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Romania National Olympiad
2014 Romania National Olympiad
4
Romania National Olympiad 2014
Romania National Olympiad 2014
Source:
March 22, 2018
complex numbers
inequalities
Problem Statement
Let
n
∈
N
,
n
≥
2
n \in \mathbb{N} , n \ge 2
n
∈
N
,
n
≥
2
and
a
0
,
a
1
,
a
2
,
⋯
,
a
n
∈
C
;
a
n
≠
0
a_0,a_1,a_2,\cdots,a_n \in \mathbb{C} ; a_n \not = 0
a
0
,
a
1
,
a
2
,
⋯
,
a
n
∈
C
;
a
n
=
0
. Then:P.
∣
a
n
z
n
+
a
n
−
1
z
z
n
−
1
+
⋯
+
a
1
z
+
a
0
∣
≤
∣
a
n
+
a
0
∣
|a_nz^n + a_{n-1}z^z{n-1} + \cdots + a_1z + a_0 | \le |a_n+a_0|
∣
a
n
z
n
+
a
n
−
1
z
z
n
−
1
+
⋯
+
a
1
z
+
a
0
∣
≤
∣
a
n
+
a
0
∣
for any
z
∈
C
,
∣
z
∣
=
1
z \in \mathbb{C}, |z|=1
z
∈
C
,
∣
z
∣
=
1
Q.
a
1
=
a
2
=
⋯
=
a
n
−
1
=
0
a_1=a_2=\cdots=a_{n-1}=0
a
1
=
a
2
=
⋯
=
a
n
−
1
=
0
and
a
0
/
a
n
∈
[
0
,
∞
)
a_0/a_n \in [0,\infty)
a
0
/
a
n
∈
[
0
,
∞
)
Prove that
P
⟺
Q
P \Longleftrightarrow Q
P
⟺
Q
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