2022 USAJMO Problems/Problem 6

Problem

Let $a_0,b_0,c_0$ be complex numbers, and define

$a_{n+1}=a_n^2+2b_nc_n$ $b_{n+1}=b_n^2+2c_na_n$ $c_{n+1}=c_n^2+2a_nb_n$ for all nonnegative integers $n$ .

Suppose that $\max{|a_n|,|b_n|,|c_n|}\leq2022$ for all $n$ . Prove that $|a_0|^2+|b_0|^2+|c_0|^2\leq 1.$

2022 USAJMO (Problems • Resources)
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