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2022 USAJMO Problems/Problem 6

Revision as of 20:37, 19 April 2022 by Wlm7 (talk | contribs) (Created page with "==Problem== Let <math>a_0,b_0,c_0</math> be complex numbers, and define <cmath>a_{n+1}=a_n^2+2b_nc_n</cmath> <cmath>b_{n+1}=b_n^2+2c_na_n</cmath> <cmath>c_{n+1}=c_n^2+2a_nb_n...")
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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 \[|a_0|^2+|b_0|^2+|c_0|^2\leq 1.\]

Solution

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