Question 6
Let $a_0,b_0,c_0$ be complex numbers and, for every integer $n\ge0$, define
$$a_{n+1}=a_n^2+2b_nc_n,\quad b_{n+1}=b_n^2+2c_na_n,\quad c_{n+1}=c_n^2+2a_nb_n.$$
Suppose $\max\{|a_n|,|b_n|,|c_n|\}\le2022$ for all $n\ge0$. Prove $|a_0|^2+|b_0|^2+|c_0|^2\le1$.
Proposed by Ankan Bhattacharya. Source: Evan Chen’s third-party compilation; https://aops.com/community/p24775314. One editorial point; manual grading.
$$a_{n+1}=a_n^2+2b_nc_n,\quad b_{n+1}=b_n^2+2c_na_n,\quad c_{n+1}=c_n^2+2a_nb_n.$$
Suppose $\max\{|a_n|,|b_n|,|c_n|\}\le2022$ for all $n\ge0$. Prove $|a_0|^2+|b_0|^2+|c_0|^2\le1$.
Proposed by Ankan Bhattacharya. Source: Evan Chen’s third-party compilation; https://aops.com/community/p24775314. One editorial point; manual grading.