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Source: 2022 USAJMO Problem 6

March 24, 2022
AMCUSA(J)MOUSAJMO

Problem Statement

Let a0,b0,c0a_0, b_0, c_0 be complex numbers, and define \begin{align*}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\end{align*}for all nonnegative integers n.n.
Suppose that max⁡{∣an∣,∣bn∣,∣cn∣}≤2022\max{\{|a_n|, |b_n|, |c_n|\}} \leq 2022 for all n.n. Prove that ∣a0∣2+∣b0∣2+∣c0∣2≤1.|a_0|^2 + |b_0|^2 + |c_0|^2 \leq 1.
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