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China National Olympiad
2022 China National Olympiad
2
2
Part of
2022 China National Olympiad
Problems
(1)
Roots of complex polynomial
Source: CMO 2022 P2
12/21/2021
Find the largest real number
λ
\lambda
λ
with the following property: for any positive real numbers
p
,
q
,
r
,
s
p,q,r,s
p
,
q
,
r
,
s
there exists a complex number
z
=
a
+
b
i
z=a+bi
z
=
a
+
bi
(
a
,
b
∈
R
)
a,b\in \mathbb{R})
a
,
b
∈
R
)
such that |b|\ge \lambda |a| \text{and} (pz^3+2qz^2+2rz+s) \cdot (qz^3+2pz^2+2sz+r) =0.
algebra
polynomial