Let $a_0, b_0, c_0$ 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 $n.$

Suppose that $\max{\{|a_n|, |b_n|, |c_n|\}} \leq 2022$ for all $n.$ Prove that$$|a_0|^2 + |b_0|^2 + |c_0|^2 \leq 1.$$

Question

Let $a_0, b_0, c_0$ 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 $n.$

Suppose that $\max{\{|a_n|, |b_n|, |c_n|\}} \leq 2022$ for all $n.$ Prove that$$|a_0|^2 + |b_0|^2 + |c_0|^2 \leq 1.$$

OronSH · Answer

@minesweeper help

OronSH · Answer

The frog does not move. I do not understand.

OronSH · Answer

you're bad you literally got a 35.7% on this test

jordanbenton · Answer

they will either be zero or one

OronSH · Answer

not true, they don't have to be integers

OronSH · Answer

oren bad give solution

OronSH · Answer

oh wait you got a 35.7% on this test and you didn't solve this question lol

cuzican · Answer

Prove:|x−1|+|x−2|+|x−3|+⋯+|x−n|≥n−1

OronSH · Answer

|x-1|+|n-x|>=|n-1| by triangle inequality which trivializes the problem

cuzican · Answer

nice dude.

OronSH · Answer

ok but can you prove that $|a_0|^2+|b_0|^2+|c_0|^2\leq1$?

cuzican · Answer

noope

cuzican · Answer

@cuzican wrote:

noope

i can try tho

cuzican · Answer

@cuzican wrote:

noope

i can try tho

(|a0|)(2)+(|b0|)(2)+(|c0|)(2)≤1 Simplifies to: 6≤1 6≤1 6≰1 False. does this help?

OronSH · Answer

it's true

cuzican · Answer

o