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Find real numbers

Source: China TST 2003

June 29, 2006

algebrapolynomialalgebra unsolved

Problem Statement

Can we find positive reals

a_1, a_2, \dots, a_{2002}

such that for any positive integer

k

, with

1 \leq k \leq 2002

, every complex root

z

of the following polynomial

f(x)

satisfies the condition

|\text{Im } z| \leq |\text{Re } z|

,

f(x)=a_{k+2001}x^{2001}+a_{k+2000}x^{2000}+ \cdots + a_{k+1}x+a_k,

where

a_{2002+i}=a_i

, for

i=1,2, \dots, 2001

.

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