F_n$ is the $n^{th}$ term of the Fibonacci sequence $1,1,2,\cdots$