CONO-SUR 2017 Problem 5

Let $a$ , $b$ and $c$ positive integers. Three sequences are defined as follows:

• $a_1=a$ , $b_1=b$ , $c_1=c$ [/*]
• $a_{n+1}=\lfloor{\sqrt{a_nb_n}}\rfloor$ , $\:b_{n+1}=\lfloor{\sqrt{b_nc_n}}\rfloor$ , $\:c_{n+1}=\lfloor{\sqrt{c_na_n}}\rfloor$ for $n \ge 1$ [/*] [/list]

[list = a]

• Prove that for any $a$ , $b$ , $c$ , there exists a positive integer $N$ such that $a_N=b_N=c_N$ .[/*]
• Find the smallest $N$ such that $a_N=b_N=c_N$ for some choice of $a$ , $b$ , $c$ such that $a \ge 2$ y $b+c=2a-1$ .[/*]