CN-TST 2012 Problem 10

Given an integer $n\ge 2$ . Prove that there only exist a finite number of n-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ which simultaneously satisfy the following three conditions:

• $a_1>a_2>\ldots>a_n$ ;
• $\gcd (a_1,a_2,\ldots,a_n)=1$ ;
• $a_1=\sum_{i=1}^{n}\gcd (a_i,a_{i+1})$ ,where $a_{n+1}=a_1$ .